nonlinear resonance in systems of conservation laws the riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed the inhomogeneity is modeled by a linearly degenerate field the solution of the riemann problem determines the nature of wave interactions and thus the riemann problem serves as a canonical form for nonlinear systems of conservation laws generic conditions on the fluxes are stated and it is proved that under these conditions the solution of the riemann problem exists is unique and has a fixed structure this demonstrates that in the above sense resonant inhomogeneous systems generically have the same canonical form the wave curves for these systems are only lipschitz continuous in a neighborhood of the states where the wave speeds coincide and so in contrast to strictly hyperbolic systems the implicit function theorem cannot be applied directly to obtain existence and uniqueness here we show that existence

a level set approach to the computation of twinning and phase transition dynamics a computational method is proposed for the dynamics of solids capable of twinning and phase transitions in a two dimensional sharp interface model of twinning the stored energy function is a nonconvex potential with multiple wells the evolution of twin interfaces is governed by field equations and jump conditions of momentum balance and by a kinetic relation expressing the interface velocity as a function of the local driving traction and interfacial orientation a regularized version of the model is constructed based on the level set method a level set function which changes signs across the interface is introduced the evolution of this function is described by a hamilton jacobi equation whose velocity coefficient is determined by the kinetic relation jump conditions are thereby eliminated allowing finite difference discretization numerical simulations exhibit complex evolution of the interface including cusp formation needle growth spontaneous tip splitting and topological changes that result in microstructure refinement the results capture experimentally observed phenomena in martensitic crystals

adaptive mesh and algorithm refinement using direct simulation monte carlo adaptive mesh and algorithm refinement amar embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement amr hierarchy the coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and coarse levels of amr direct simulation monte carlo dsmc is used as the particle algorithm embedded within a godunov type compressible navier stokes solver several examples are presented and compared with purely continuum calculations

kinetic solvers with adaptive mesh in phase space an adaptive mesh in phase space amps methodology has been developed for solving multidimensional kinetic equations by the discrete velocity method a cartesian mesh for both configuration r and velocity v spaces is produced using a tree of trees tot data structure the r mesh is automatically generated around embedded boundaries and is dynamically adapted to local solution properties the v mesh is created on the fly in each r cell mappings between neighboring v space trees is implemented for the advection operator in r space we have developed algorithms for solving the full boltzmann and linear boltzmann equations with amps several recent innovations were used to calculate the discrete boltzmann collision integral with dynamically adaptive v mesh the importance sampling multipoint projection and variance reduction methods we have developed an efficient algorithm for calculating the linear boltzmann collision integral for elastic and inelastic collisions of hot light particles in a lorentz gas our amps technique has been demonstrated for simulations of hypersonic rarefied gas flows ion and electron kinetics in weakly ionized plasma radiation and light particle transport through thin films and electron streaming in semiconductors we have shown that amps allows minimizing the number of cells in phase space to reduce the computational cost and memory usage for solving challenging kinetic problems

kinetic relations and the propagation of phase boundaries in solids this paper treats the dynamics of phase transformations in elastic bars the specific issue studied is the compatibility of the field equations and jump conditions of the one dimensional theory of such bars with two additional constitutive requirements a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition a special elastic material with a piecewise linear non monotonic stress strain relation is considered and the riemann problem for this material is analyzed for a large class of initial data it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities

a fully discrete scheme for diffusive dispersive conservation laws we introduce a fully discrete in both space and time scheme for the numerical approximation of diffusive dispersive hyperbolic conservation laws in one space dimension this scheme extends an approach by lefloch and rohde 4 it satisfies a cell entropy inequality and as a consequence the space integral of the entropy is a decreasing function of time this is an important stability property shared by the continuous model as well following hayes and lefloch 2 we show that the limiting solutions generated by the scheme need not coincide with the classical oleinik kruzkov entropy solutions but contain nonclassical undercompressive shock waves investigating the properties of the scheme we stress various similarities and differences between the continuous model and the discrete scheme dynamics of nonclassical shocks nucleation etc

