on computing canonical forms using flows let g be a lie group acting on a vector space v i say that a vector field u v v is orbital if for all p in v u p is tangent to the orbit of p at p now let the vector space v and the vector space tangent to g at the identity have inner products in this setting i define a simple map which i call quasi projection which transforms any vector field on v into an orbital one i use the quasi projection map to define flows which compute canonical forms

modeling hyperelasticity in non equilibrium multiphase flows the aim of this article is the construction of a multiphase hyperelastic model the eulerian formulation of the hyperelasticity represents a system of 14 conservative partial differential equations submitted to stationary differential constraints this model is constructed with an elegant approach where the specific energy is given in separable form the system admits 14 eigenvalues with 7 characteristic eigenfields the associated riemann problem is not easy to solve because of the presence of 7 waves the shear waves are very diffusive when dealing with the full system in this paper we use a splitting approach to solve the whole system using 3 sub systems this method reduces the diffusion of the shear waves while allowing to use a classical approximate riemann solver the multiphase model is obtained by adapting the discrete equations method this approach involves an additional equation governing the evolution of a phase function relative to the presence of a phase in a cell the system is integrated over a multiphase volume control finally each phase admits its own equations system composed of three sub systems one and three dimensional test cases are presented

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

a new level set model for multimaterial flows we propose a new level set model for representing multimaterial flows in multiple space dimensions instead of associating a level set function with a specific fluid material the function is associated with a pair of materials and the interface that separates them a voting algorithm collects sign information from all level set functions and determines material designation to represent a general m material configuration m m 1 2 level set functions need to be accounted for problems of practical interest use far fewer functions as not all pairs of materials share an interface and level set functions that coincide are grouped together under this model regions of potential material ambiguity i e overlaps or vacuum are markedly reduced in size in 2d ambiguous regions are points as opposed to lines in material based level set models in 3d they are lines as opposed to surfaces the model produces excellent results without the need for reinitialization thereby avoiding additional computational costs and preventing excessive numerical diffusion

reality based algebras with a two dimensional representation abstract it is shown that a two dimensional irreducible representation of a reality based algebra with a standard degree map is determined by the other irreducible representations of the algebra and the action of its given anti automorphism on the distinguished basis this generalizes known results for algebras of dimensions 5 6 7 and 8 explicit conclusions equivalent to those for dimensions 5 and 6 are recovered here by specializing the procedure of proof of the main theorem

fourier transforms and frobenius eigenvalues for finite coxeter groups abstract lusztig s classification of the unipotent characters of a finite chevalley or steinberg group involves a certain non abelian fourier transformation we construct analogous transformations for the suzuki and ree groups based on a set of axioms derived from lusztig s theory of character sheaves we also determine fourier matrices for the spetses in the sense of broue michel and the second author associated with twisted dihedral groups this completes the determination of fourier matrices for all spetses associated with finite coxeter groups we end by collecting common properties of these fourier matrices and the eigenvalues of frobenius of character sheaves and unipotent characters

on matrix structures invariant under toda like isospectral flows abstract we work in the space r n n of n by n real matrices we say that a linear transformation on this space is toda like if it maps symmetric matrices to skew symmetric matrices with such a transformation we associate a bilinear operation defined by x y x y y x where u v uv vu then r n n together with this operation is a usually not associative algebra we call any subalgebra of such an algebra a toda like algebra we classify these algebras which arise in connection with eigenvalue computations we determine the toda like algebras which are of primary interest for this application we accomplish this classification by studying certain difference weighted graphs which we associate with the algebras

on the flexibility of agglomeration based physical space discontinuous galerkin discretizations in this work we show that the flexibility of the discontinuous galerkin dg discretization can be fruitfully exploited to implement numerical solution strategies based on the use of elements with very general shapes thanks to the freedom in defining the mesh topology we propose a new h adaptive technique based on agglomeration coarsening of a fine mesh the possibility to enhance the error distribution over the computational domain is investigated on a poisson problem with the goal of obtaining a mesh independent discretization the main building block of our dg method consists of defining discrete polynomial spaces directly on physical frame elements for this purpose we orthonormalize with respect to the l 2 product a set of monomials relocated in a specific element frame and we introduce an easy way to reduce the cost related to numerical integration on agglomerated meshes to complete the dg formulation for second order problems two extensions of the br2 scheme to arbitrary polyhedral grids including an estimate of the stabilization parameter ensuring the coercivity property are here proposed

a relaxation projection method for compressible flows part i the numerical equation of state for the euler equations a new projection method is developed for the euler equations to determine the thermodynamic state in computational cells it consists in the resolution of a mechanical relaxation problem between the various sub volumes present in a computational cell these sub volumes correspond to the ones traveled by the various waves that produce states with different pressures velocities densities and temperatures contrarily to godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium the pressure computation with this relaxation process replaces the use of the conventional equation of state eos a simplified relaxation method is also derived and provides a specific eos named the numerical eos the use of the numerical eos gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas eos it is then extended to the computation of interface problems separating fluids with different eos liquid gas interface for example with the euler equations the resulting method is very robust accurate oscillation free and conservative for the sake of simplicity and efficiency the method is developed in a lagrange projection context and is validated over exact solutions in a companion paper f petitpas e franquet r saurel a relaxation projection method for compressible flows part ii computation of interfaces and multiphase mixtures with stiff mechanical relaxation j comput phys submitted for publication the method is extended to the numerical approximation of a non conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures

why the muscl hancock scheme is l 1 stable the finite volume methods are one of the most popular numerical procedure to approximate the weak solutions of hyperbolic systems of conservation laws they are developed in the framework of first order numerical schemes several approaches are proposed to increase the order of accuracy the van leer methods are interesting ways one of them namely the muscl hancock scheme is full time and space second order accuracy in the present work we exhibit relevant conditions to ensure the l1 stability of the method a cfl like condition is established and a suitable limitation procedure for the gradient reconstruction is developed in order to satisfy the stability criterion in addition we show that the conservative variables are not useful within the gradient reconstruction and the procedure is extended in the framework of the primitive variables numerical experiments are performed to show the interest and the robustness of the method