block structured adaptive mesh refinement algorithms for vlasov simulation direct discretization of continuum kinetic equations like the vlasov equation are under utilized because the distribution function generally exists in a high dimensional 3d space and computational cost increases geometrically with dimension we propose to use high order finite volume techniques with block structured adaptive mesh refinement amr to reduce the computational cost the primary complication comes from a solution state comprised of variables of different dimensions we develop the algorithms required to extend standard single dimension block structured amr to the multi dimension case specifically algorithms for reduction and injection operations that transfer data between mesh hierarchies of different dimensions are explained in detail in addition modifications to the basic amr algorithm that enable the use of high order spatial and temporal discretizations are discussed preliminary results for a standard 1d 1v vlasov poisson test problem are presented results indicate that there is potential for significant savings for some classes of vlasov problems

lattices for the lattice boltzmann method a recently introduced theory of higher order lattice boltzmann models chikatamarla and karlin phys rev lett 97 190601 2006 is elaborated in detail a general theory of the construction of lattice boltzmann models as an approximation to the boltzmann equation is presented new lattices are found in all three dimensions and are classified according to their accuracy degree of approximation of the boltzmann equation the numerical stability of these lattices is argued based on the entropy principle the efficiency and accuracy of many new lattices are demonstrated via simulations in all three dimensions

higher order galilean invariant lattice boltzmann model for microflows single component gas we introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice boltzmann model we show that an off lattice d3q27 model exists with correct equilibrium to recover galilean invariant form of navier stokes equation without any cubic error in the first part of this work we show that the present model can capture two important features of the microflow in a single component gas knudsen boundary layer and knudsen paradox finally we present numerical results corresponding to couette flow for two representative knudsen numbers we show that the off lattice d3q27 model exhibits better accuracy as compared to more widely used on lattice d3q19 or d3q27 model finally our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature based procedure for the construction of the lattice boltzmann model

coupling of general lagrangian systems this work is devoted to the coupling of two fluid models such a s two euler systems in lagrangian coordinates at a fixed interface we define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled riemann problem in the present setting where the interface is characteristic a particula r choice of dependent variables which are transmitted ensures the overall conservativity

choice of measure source terms in interface coupling for a model problem in gas dynamics this paper is devoted to the mathematical and numerical analysis of a coupling procedure for one dimensional euler systems the two systems have different closure laws and are coupled through a thin fixed interface following the work of 5 we propose to couple these systems by a bounded vector valued dirac measure concentrated at the coupling interface which in the applications may have a physical meaning we show that the proposed framework allows to control the coupling conditions and we propose an approximate riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem numerical experiments in constrained optimization problems are then presented to assess the performances of the present method 1 introduction the study of large scale and complex problems exhibiting a wide range of physical space and time scales see for instance 62 35 14 usually requires separate solvers adapted to the resolution of specific scales this is the case of many industrial flows let us quote for example the numerical simulation of two phase flows applied to the burning liquid oxygen hydrogen gas in rocket engines 58 this kind of flow contains both separated and dispersed two phase flows due to atomization and evaporation phenomena this requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled another example concerns turbomachine flows which can be modeled by the euler equations of gas dynamics with different closure laws between the stages of the turbine where the conditions of temperature and pressure are strongly heterogeneous the coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine the method of interface coupling allows to represent the evolution of such flows where different models are separated by fixed interfaces first coupling conditions are specified at the interface to exchange information between the systems the definition of transmission conditions generally results from physical consideration e g the conservation or the continuity of given variables then the transmission conditions are represented at the discrete level the study of interface coupling for nonlinear hyperbolic systems has received attention for several years in 43 the authors study the scalar case from both mathematical and numerical points of view

a new conservative unsplit method for the solution of the vlasov equation we have developed a new conservative method for solving the vlasov equation without using any splitting technique our goal is to maintain the positivity of the distribution function and to avoid un physical oscillations which might lead to numerical instabilities based on a finite volume conservative discretization of the conservative form of the vlasov equation we implemented a highly accurate second order upwind scheme in order to avoid un physical oscillations and their possible numerical instability we apply a flux limiter in the second order we validate our new vlasov solver by considering standard cases of one dimensional current driven ion acoustic instabilities solving a vlasov ampere set of equations