an interface capturing method for the simulation of multi phase compressible flows a novel finite volume interface contact capturing method is presented for simulation of multi component compressible flows with high density ratios and strong shocks in addition the materials on the two sides of interfaces can have significantly different equations of state material boundaries are identified through an interface function which is solved in concert with the governing equations on the same mesh for long simulations the method relies on an interface compression technique that constrains the thickness of the diffused interface to a few grid cells throughout the simulation this is done in the spirit of shock capturing schemes for which numerical dissipation effectively preserves a sharp but mesh representable shock profile for contact capturing the formulation is modified so that interface representations remain sharp like captured shocks countering their tendency to diffuse via the same numerical diffusion needed for shock capturing special techniques for accurate and robust computation of interface normals and derivatives of the interface function are developed the interface compression method is coupled to a shock capturing compressible flow solver in a way that avoids the spurious oscillations that typically develop at material boundaries convergence to weak solutions of the governing equations is proved for the new contact capturing approach comparisons with exact riemann problems for model one dimensional multi material flows show that the interface compression technique is accurate the method employs cartesian product stencils and therefore there is no inherent obstacles in multiple dimensions examples of two and three dimensional flows are also presented including a demonstration with significantly disparate equations of state a shock induced collapse of three dimensional van der waal s bubbles air in a stiffened equation of state liquid water adjacent to a mie gruneisen equation of state wall copper

a relaxation projection method for compressible flows part ii artificial heat exchanges for multiphase shocks the relaxation projection method developed in saurel et al r saurel e franquet e daniel o le metayer a relaxation projection method for compressible flows part i the numerical equation of state for the euler equations j comput phys 2007 822 845 is extended to the non conservative hyperbolic multiphase flow model of kapila et al a k kapila menikoff j b bdzil s f son d s stewart two phase modeling of deflagration to detonation transition in granular materials reduced equations physics of fluids 13 10 2001 3002 3024 this model has the ability to treat multi temperatures mixtures evolving with a single pressure and velocity and is particularly interesting for the computation of interface problems with compressible materials as well as wave propagation in heterogeneous mixtures the non conservative character of this model poses however computational challenges in the presence of shocks the first issue is related to the riemann problem resolution that necessitates shock jump conditions thanks to the rankine hugoniot relations proposed and validated in saurel et al r saurel o le metayer j massoni s gavrilyuk shock jump conditions for multiphase mixtures with stiff mechanical relaxation shock waves 16 3 2007 209 232 exact and approximate 2 shocks riemann solvers are derived however the riemann solver is only a part of a numerical scheme and non conservative variables pose extra difficulties for the projection or cell average of the solution it is shown that conventional godunov schemes are unable to converge to the exact solution for strong multiphase shocks this is due to the incorrect partition of the energies or entropies in the cell averaged mixture to circumvent this difficulty a specific lagrangian scheme is developed the correct partition of the energies is achieved by using an artificial heat exchange in the shock layer with the help of an asymptotic analysis this heat exchange takes a similar form as the pseudoviscosity introduced by von neumann and richtmyer j von neumann r d richtmyer a method for the numerical calculation of hydrodynamic shocks j appl phys 21 1950 232 237 the present lagrangian numerical scheme thus combines riemann solvers and artificial heat exchanges an eulerian variant is then obtained by using the relaxation projection method developed earlier by the authors for the euler equations the method is validated against exact solutions based on the multiphase shock relations as well as exact solutions of the euler equations in the context of interface problems the method is able to solve interfaces separating pure fluids or heterogeneous mixtures with very large density ratio and with very strong shocks

finite volume approximation of two phase fluid flows based on an approximate roe type riemann solver abstract we introduce an approximate roe type riemann solver for the numerical simulation of two phase fluid flows composed of liquid droplets suspended in gas we compute a roe linearization of some well conditioned approximate rankine hugoniot relations in non conservation form the computed solutions are found to be in good agreement with the exact solution in one dimension slab geometry we extend this solver to two dimensional geometries using a finite volume formulation

a positive and entropy satisfying finite volume scheme for the baer nunziato model we present a relaxation scheme for approximating the entropy dissipating weak solutions of the baer nunziato two phase flow model this relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in 16 for the isentropic baer nunziato model and consequently inherits its main properties to our knowledge this is the only existing scheme for which the approximated phase fractions phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable cfl condition for ideal gas and stiffened gas equations of state real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme it is also the only scheme for which a discrete entropy inequality is proven under a cfl condition derived from the natural sub characteristic condition associated with the relaxation approximation this last property which ensures the non linear stability of the numerical method is satisfied for any admissible equation of state we provide a numerical study for the convergence of the approximate solutions towards some exact riemann solutions the numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the baer nunziato model namely schwendeman wahle kapila s godunov type scheme 39 and tokareva toro s hllc scheme 44 the relaxation scheme also shows a higher precision and a lower computational cost for comparable accuracy than a standard numerical scheme used in the nuclear industry namely rusanov s scheme finally we assess the good behavior of the scheme when approximating vanishing phase solutions

hllc type riemann solver for the baer nunziato equations of compressible two phase flow we first construct an approximate riemann solver of the hllc type for the baer nunziato equations of compressible two phase flow for the subsonic wave configuration the solver is fully nonlinear it is also complete that is it contains all the characteristic fields present in the exact solution of the riemann problem in particular stationary contact waves are resolved exactly we then implement and test a new upwind variant of the path conservative approach such schemes are suitable for solving numerically nonconservative systems finally we use locally the new hllc solver for the baer nunziato equations in the framework of finite volume discontinuous galerkin finite element and path conservative schemes we systematically assess the solver on a series of carefully chosen test problems

implicit solution of the unsteady euler equations for high order accurate discontinuous galerkin discretizations efficient solution techniques for high order accurate time dependent problems are investigated for solving the two dimensional non linear euler equations in this work the spatial discretization consists of a high order accurate discontinuous galerkin dg approach implicit time integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods standard backwards differencing methods bdf1 and bdf2 as well as a second order crank nicholson cn2 and a fourth order implicit runge kutta irk4 scheme are considered in an attempt to balance the spatial and temporal accuracy of the overall approach the implicit system arising at each time step is solved using a p multigrid approach which is shown to produce h independent convergence rates while remaining relatively insensitive to the time step size the crank nicholson methodology although not l stable demonstrates superior performance compared to the bdf2 scheme for the problems chosen in this work however the fourth order accurate implicit runge kutta scheme is found to be the most efficient in terms of computational cost for a given accuracy level as compared to the lower order schemes in spite of the added cost per time step and the benefits of this scheme increase for tighter error tolerances