lattice boltzmann method on unstructured grids further developments we discuss further developments of the finite volume lattice boltzmann formulation on unstructured grids it is shown that the method tolerates significant grid distortions without showing any appreciable numerical viscosity effects at second order in the mesh size a theoretical argument of plausibility for such a property is presented in addition a set of boundary conditions which permit to handle flows with open boundaries is also introduced and numerically demonstrated for the case of channel flows and driven cavity flows

high order schemes entropy inequalities and nonclassical shocks we are concerned with the approximation of undercompressive regularization sensitive nonclassical solutions of hyperbolic systems of conservation laws by high order accurate conservative and semidiscrete finite difference schemes nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance particular attention is given here to a class of systems of conservation laws including the scalar equations and the system of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables first we investigate the existence and the properties of entropy conservative schemes a notion due to tadmor math comp 49 1987 pp 91 103 in particular we exhibit a new five point scheme which is third order accurate at least second we study a class of entropy stable and high order accurate schemes satisfying a single cell entropy inequality they are built from any high order entropy conservative scheme by adding to it a mesh independent numerical viscosity which preserves the order of accuracy of the base scheme these schemes can only converge to solutions of the system of conservation laws satisfying the entropy inequality these entropy stable schemes exhibit mild oscillations near shocks and interestingly may converge to classical or nonclassical entropy solutions depending on the sign of their dispersion coefficient then based on a third order entropy conservative scheme we propose a general scheme for the numerical computation of nonclassical shocks importantly our scheme satisfies a cell entropy inequality following hayes and lefloch siam j numer anal 35 1998 pp 2169 2194 we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws our results compare favorably with previous analytical and numerical results finally we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete lax wendroff type schemes

comparison of the shakhov kinetic equation and dsmc method as applied to space vehicle aerothermodynamics abstract the paper is devoted to the comparative study of shakhov model kinetic equation and direct simulation monte carlo dsmc solutions as applied to high speed flow of a monatomic gas over generic three dimensional space vehicle under angle of attack the corresponding calculations are carried out using nesvetay and smile software packages developed by the authors satisfactory agreement is shown for surface distribution of pressure friction and heat transfer coefficients despite very different modeling approaches and computational meshes employed it is thus shown that the use of the model kinetic equations with approximate collision integrals provides accurate results for high speed external monatomic gas flows over complex three dimensional geometries at acceptable computational cost

stability and accuracy of spatial approximations for wave equation tidal models amplitude and phase characteristics for numerical approximations to the shallow water wave equation are obtained for linear and quadratic finite elements for finite difference approximations for non constant bathemetry and for uneven node spacing stability is shown to require non zero friction as well as satisfaction of a courant constraint lumping is shown to reduce the courant constraint for stability while higher order and quadratic finite element approximations require a more restrictive constraint than their second order and linear finite element counterparts the amplitude of the propagation factor for stable schemes and propagating waves is seen to be independent of the courant number and type of numerical approximation although the higher order and quadratic schemes provide better propagation of the low and moderate frequency waves the highest frequency waves 2 x are better propagated by low order numerical methods

numerical approximation of a macroscopic model of pedestrian flows this paper is concerned with the numerical approximation of the solutions of a macroscopic model for the description of the flow of pedestrians solutions of the associated riemann problem are known to be possibly nonclassical in the sense that the underlying discontinuities may well violate the lax inequalities which makes their numerical approximation very sensitive this study proposes to extend the transport equilibrium strategy proposed in c chalons c r acad sci paris ser i 342 2006 pp 623 626 and c chalons transport equilibrium schemes for computing nonclassical shocks i scalar conservation laws submitted for computing the nonclassical solutions of scalar conservation laws with either a concave convex or a convex concave flux function and supplemented with an invertible kinetic function these strong properties are not fulfilled in the present setting since the flow function admits several inflection points and the kinetic function is not invertible we nevertheless succeed in