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

isospectral flows and abstract matrix factorizations a general framework for constructing isospectral flows in the space operatorname gl n of n by n matrices is proposed depending upon how operatorname gl n is split this framework gives rise to different types of abstract matrix factorizations when sampled at integer times these flows naturally define special iterative processes and each flow is associated with the sequence generated by the corresponding abstract factorization the proposed theory unifies as special cases the well known matrix decomposition techniques used in numerical linear algebra and is likely to offer a broader approach to the general matrix factorization problem

schemes to compute unsteady flashing flows some ways to compute flashing flows in variable cross section ducts are provided focusing on the homogeneous relaxation model the basic numerical method relies on a splitting technique that is consistent with the overall entropy inequality the cross section is assumed to he continuous and the finite volume approach is applied to approximate homogeneous equations several suitable schemes to account for complex equation of state are discussed namely the rusanov scheme an approximate form of the roe scheme and the volumes finis roe vfroe scheme with the help of nonconservative variables to evaluate respective accuracy the homogeneous euler equations are computed first and the l1 error norm of transient solutions of shock tube experiments are plotted it is shown that the rusanov scheme is indeed less accurate which balances its interesting properties inasmuch as it preserves the positivity of the mean density and the maximum principle for the vapor quality computations of real cases are presented which account for the mass transfer term and the time space dependent cross sections

a discontinuous galerkin method for the simulation of compressible gas gas and gas water two medium flows abstract in this paper we develop a new discontinuous galerkin method for the simulation of shocks and interfaces in compressible gas gas and gas water two medium flows by solving the based model the spatial discretization is carefully designed to possess the following features discrete conservation in terms of the total mass total momentum and total energy high order accuracy and consistency for smooth flows free of oscillations at an isolated material interface in order to handle potential discontinuities arising in the simulation a nonlinear limiter based on the weighted essentially non oscillatory weno strategy is employed to suppress numerical oscillations and to preserve high order of accuracy in regions of smooth flows the weno reconstruction is imposed on suitably selected quantities rather than the conserved ones in case for discontinuities with large pressure ratio low density and dramatic change of material property where unphysical variables may be encountered a posteriori solution correction on the subcell level is locally adopted to enhance the robustness a series of typical test cases for both one and two dimensional problems are provided to demonstrate the performance of the proposed method

the riemann problem for the baer nunziato two phase flow model we consider the riemann problem for the two phase flow model proposed by baer and nunziato int j multiphase flows 12 1986 861 it describes the flame spread and the deflagration to detonation transition ddt in gaspermeable reactive granular materials the model is given by a non strictly hyperbolic non conservative system of partial differential equations we investigate the structure of the riemann problem and construct the exact solution for it we establish that the solution across one wave is not unique and propose to use the evolutionarity criterion to single out the admissible solution due to special structure of the riemann problem for the baer nunziato model we are able to introduce a notion of a weak solution for it finally we propose a number of test cases based on the exact solution to the riemann problem for the baer nunziato model

multicomponent flow calculations by a consistent primitive algorithm the dynamics of inviscid multicomponent fluids may be modelled by the euler equations augmented by one or more additional species equation s attempts to compute solutions for extended euler models in conservation form show strong oscillations and other computational inaccuracies near material interfaces these are due to erroneous pressure fluctuations generated by the conservative wave model this problem does not occur in single component computations and arises only in the presence of several species a nonconservative primitive euler formulation is proposed which results in complete elimination of the oscillations the numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation numerical experiments show clean monotonic solution profiles with acceptably small conservation error for shocks of weak to moderate strengths

efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes this paper presents efficient second order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows for compressible unsteady flows a time dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables as a result a compact scheme on the unstructured meshes is developed for incompressible steady flows a continuous second order gas kinetic bgk type scheme is presented for which the time dependent gas distribution function with a continuous particle velocity is used on unstructured meshes the efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells therefore even with the stencil of a first order scheme a high resolution method is constructed numerical examples are presented which are compared with the benchmark solutions and the experimental measurements

mathematical and numerical modeling of two phase compressible flows with micro inertia a new model with full coupling between micro and macroscale motion is developed for compressible multiphase mixtures the equations of motion and the coupling microstructural equation an analogue of the rayleigh lamb equation are obtained by using the hamilton principle of stationary action in the particular case of bubbly fluids the resulting model contains eight partial differential equations one dimensional case and is unconditionally hyperbolic the equations are solved numerically by an adapted godunov method the model and methods are validated for two very different test problems the first one consists of a wave propagating in a liquid containing a small quantity of gas bubbles computed oscillating shock waves fit perfectly the experimental data then the one dimensional multiphase model is used as a reduction tool for the multidimensional interaction of a shock wave with a large bubble good agreement is again obtained

discrete equations for physical and numerical compressible multiphase mixtures we have recently proposed in 21 a compressible two phase unconditionally hyperbolic model able to deal with a wide range of applications interfaces between compressible materials shock waves in condensed multiphase mixtures homogeneous two phase flows bubbly and droplet flows and cavitation in liquids one of the difficulties of the model as always in this type of physical problems was the occurrence of non conservative products in 21 we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces not in the case of shock waves this model was extended to several space dimensions in 24 in this paper thanks to a deeper analysis of the model we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities this analysis provides a more accurate estimate of closure terms but also an accurate resolution method for the conservative fluxes as well as non conservative terms even for situations involving discontinuous solutions the accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems

a finite volume hllc based scheme for compressible interfacial flows with surface tension shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization similar behavior occurs inside of supersonic combustors scramjets under startup conditions but it is challenging to study these conditions experimentally in order to investigate this phenomenon further a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces the flow field is solved via the compressible multicomponent euler equations i e the five equation model discretized with the finite volume method on a uniform cartesian grid the solver utilizes a total variation diminishing tvd third order rungekutta method for time marching and second order tvd spatial reconstruction surface tension is incorporated using the continuum surface force csf model fluxes are upwinded with a modified hartenlaxvan leer contact hllc approximate riemann solver an interface compression scheme is employed to counter numerical diffusion of the interface the present work includes modifications to both the hllc solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gasliquid interface a simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non dimensionalization of the model the model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and mach 1 47 and 3 shock droplet interaction problems the general characteristics of secondary atomization at a range of weber numbers are also captured in a series of simulations

adaptive osher type scheme for the euler equations with highly nonlinear equations of state abstract for the numerical simulation of detonation of condensed phase explosives a complex equation of state eos such as the jones wilkins lee jwl eos or the cochran chan c c eos are widely used however when a conservative scheme is used for solving the euler equations with such equations of state a spurious solution across the contact discontinuity a well known phenomenon in multi fluid systems arises even for single materials in this work we develop a generalised osher type scheme in an adaptive primitive conservative framework to overcome the aforementioned difficulties resulting numerical solutions are compared with the exact solutions and with the numerical solutions from the godunov method in conjunction with the exact riemann solver for the euler equations with mie gruneisen form of equations of state such as the jwl and the c c equations of state the adaptive scheme is extended to second order and its empirical convergence rates are presented verifying second order accuracy for smooth solutions through a suite of several tests problems in one and two space dimensions we illustrate the failure of conservative schemes and the capability of the methods of this paper to overcome the difficulties

zero preserving iso spectral flows based on parallel sums abstract driessel k r driessel computing canonical forms using flows linear algebra appl 379 2004 353 379 introduced the notion of quasi projection onto the range of a linear transformation from one inner product space into another inner product space here we introduce the notion of quasi projection onto the intersection of the ranges of two linear transformations from two inner product spaces into a third inner product space as an application we design a new family of iso spectral flows on the space of symmetric matrices that preserves zero patterns we discuss the equilibrium points of these flows we conjecture that these flows generically converge to diagonal matrices we perform some numerical experiments with these flows which support this conjecture we also compare our zero preserving flows with the toda flow

numerical resolution for a compressible two phase flow model based on the theory of thermodynamically compatible systems this paper is concerned with the numerical solution of the equations governing two phase gas solid mixture in the framework of thermodynamically compatible systems theory the equations constitute a non homogeneous system of nonlinear hyperbolic conservation laws a total variation diminishing tvd slope limiter centre slic numerical scheme based on the splitting approach is presented and applied for the solution of the initial boundary value problem for the equations the model equations and the numerical methods are systematically assessed through a series of numerical test cases strong numerical evidence shows that the model and the methods are accurate robust and conservative the model correctly describes the formations of shocks and rarefactions in two phase gas solid flow

runge kutta discontinuous galerkin methods for compressible two medium flow simulations one dimensional case the runge kutta discontinuous galerkin rkdg method for solving hyperbolic conservation laws is a high order finite element method which utilizes the useful features from high resolution finite volume schemes such as the exact or approximate riemann solvers tvd runge kutta time discretizations and limiters in this paper we investigate using the rkdg finite element method for compressible two medium flow simulation with conservative treatment of the moving material interfaces numerical results for both gas gas and gas water flows in one dimension are provided to demonstrate the characteristic behavior of this approach

numerical simulation for two phase flows using hybrid scheme the present paper is devoted to the computation of two phase flows using two fluid approach two difficulties arise for the model one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations to deal with the aforementioned difficulties the non conservative volume fraction equation is discretized by weno scheme and the cweno type central upwind scheme is applied to solve the mass momentum and energy equations computational results are eventually provided and discussed

how to preserve the mass fractions positivity when computing compressible multi component flows abstract we are interested in the numerical investigation of the compressible flow of a gaseous mixture considering a hyperbolic system including the euler equations for the mixture and a mass conservation equation for each species we propose a new approximation scheme for the convective term of the species equations this approximation relies on some properties of the exact solution of the riemann problem for the multi component system and applies when an upwind godunov type scheme is used for the euler equations its main interest lies in the fact that it preserves the positivity and monotonicity of the mass fractions of all species

steady and unsteady shock waves on overlapping grids the interaction between steady and unsteady shock waves and interfaces of overlapping grids is studied conditions permitting shock penetration through an interface and ensuring solution uniqueness are derived easily usable corrections ensuring shock interface penetration and convergence are devised

preventing numerical errors generated by interface capturing schemes in compressible multi material flows in the present work errors generated in computations of compressible multi material flows using shock capturing schemes are examined specifically pressure oscillations when the specific heats ratio is variable but also temperature spikes and species conservation errors these numerical errors are generated at material discontinuities due to an inconsistent treatment of the convective terms though temperature errors are irrelevant to solutions to the euler equations it is shown that they have the potential to lead to problems when physical diffusion is included i e for the navier stokes equations these errors are studied analytically and numerically by considering the one dimensional advection of isolated material discontinuities a methodology preventing such errors for weighted essentially non oscillatory weno schemes is presented in which modified weno weights are used to solve the transport equation for mass fraction in conservative form to prevent temperature and species conservation errors pressure errors are prevented by solving an additional transport equation for a given function of the ratio of specific heats several multi dimensional problems with various discontinuities shocks material interfaces and contact discontinuities including the single mode richtmyer meshkov instability and turbulence are considered to test the method

fusion algebras with negative structure constants abstract we introduce fusion algebras with not necessarily positive structure constants and without identity element we prove that they are semisimple when tensored with c and that their characters satisfy orthogonality relations then we define the proper notion of subrings and factor rings for such algebras for certain algebras r we prove the existence of a ring r with nonnegative structure constants such that r is a factor ring of r we give some examples of interesting factor rings of the representation ring of the quantum double of a finite group then we investigate the algebras associated to hadamard matrices for an n n matrix the corresponding algebra is a factor ring of a subalgebra of z z 2 z n 2