relaxation approximation of the euler equations the aim of this paper is to show how solutions to the one dimensional compressible euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term the enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate riemann solver from a theoretical point of view the convergence of solutions to the enlarged system towards solutions to the euler equations is proved for local in time smooth solutions we also show that arbitrarily large shock waves for the euler equations admit smooth shock profiles for the enlarged relaxation system in the end we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system we test it on various cases

the numerical interface coupling of nonlinear hyperbolic systems of conservation laws i the scalar case we study the theoretical and numerical coupling of two general hyperbolic conservation laws the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model in order to analyze the convergence of the coupled numerical scheme we first revisit the approximation of the boundary value problems we then prove the convergence and characterize the limit solution of the coupled schemes in a few simple but significative coupling situations the general coupling problem is analyzed for riemann initial data and illustrated by numerical simulations resume nous nous interessons a une nouvelle forme de couplage de deux systemes hyperboliques de lois de conservation ce couplage assure de facon faible la continuite de la solution a l interface sans imposer la conservativite du modele couple pour etudier la convergence du schema d approximation numerique nous commencons par reprendre les resultats concernant l approximation du probleme aux limites nous demontrons ensuite la convergence du schema couple dans un certain nombre de cas interessants le cas general du couplage est etudie et illustre numeriquement pour une donnee initiale de riemann

propagating phase boundaries formulation of the problem and existence via the glimm method this paper treats the hyperbolic elliptic system of two conservation laws which describes the dynamics of an elastic material having a non monotone strain stress function followingabeyaratne knowles we propose a notion of admissible weak solution for this system in the class of functions of bounded variation the formulation includes an entropy inequality a kinetic relation imposed along any subsonic phase boundary and an initiation criterion for the appearance of new phase boundaries we prove thel1 continuous dependence of the solution to the riemann problem our main result yields the existence and the stability of propagating phase boundaries the proofs are based onglimm s scheme and in particular on the techniques ofglimm andlax in order to deal with the kinetic relation we prove a result of pointwise convergence of the phase boundary

an euler solver based on locally adaptive discrete velocities a new discrete velocity model is presented to solve the three dimensional euler equations the velocities in the model are of an adaptive nature both the origin of the discrete velocity space and the magnitudes of the discrete velocities are dependent on the local flow and are used in a finite volume context the numerical implementation of the model follows the near equilibrium flow method of nadiga and pullin and results in a scheme which is second order in space in the smooth regions and between first and second order at discontinuities and second order in time the three dimensional code is included for one choice of the scaling between the magnitude of the discrete velocities and the local internal energy of the flow the method reduces to a flux splitting scheme based on characteristics as a preliminary exercise the result of the sod shock tube simulation is compared to the exact solution

computational methods for propagating phase boundaries this paper considers numerical methods for computing propagating phase boundaries in solids described by the physical model introduced by abeyaratne and knowles the model under consideration consists of a set of conservation laws supplemented with a kinetic relation and a nucleation criterion discontinuities between two different phases are undercompressive crossing waves in the general terminology of nonstrictly hyperbolic systems of conservation laws this paper studies numerical methods designed for the computation of such crossing waves we propose a godunov type method combining front tracking with a capturing method we also consider glimm s random choice scheme both methods share the property that the phase boundaries are sharply computed in the sense that there are no numerical interior points for the description of a phase boundary this property is well known for the glimm s scheme on the other hand our front tracking algorithm is designed so that it tracks phase boundaries but captures shock waves phase boundaries are sensitive to numerical dissipation effects so the above property is essential to ensure convergence toward the correct entropy weak solution convergence of the godnuov type method is demonstrated numerically extensive numerical experiments show the practical interest of both approaches for computations of undercompressive crossing waves