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

the ghost fluid method for compressible gas water simulation an analysis is carried out for the ghost fluid method gfm based algorithm as applied to the gas water riemann problems which can be construed as two single medium gfm riemann problems it is found that the inability to provide correct and consistent riemann waves in the respective real fluids by these two gfm riemann problems may lead to inaccurate numerical results based on this finding two conditions are suggested and imposed for the ghost fluid status in order to ensure that correct and consistent riemann waves are provided in the respective real fluids during the numerical decomposition of the singularity using these two conditions to analyse some of the existing gfm based algorithms such as the original gfm j comput phys 152 1999 457 the new version gfm j comput phys 166 2001 1 j comput phys 175 2002 200 and the modified gfm mgfm j comput phys 190 2003 651 it is found that there are ranges of conditions for each type of solution where either the original gfm or the new version gfm or both are unable to provide correct or consistent riemann waves in one of the real fluids within these ranges examples can be found such that either the original gfm or the new version gfm or both are unable to provide accurate results the mgfm is also found to encounter difficulties when applied to nearly cavitating flow various examples are presented to demonstrate the conclusions obtained the mgfm with proposed modification when applied to nearly cavitating flow is then found to be quite robust and can provide relatively reasonable results

a five equation reduced model for compressible two phase flow problems this paper studies an eulerian diffuse interface model for the simulation of compressible multifluid and two phase flow problems we first show how to derive this model from a seven equation two pressure two velocity model of baer nunziato type using an asymptotic analysis in the limit of zero relaxation time we then study the mathematical properties of the system the structure of the waves the expression of the riemann s invariants and the existence of a mathematical entropy we also describe two different numerical approximation schemes for this system the first one relies on a linearized riemann solver while the second uses more heavily the mathematical structure of the system and relies on a linearization of the characteristic relations finally we present some numerical experiments and comparisons with the results obtained by the two pressure two velocity model as well as some test cases and comparisons with another five equation model recently proposed for interface computations between compressible fluids

a cell centered lagrangian scheme for two dimensional compressible flow problems we present a new lagrangian cell centered scheme for two dimensional compressible flows the primary variables in this new scheme are cell centered i e density momentum and total energy are defined by their mean values in the cells the vertex velocities and the numerical fluxes through the cell interfaces are not computed independently contrary to standard approaches but are evaluated in a consistent manner due to an original solver located at the nodes the main new feature of the algorithm is the introduction of four pressures on each edge two for each node on each side of the edge this extra degree of freedom allows us to construct a nodal solver which fulfills two properties first the conservation of momentum and total energy is ensured second a semidiscrete entropy inequality is provided in the case of a one dimensional flow the solver reduces to the classical godunov acoustic solver it can be considered as its two dimensional generalization many numerical tests are presented they are representative test cases for compressible flows and demonstrate the robustness and the accuracy of this new solver

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

viscous shock profiles and primitive formulations weak solutions of hyperbolic systems in primitive nonconservative form for which a consistent conservation form exists are considered it is shown that for primitive formulations shock relations are not uniquely defined by the states to either side of the shock but also depend on the viscous path connecting the two consistent viscous shock profiles are enforced by adding scheme dependent small viscous perturbations that account for leading order conservation errors the resulting primitive algorithm is conservative to the order of the approximation one dimensional euler calculations of flows containing weak to moderate shocks show that conservation errors in primitive calculations are substantially reduced by including the viscous perturbation terms while not eliminating conservation errors entirely it is found that for a wide range of problems both conservative and primitive flow calculations are of comparable quality

a numerical method for two phase flow consisting of separate compressible and incompressible regions abstract we propose a numerical method for modeling two phase flow consisting of separate compressible and incompressible regions this is of interest for example when the combustion of fuel droplets or the shock induced mixing of liquids is numerically modeled we use the level set method to track the interface between the compressible and incompressible regions as well as the ghost fluid method gfm to create accurate discretizations across the interface the gfm is particularly effective here since the equations differ in both number and type across the interface the numerical method is presented in two spatial dimensions with numerical examples in both one and two spatial dimensions while three dimensional extensions are straightforward

the riemann problem and a high resolution godunov method for a model of compressible two phase flow this paper considers the riemann problem and an associated godunov method for a model of compressible two phase flow the model is a reduced form of the well known baer nunziato model that describes the behavior of granular explosives in the analysis presented here we omit source terms representing the exchange of mass momentum and energy between the phases due to compaction drag heat transfer and chemical reaction but retain the non conservative nozzling terms that appear naturally in the model for the riemann problem the effect of the nozzling terms is confined to the contact discontinuity of the solid phase treating the solid contact as a layer of vanishingly small thickness within which the solution is smooth yields jump conditions that connect the states across the contact as well as a prescription that allows the contribution of the nozzling terms to be computed unambiguously an iterative method of solution is described for the riemann problem that determines the wave structure and the intermediate states of the flow for given left and right states a godunov method based on the solution of the riemann problem is constructed it includes non conservative flux contributions derived from an integral of the nozzling terms over a grid cell the godunov method is extended to second order accuracy using a method of slope limiting and an adaptive riemann solver is described and used for computational efficiency numerical results are presented demonstrating the accuracy of the numerical method and in particular the accurate numerical description of the flow in the vicinity of a solid contact where phases couple and nozzling terms are important the numerical method is compared with other methods available in the literature and found to give more accurate results for the problems considered