non classical riemann solvers with nucleation we introduce a new nonclassical riemann solver for scalar conservation laws with concave convex flux function this solver is based on both a kinetic relation which determines the propagation speed of undercompressive nonclassical shock waves and a nucleation crite rion which makes a choice between a classical riemann solution and a nonclassical one we establish the existence of nonclassical entropy solutions of the cauchy problem and discuss several examples of wave interactions we also show the existence of a class of solutions called splitting merging solutions which are made of two large shocks and small bv bounded variation perturbations the nucleation solvers as we call them are applied to and actually motivated by the theory of thin film flows they help explain numerical results observed for such flows

modelling the transition between fixed and mobile bed conditions in two phase free surface flows in a two phase free surface flow the transition from a mobile bed condition to a fixed bed one and vice versa occurs at a sharp interface across which the relevant system of partial differential equations changes abruptly this leads to the possibility of conceiving a new type of riemann problem rp which we have called composite riemann problem crp where not only the initial constant values of the variables but also the system of equations change from left to right of a discontinuity in this paper we present a strategy for solving a crp by reducing it to a standard rp of a single composite system of equations this can be obtained by combining the two original systems by means of a suitable weighting function namely the erodibility variable and the introduction of an appropriate differential equation for this quantity in this way the crp problem can be analyzed theoretically with standard methods and the features of the solutions can be clearly identified in particular a stationary contact wave is able to correctly describe the sharp transition between mobile and fixed bed conditions a finite volume scheme based on the multiple averages generalized roe approach rosatti and begnudelli 2013 22 was used to numerically solve the fixed mobile crp several test cases demonstrate the effectiveness exact well balanceness and high accuracy of the scheme when applied to problems that fall within the physical range of applicability of the relevant mathematical model

solution of the shallow water equations by least squares collocation a method of solution of the shallow water equations in their primitive form that yield solutions free of numerical oscillations is presented the method is based on the least squares collocation concept the focus is on the accuracy of the solution procedure comparison of the numerical solution with analytic solutions for problems of irregular geometry are presented key features are the ability of the method to solve the equations in an irregular domain with a completely orthogonal computational mesh and the inherent nondissipative noise control of the temporal and spatial approximations

some response characteristics of finite element tidal models abstract small scale error can arise in finite element tidal models from different aspects of the spatial discretization one source is incompatibility in smoothness of interpolation of the pressure and velocity fields noise of this type can be eliminated completely by using stokes helmholtz potentials to represent the velocity and interpolating these potentials with the same smoothness as is given to the pressure another source is inhomogeneous triangulation in which the number of links is not the same at all nodes noise of this type can be suppressed by using a high resolution grid the resolution in question being that of the response rather than of the domain these points are illustrated by examining the response characteristics of idealized models it is found that in the models considered small scale noise can be ascribed to resonance anomalies associated with the method of spatial discretization in this analysis the time is continuous but some remarks are appended on the effect of time discretization and a formal proof is given of the time stepping stability of a general discretized form of laplace s tidal equations

navier stokes equations with several independent pressure laws and explicit predictor corrector schemes this work is concerned with the numerical capture of stiff viscous shock solutions of navier stokes equations for complex compressible materials in the regime of large reynolds numbers after 2 and 6 a relevant numerical capture is known to require the satisfaction of an extended set of non classical rankine hugoniot conditions due to the non conservation form of the governing pde model here we show how to enforce their validity at the discrete level without the need for solving local non linear algebraic problems non linearities are bypassed when introducing new averaging techniques which are proved to satisfy all the desirable stability properties when invoking suitable approximate riemann solutions a relaxation procedure is proposed to that purpose with the benefit of a fairly simple overall numerical method

nonclassical shocks and kinetic relations finite difference schemes we consider hyperbolic systems of conservation laws that are not genuinely nonlinear the solutions generated by diffusive dispersive regularizations may include nonclassical n c shock waves that do not satisfy the classical liu entropy criterion we investigate the numerical approximation of n c shocks via conservative difference schemes constrained only by a single entropy inequality the schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms limits of these schemes are characterized via the kinetic relation introduced earlier by the authors we determine the kinetic function numerically for several examples of systems and schemes this study demonstrates that the kinetic relation is a suitable tool for the selection of unique n c solutions and for the study of their sensitive dependence on the critical parameters the ratios of diffusion dispersion and diffusion mesh size the shock strength and the order of discretization of the flux