a robust and accurate numerical method for transcritical turbulent flows at supercritical pressure with an arbitrary equation of state this paper addresses issues in high fidelity numerical simulations of transcritical turbulent flows at supercritical pressure the proposed strategy builds on a tabulated look up table method based on refprop database for an accurate estimation of non linear behaviors of thermodynamic and fluid transport properties at the transcritical conditions based on the look up table method we propose a numerical method that satisfies high order spatial accuracy spurious oscillation free property and capability of capturing the abrupt variation in thermodynamic properties across the transcritical contact surface the method introduces artificial mass diffusivity to the continuity and momentum equations in a physically consistent manner in order to capture the steep transcritical thermodynamic variations robustly while maintaining spurious oscillation free property in the velocity field the pressure evolution equation is derived from the full compressible navier stokes equations and solved instead of solving the total energy equation to achieve the spurious pressure oscillation free property with an arbitrary equation of state including the present look up table method flow problems with and without physical diffusion are employed for the numerical tests to validate the robustness accuracy and consistency of the proposed approach

quotient structures homomorphisms and partial wreath products in reality based algebras abstract homomorphisms and quotient structures of reality based algebras are presented and described related central idempotents and quotient sets are characterized and the extent to which a homomorphism preserves the given anti automorphisms of the algebras is determined the notion of a partial wreath product an algebraic abstraction of the wedge product of association schemes is developed prototypical results previously obtained for various special cases including table algebras and commutative c algebras are thus uniformly generalized

space time discontinuous galerkin finite element method for two fluid flows a novel numerical method for two fluid flow computations is presented which combines the space time discontinuous galerkin finite element discretization with the level set method and cut cell based interface tracking the space time discontinuous galerkin stdg finite element method offers high accuracy an inherent ability to handle discontinuities and a very local stencil making it relatively easy to combine with local hp refinement the front tracking is incorporated via cut cell mesh refinement to ensure a sharp interface between the fluids to compute the interface dynamics the level set method lsm is used because of its ability to deal with merging and breakup also the lsm is easy to extend to higher dimensions small cells arising from the cut cell refinement are merged to improve the stability and performance the interface conditions are incorporated in the numerical flux at the interface and the stdg discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative the numerical method is applied to one and two dimensional two fluid test problems using the euler equations

multicomponent gas flow computations by a discontinuous galerkin scheme using l2 projection of perfect gas eos a new formulation of multicomponent gas flow computation suited to a discontinuous galerkin discretization is here presented and discussed the original key feature is the use of l 2 projection form of the perfect gas equation of state that allows all thermodynamic variables to span the same functional space this choice greatly mitigates problems encountered by the front capturing schemes in computing discontinuous flow field retaining at the same time their conservation properties at the discrete level and ease of use this new approach combined with an original residual based artificial dissipation technique shows itself capable through a series of tests illustrated in the paper to both control the spurious oscillations of flow variables occurring in high order accurate computations and reduce them increasing the degree of the polynomial representation of the solution this result is of great importance in computing reacting gaseous flows where the local accuracy of temperature and species mass fractions is crucial to the correct evaluation of the chemical source terms contained in the equations even if the presence of the physical diffusivities somewhat brings relief to these problems the present work can therefore also be considered among many others already presented in the literature as the authors first step toward the construction of a new discontinuous galerkin scheme for reacting gas mixture flows

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

multi solid and multi fluid diffuse interface model we extend the model of diffuse solid fluid interfaces developed earlier by authors of this paper to the case of arbitrary number of interacting hyperelastic solids plastic transformations of solids are taken into account through a maxwell type model the specific energy of each solid is given in separable form it is the sum of a hydrodynamic part of the energy depending only on the density and the entropy and an elastic part of the energy which is unaffected by the volume change it allows us to naturally pass to the fluid description in the limit of vanishing shear modulus in spite of a large number of governing equations the model has a quite simple mathematical structure it is a duplication of a single visco elastic model the model is well posed both mathematically and thermodynamically it is hyperbolic and compatible with the second law of thermodynamics the resulting model can be applied in the situations involving an arbitrary number of fluids and solids in particular we show the ability of the model to describe spallation and penetration phenomena occurring during high velocity impacts

the modified ghost fluid method for coupling of fluid and structure constituted with hydro elasto plastic equation of state in this work the modified ghost fluid method mgfm t g liu b c khoo and k s yeo j comput phys 190 2003 pp 651 681 is further developed and applied to treat the compressible fluid compressible structure coupling to facilitate theoretical analysis the structure is modeled as elastic plastic material with perfect plasticity and constituted with the hydro elasto plastic equation of state h s tang and f sotiropoulos j comput phys 151 1999 pp 790 815 under strong impact this results in the coupled compressible fluid compressible structure system which is fully hyperbolic to understand the effect of structure deformation on the interfacial and flow status the compressible fluid compressible structure riemann problem is analyzed in the consideration of material deformation with an approximate riemann problem solver proposed to take into account the effect of material elastic plastic deformation we clearly show the ghost fluid method can be applied to treat the flow deformable structure coupling under strong impact provided that a proper riemann problem solver is used to predict the ghost fluid states and the resultant mgfm can work effectively and efficiently in such situations various examples are presented to validate and support the conclusions reached

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

an interface treating technique for compressible multi medium flow with runge kutta discontinuous galerkin method the high order accurate runge kutta discontinuous galerkin rkdg method is applied to the simulation of compressible multi medium flow generalizing the interface treating method given in chertock et al 2008 9 in mixed cells where the interface is located riemann problems are solved to define the states on both sides of the interface the input states to the riemann problem are obtained by extrapolation to the cell boundary from solution polynomials in the neighbors of the mixed cell the level set equation is solved by using a high order accurate rkdg method for hamilton jacobi equations resulting in a unified dg solver for the coupled problem the method is conservative if we include the states in the mixed cells which are however not used in the updating of the numerical solution in other cells the states in the mixed cells are plotted to better evaluate the conservation errors manifested by overshoots undershoots when compared with states in neighboring cells these overshoots undershoots in mixed cells are problem dependent and change with time numerical examples show that the results of our scheme compare well with other methods for one and two dimensional problems in particular the algorithm can capture well complex flow features of the one dimensional shock entropy wave interaction problem and two dimensional shock bubble interaction problem

runge kutta discontinuous galerkin method for the approximation of baer and nunziato type multiphase models a high order numerical method is developed for the computation of compressible multiphase flows the model we use is based on the baer and nunziato type systems 4 among all the other available models in the literature these systems present the advantage to be able to simulate either interface or mixture problems nevertheless they still raise some issues mainly based on their non conservative feature the numerical method we propose is a discontinuous galerkin type in this work the interior side integrals are computed thanks to 2 robustness and high order of accuracy of the method are proved on classical interface problems but also on suitably derived analytical solutions

diffuse interface model for compressible fluid compressible elastic plastic solid interaction an eulerian hyperbolic diffuse interface model for elastic plastic solid fluid interaction is constructed the system of governing equations couples euler equations of compressible fluids and a visco plastic model of maxwell type materials the deviatoric part of the stress tensor decreases during plastic deformations in the same manner as models of multicomponent fluids in particular the model is able to create interfaces which were not present initially the model is thermodynamically compatible it verifies the entropy inequality however a numerical treatment of the model is particularly challenging indeed the model is non conservative so a special numerical splitting is proposed to overcome this difficulty the numerical algorithm contains two relaxation procedures one of them is physical and is related to the plastic relaxation mechanism relaxation toward the yield surface the second one is numerical it consists in replacing the algebraic equation expressing a mechanical equilibrium between components by a partial differential equation with a short relaxation time the numerical method was tested in 1d case wilkins flying plate problem 2d plane case impact of a projectile on a plate and axisymmetrical case taylor test problem impact with penetration effects etc numerical examples show the ability of the model to deal with real physical phenomena

discontinuous galerkin finite element methods for hyperbolic nonconservative partial differential equations we present space and space time discontinuous galerkin finite element dgfem formulations for systems containing nonconservative products such as occur in dispersed multiphase flow equations the main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form then the formulation must reduce to that for conservative systems standard dgfem formulations cannot be applied to nonconservative systems of partial differential equations we therefore introduce the theory of weak solutions for nonconservative products into the dgfem formulation leading to the new question how to define the path connecting left and right states across a discontinuity the effect of different paths on the numerical solution is investigated and found to be small we also introduce a new numerical flux that is able to deal with nonconservative products our scheme is applied to two different systems of partial differential equations first we consider the shallow water equations where topography leads to nonconservative products in which the known possibly discontinuous topography is formally taken as an unknown in the system second we consider a simplification of a depth averaged two phase flow model which contains more intrinsic nonconservative products

a riemann problem based method for solving compressible and incompressible flows a riemann problem based method for solving two medium flow including compressible and incompressible regions is presented the material interface is advanced by front tracking method and the material interface boundary conditions are defined by modified ghost fluid method a coupled compressible and incompressible riemann problem constructed in the normal direction of the material interface is proposed to predict the interfacial states with the ghost fluid states the compressible and incompressible flows are solved by discontinuous galerkin method an incompressible discontinuous galerkin method with nonuniform time step is also deduced for shock wave formed in compressible flow the numerical errors for the ghost fluid method in earlier works are analyzed and discussed in the numerical examples it shows that the proposed method can provide reasonable results including shock wave location two medium flow including compressible and incompressible regions is investigated an incompressible discontinuous galerkin method with nonuniform time step is derived a coupled compressible and incompressible riemann problem is proposed reasonable results are obtained for strong shock wave formed in compressible flow

a five equation model for the simulation of miscible and viscous compressible fluids abstract typical multispecies compressible navier stokes computations employ conservative equations for mass fraction transport upwind discretisations of these governing equations produce spurious pressure oscillations at diffuse contact surfaces between gases of differing ratio of specific heat capacities which degrade the convergence rate of the algorithm adding quasi conservative equations for volume fraction can solve this error however this approach has been derived only for immiscible fluids here a five equation quasi conservative model is proposed that includes the effects of species diffusion viscosity and thermal conductivity the derivation of the model is presented along with a numerical method to solve the governing equations at second order accuracy in space and time formal convergence studies demonstrate the expected order of accuracy is achieved for three benchmark problems cross validated against two standard mass fraction models in these test cases the new model has between 2 and 10 times lower error for a given grid size simulations of a two dimensional air sf6 richtmyer meshkov instability demonstrate that the new model converges to the solution with four times fewer points in each direction when compared to the mass fraction model in an identical numerical framework this represents an 40 times lower computational cost for an equivalent error in two dimensional computations the proposed model is thus very suitable for direct numerical simulation and large eddy simulation of compressible mixing

fusion algebras for imprimitive complex reflection groups abstract we prove that the fourier matrices for the imprimitive complex reflection groups introduced by malle in gunter malle unipotente grade imprimitiver komplexer spiegelungsgruppen j algebra 177 3 1995 768 826 define fusion algebras with not necessarily positive but integer structure constants hence they define z algebras as a result we obtain that all known fourier matrices belonging to spetses define algebras with integer structure constants

riemann problem and level set approaches for homentropic two fluid flow computations a finite volume method is presented for the computation of compressible flows of two immiscible fluids at very different densities a novel ingredient in the method is a linearized two fluid osher scheme allowing for flux computations in the case of different fluids e g water and air left and right of a cell face a level set technique is employed to distinguish between the two fluids the level set equation is incorporated into the system of hyperbolic conservation laws fixes are presented for the solution errors pressure oscillations that may occur near two fluid interfaces when applying a capturing method the fixes are analyzed and tested for two fluid flows with arbitrarily large density ratios a simple variant of the ghost fluid method appears to be a perfect remedy computations for compressible water air flows yield perfectly sharp pressure oscillation free interfaces the masses of the separate fluids appear to be conserved up to first order accuracy