lagrangian systems of conservation laws we study the mathematical structure of 1d systems of conservation laws written in the lagrange variable modifying the symmetrization proof of systems of conservation laws with three hypothesis we prove that these models have a canonical formalism these hypothesis are i the entropy flux is zero ii galilean invariance iii reversibility for smooth solutions then we study a family of numerical schemes for the solution of these systems we prove that they are entropy consistent we also prove from general considerations the symmetry of the spectrum of the jacobian matrix

discontinuous galerkin algorithms for fully kinetic plasmas abstract we present a new algorithm for the discretization of the non relativistic vlasov maxwell system of equations for the study of plasmas in the kinetic regime using the discontinuous galerkin finite element method for the spatial discretization we obtain a high order accurate solution for the plasma s distribution function time stepping for the distribution function is done explicitly with a third order strong stability preserving runge kutta method since the vlasov equation in the vlasov maxwell system is a high dimensional transport equation up to six dimensions plus time we take special care to note various features we have implemented to reduce the cost while maintaining the integrity of the solution including the use of a reduced high order basis set a series of benchmarks from simple wave and shock calculations to a five dimensional turbulence simulation are presented to verify the efficacy of our set of numerical methods as well as demonstrate the power of the implemented features

coupling techniques for nonlinear hyperbolic equations iv well balanced schemes for scalar multi dimensional and multi component laws this series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations the coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations in an earlier work this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations in the present paper we significantly extend this framework and in addition encompass equations in several space variables this new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space our main contributions are on one hand the design and analysis of a well balanced finite volume method on general triangulations and on the other hand a proof of convergence of this method toward entropy solutions extending coquel cockburn and lefloch s theory restricted to a single conservation law without coupling the core of our analysis is first the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and second a proof of convergence toward the entropy solution of the coupling problem

mixed finite elements for global tide models we study mixed finite element methods for the linearized rotating shallow water equations with linear drag and forcing terms by means of a strong energy estimate for an equivalent second order formulation for the linearized momentum we prove long time stability of the system without energy accumulation the geotryptic state a priori error estimates for the linearized momentum and free surface elevation are given in l 2 l2 as well as for the time derivative and divergence of the linearized momentum numerical results confirm the theoretical results regarding both energy damping and convergence rates

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement this paper describes a unified flow solver ufs for rarefied and continuum gas flows the ufs separates the rarefied and continuum flow domains and selects appropriate solvers to combine the efficiency of continuum models with the accuracy of kinetic models the direct numerical solution of the boltzmann transport equation is used in rarefied regions while kinetic schemes of continuum fluid dynamics are used elsewhere using similar computational techniques for the kinetic and continuum solvers and employing intelligent domain decomposition algorithms attain the efficiency and numerical stability of the ufs solutions of test problems are presented to illustrate the capabilities of the ufs for high and low speed flows it is shown that the ufs can dynamically adapt the computational mesh and automatically introduce and remove kinetic patches to provide significant savings by limiting molecular scale solutions only to the regions where they are needed

locally refined discrete velocity grids for stationary rarefied flow simulations most of deterministic solvers for rarefied gas dynamics use discrete velocity or discrete ordinate approximations of the distribution function on a cartesian grid this grid must be sufficiently large and fine to describe the distribution functions at every space position in the computational domain for 3 dimensional hypersonic flows like in re entry problems this induces much too dense velocity grids that cannot be practically used for memory storage requirements in this article we present an approach to generate automatically a locally refined velocity grid adapted to a given simulation this grid contains much less points than a standard cartesian grid and allows us to make realistic 3 dimensional simulations at a reduced cost with a comparable accuracy

the δ e method for the boltzmann equation a new numerical method is proposed to solve the boltzmann equation a frame is set up by using a discrete velocity approximation in the infinite velocity space but by considering only those distribution function points which are not too small the distribution function points may occur anywhere in the infinite discrete velocity space and are not constrained to a pre specified region a fourth order finite difference is used for the convection terms a monte carlo like method is applied to the discrete velocity model of the collision integral the effort of the method is proportional to the number of discrete points numerical examples are given for the full boltzmann equation and results for some benchmark problems are compared with analytical or prior solutions

an accuracy analysis of boundary conditions for the forced shallow water equations the accuracy of numerical boundary conditions for the linearized one dimensional shallow water equations is determined by extending the dispersion fourier analysis originally presented by c k chu and a sereny j comput phys 15 476 1974 accuracy is measured by calculating steady state amplitudes of incoming and outgoing numerical waves at a boundary as functions of a forcing frequency radiating closed and forced radiating boundary conditions are studied in combination with two numerical schemes the richardson sielecki explicit finite difference scheme and the galerkin finite element method with piecewise linear basis functions and crank nicolson time stepping contamination of the finite element solution with short waves is seen to vary with the choice of extraneous boundary conditions an example of l n trefethen s j comput phys 49 199 1983 interpretation of the b gustafsson h kreiss and a sundstrom math comput 26 649 1972 instability is also given for the finite element method

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

high order entropy conservative schemes and kinetic relations for van der waals fluids abstract we consider the mixed hyperbolic elliptic system of two conservation laws modeling the dynamics of van der waals fluids viscosity and capillarity effects are taken into account we introduce a new class of semidiscrete high order schemes which are entropy conservative in the sense of tadmor when the viscosity is neglected and otherwise dissipate the associated mathematical entropy our numerical schemes generalize the works by e tadmor 1987 math comput 49 91 and p g lefloch and c rohde 2000 siam j numer anal 37 2023 who proposed second order and third order entropy conservative schemes respectively following b t hayes and p g lefloch 1998 siam j numer anal 35 2169 we demonstrate numerically that balanced viscosity and capillarity terms in van der waals fluids may generate nonclassical shock waves or subsonic propagating phase transitions such waves are undercompressive and do not satisfy standard entropy criteria they must be characterized by a kinetic function which we determine numerically in this paper from vanishing viscosity and capillarity the kinetic relation is an efficient tool to discuss the interplay among the viscosity capillarity and discretization parameters in van der waals fluids

a discrete least squares collocation method for two dimensional nonlinear time dependent partial differential equations abstract in this paper we develop regularized discrete least squares collocation and finite volume methods for solving two dimensional nonlinear time dependent partial differential equations on irregular domains the solution is approximated using tensor product cubic spline basis functions defined on a background rectangular interpolation mesh which leads to high spatial accuracy and straightforward implementation and establishes a solid base for extending the computational framework to three dimensional problems a semi implicit time stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem a key finding of our study is that the newly proposed mesh free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero both methods produce a large constrained least squares problem that must be solved at each time step in the advancement of the solution we have found that regularization yields a relatively well conditioned system that can be solved accurately using qr factorization an extensive numerical investigation is performed to illustrate the effectiveness of the present methods including the application of the new method to a coupled system of time fractional partial differential equations having different fractional indices in different irregularly shaped regions of the solution domain

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

diffusive dispersive travelling waves and kinetic relations v singular diffusion and nonlinear dispersion we consider scalar hyperbolic conservation laws with non convex flux and vanishing nonlinear and possibly singular diffusion and dispersion terms the diffusion has the form r u u x x and we cover for instance the singular diffusion u x p u x x where p 0 is arbitrary we investigate the existence uniqueness and various properties of classical and non classical travelling waves and of the kinetic function the latter serves to characterize non classical shock waves via an additional algebraic constrain called a kinetic relation we discover that p is a somewhat unexpected critical value for p we obtain properties that are qualitatively similar to those we established earlier for regular and linear diffusion however for p the behaviour of the kinetic function is very different as for instance non classical shocks can have arbitrary small strength the behaviour of the kinetic function near the origin is carefully investigated and depends on whether p p or p in particular in the special case of the cubic flux function and for the regularization u x p u x x with p 0 or 1 the kinetic function can be computed explicitly when p the kinetic function is simply a linear function of its argument