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

an approximate linearized riemann solver for a two fluid model an approximate linearized riemann solver is presented for the numerical simulation of two phase flows this new solver is based on a linearization of nonconservative products and uses an extension of roe s approximate riemann solver the scheme is applied to shock tube problem and to a standard test for two fluid codes

accurate computation of contact discontinuities in flows with general equations of state abstract the evolution of a contact discontinuity in a real fluid flow is particularly difficult to compute with a conservative scheme indeed godunov type schemes involve an averaging step which may create an intermediate state with an incorrect pressure spurious acoustic waves are then generated these waves affect transient simulations as well as implicit time marching procedures for steady state computations however a formulation involving the pressure as a dependent variable can be used to restore a fast convergence for steady state computations

a model and numerical method for compressible flows with capillary effects a new model for interface problems with capillary effects in compressible fluids is presented together with a specific numerical method to treat capillary flows and pressure waves propagation this new multiphase model is in agreement with physical principles of conservation and respects the second law of thermodynamics a new numerical method is also proposed where the global system of equations is split into several submodels each submodel is hyperbolic or weakly hyperbolic and can be solved with an adequate numerical method this method is tested and validated thanks to comparisons with analytical solutions laplace law and with experimental results on droplet breakup induced by a shock wave

a high resolution godunov method for compressible multi material flow on overlapping grids a numerical method is described for inviscid compressible multi material flow in two space dimensions the flow is governed by the multi material euler equations with a general mixture equation of state composite overlapping grids are used to handle complex flow geometry and block structured adaptive mesh refinement amr is used to locally increase grid resolution near shocks and material interfaces the discretization of the governing equations is based on a high resolution godunov method but includes an energy correction designed to suppress numerical errors that develop near a material interface for standard conservative shock capturing schemes the energy correction is constructed based on a uniform pressure velocity flow and is significant only near the captured interface a variety of two material flows are presented to verify the accuracy of the numerical approach and to illustrate its use these flows assume an equation of state for the mixture based on the jones wilkins lee jwl forms for the components this equation of state includes a mixture of ideal gases as a special case flow problems considered include unsteady one dimensional shock interface collision steady interaction of a planar interface and an oblique shock planar shock interaction with a collection of gas filled cylindrical inhomogeneities and the impulsive motion of the two component mixture in a rigid cylindrical vessel

hybrid multifluid algorithms extensions of many successful single component schemes to compute multicomponent gas dynamics suffer from oscillations and other computational inaccuracies near material interfaces that are caused by the failure of the schemes to maintain pressure equilibrium between the fluid components a new algorithm based on the compressible euler equations for multicomponent fluids augmented by the pressure evolution equation is presented the extended set of equations offers two alternative ways to update the pressure field i using the equation of state or ii using the pressure evolution equation in a numerical implementation these two procedures generally yield different answers the former is a standard conservative update but may produce oscillations near material interfaces the latter is nonconservative but becomes exact near interfaces and automatically maintains pressure equilibrium a hybrid scheme which selects from the two pressure update procedures is presented the scheme perfectly conserves total mass and momentum and conserves total energy everywhere except at a finite very small number of grid cells computed solutions exhibit oscillation free interfaces and have em negligible relative conservation errors in total energy even for very strong shocks the proposed hybrid approach and switching strategies are independent of the numerical implementation and may provide a simple framework within which to extend one s favourite scheme to solve multifluid dynamics

direct numerical simulation of interfacial instabilities a consistent conservative all speed sharp interface method we present a conservative and consistent numerical method for solving the navier stokes equations in flow domains that may be separated by any number of material interfaces at arbitrarily high density viscosity ratios and acoustic impedance mismatches subjected to strong shock waves and flow speeds that can range from highly supersonic to near zero mach numbers a principal aim is prediction of interfacial instabilities under superposition of multiple potentially active modes rayleigh taylor kelvin helmholtz richtmyer meshkov as found for example with shock driven immersed fluid bodies locally oblique shocks accordingly we emphasize fidelity supported by physics based validation including experiments consistency is achieved by satisfying the jump discontinuities at the interface within a conservative 2nd order scheme that is coupled in a conservative manner to the bulk fluid motions the jump conditions are embedded into a riemann problem solved exactly to provide the pressures and velocities along the interface which is tracked by a level set function to accuracy of o dx 5 dt 4 subgrid representation of the interface is achieved by allowing curvature of its constituent interfacial elements to obtain o dx 3 accuracy in cut cell volume with attendant benefits in calculating cell geometric features and interface curvature o dx 3 overall the computation converges at near theoretical o dx 2 spurious currents are down to machine error and there is no time step restriction due to surface tension our method is built upon a quadtree like adaptive mesh refinement infrastructure when necessary this is supplemented by body fitted grids to enhance resolution of the gas dynamics including flow separation shear layers slip lines and critical layers comprehensive comparisons with exact solutions for the linearized rayleigh taylor and kelvin helmholtz problems demonstrate excellent performance sample simulations of liquid drops subjected to shock waves demonstrate for the first time ab initio numerical prediction of the key interfacial features and phenomena found in recent experimental and theoretical studies of this class of problems t g theofanous aerobreakup of newtonian and viscoelastic liquids ann rev fluid mech 43 2011 661 690

an exact riemann solver for compressible two phase flow models containing non conservative products in this article we present a new numerical procedure for solving exactly the riemann problem of compressible two phase flow models containing non conservative products these products appear in the expressions for the interactions between the two phases thus in the compressible limit the governing equations are hyperbolic but can not be written as conservation laws i e in divergence form in general the solution to the riemann problem of these models contains six distinct centered waves according to the relative position of these waves in the x t plane the possible solutions can be classified into four principal configurations the riemann solver we propose herein investigates sequentially each of these configurations until an admissible solution is calculated special configurations corresponding to coalescence of waves are also analyzed and included in the solver further we examine the accuracy and robustness of three known methods for the integration of the non conservative products via a series of numerical tests finally the issue of existence and uniqueness of solutions to the riemann problem is discussed

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients