a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

coupling an eulerian fluid calculation to a lagrangian solid calculation with the ghost fluid method we propose a numerical method for modeling multimaterial flows where the domain is decomposed into separate eulerian and lagrangian subdomains that is the equations are written in eulerian form in one subdomain and in lagrangian form in the other subdomain this is of interest for example when considering the effect of underwater explosions on the hull of a ship or the impact of a low speed projectile on a soft explosive target on the one hand high speed fluid flows are traditionally modeled by applying shock capturing schemes to the compressible euler equations to avoid problems associated with tangling of a lagrangian mesh on the other hand solid dynamics calculations are traditionally carried out using lagrangian numerical methods to avoid problems associated with numerical smearing in eulerian calculations we use the ghost fluid method to create accurate discretizations across the eulerian lagrangian interface the numerical method is presented in both one and two spatial dimensions three dimensional extensions to the interface coupling method are straightforward

a robust and efficient polyhedron subdivision and intersection algorithm for three dimensional mmale remapping the multi material arbitrary lagrangian eulerian mmale method is an effective way to simulate the multi material flow with severe surface deformation comparing with the traditional arbitrary lagrangian eulerian ale method the mmale method allows for multiple materials in a single cell which overcomes the difficulties in grid refinement process in recent decades many researches have been conducted for the lagrangian rezoning and surface reconstruction phases but less attention has been paid to the multi material remapping phase especially for the three dimensional problems due to two complex geometric problems the polyhedron subdivision and the polyhedron intersection in this paper we propose a clipping and projecting algorithm for polyhedron intersection whose basic idea comes from the commonly used method by grandy 1999 29 and jia et al 2013 34 our new algorithm solves the geometric problem by an incremental modification of the topology based on segmentplane intersections a comparison with jia et al 2013 34 shows our new method improves the efficiency by 55 to 65 when calculating polyhedron intersections moreover the instability caused by the geometric degeneracy can be thoroughly avoided because the geometry integrity is preserved in the new algorithm we also focus on the polyhedron subdivision process and describe an algorithm which could automatically and precisely tackle the various situations including convex non convex and multiple subdivisions numerical studies indicate that by using our polyhedron subdivision and intersection algorithm the volume conversation of the remapping phase can be exactly preserved in the mmale simulation

the mimetic finite difference method for the 3d magnetostatic field problems on polyhedral meshes we extend the mimetic finite difference mfd method to the numerical treatment of magnetostatic fields problems in mixed div curl form for the divergence free magnetic vector potential to accomplish this task we introduce three sets of degrees of freedom that are attached to the vertices the edges and the faces of the mesh and two discrete operators mimicking the curl and the gradient operator of the differential setting then we present the construction of two suitable quadrature rules for the numerical discretization of the domain integrals of the div curl variational formulation of the magnetostatic equations this construction is based on an algebraic consistency condition that generalizes the usual construction of the inner products of the mfd method we also discuss the linear algebraic form of the resulting mfd scheme its practical implementation and discuss existence and uniqueness of the numerical solution by generalizing the concept of logically rectangular or cubic meshes by hyman and shashkov to the case of unstructured polyhedral meshes the accuracy of the method is illustrated by solving numerically a set of academic problems and a realistic engineering problem

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

a free lagrange augmented godunov method for the simulation of elastic plastic solids a lagrangian finite volume godunov scheme is extended to simulate two dimensional solids in planar geometry the scheme employs an elastic perfectly plastic material model implemented using the method of radial return and either the stiffened gas or osborne equation of state to describe the material the problem of mesh entanglement common to conventional two dimensional lagrangian schemes is avoided by utilising the free lagrange method the lagrangian formulation enables features convecting at the local velocity such as material interfaces to be resolved with minimal numerical dissipation the governing equations are split into separate subproblems and solved sequentially in time using a time operator split procedure local riemann problems are solved using a two shock approximate riemann solver and piecewise linear data reconstruction is employed using a muscl based approach to improve spatial accuracy to illustrate the effectiveness of the technique numerical simulations are presented and compared with results from commercial fixed connectivity lagrangian and smooth particle hydrodynamics solvers autodyn 2d the simulations comprise the low velocity impact of an aluminium projectile on a semi infinite target the collapse of a thick walled beryllium cylinder and the high velocity impact of cylindrical aluminium and steel projectiles on a thin aluminium target the analytical solution for the collapse of a thick walled cylinder is also presented for comparison

using a curvilinear grid to construct symmetry preserving discretizations for lagrangian gas dynamics the goal of this paper is to construct discretizations for the equations of lagrangian gas dynamics that preserve plane cylindrical and spherical symmetry in the solution of the original differential equations the new method uses a curvilinear grid that is reconstructed from a given logically rectangular distribution of nodes the sides of the cells of the reconstructed grid can be segments of straight lines or arcs of local circles our procedure is exact for straight lines and circles that is it reproduces rectangular and polar grids exactly we use the method of support operators to construct a conservative finite difference method that we demonstrate will preserve spatial symmetries for certain choices of the initial grid we also introduce a curvilinear version of artificial edge viscosity that also preserves symmetry we present numerical examples to demonstrate our theoretical considerations and the robustness of the new method

a nonconforming virtual element method for the stokes problem on general meshes abstract in this paper a nonconforming virtual element method for the stokes problem is proposed and investigated this method gives a unified scheme for two and three dimensions simultaneously in addition it provides an attractive computational feature treatment of general elements including non convex and degenerate elements moreover by choosing a proper low order velocity and pressure pair we prove the discrete inf sup condition for this method to obtain its solvability and stability furthermore we show optimal energy norm error estimates for velocity and l 2 norm error estimates for both velocity and pressure finally a series of numerical experiments are performed to validate that this method has good stability and accuracy for the stokes problem

on preservation of symmetry in r z staggered lagrangian schemes in the focus of this work are symmetry preservation conservation of energy and volume and other important properties of staggered lagrangian hydrodynamic schemes in cylindrical r z geometry it is well known that on quadrilateral cells in r z preservation of spherical symmetry perfect satisfaction of the geometrical conservation law gcl and total energy conservation are incompatible even on conforming grids this paper suggests a novel staggered grid approach that preserves symmetry conserves total energy by construction and tries to do its best by diminishing the gcl error to the order of entropy error in particular the forces from an existing volume consistent scheme are corrected so that spherical symmetry is preserved the incorporation of subcell pressure mechanism to reduce spurious grid deformations is described and the relation of the new scheme to popular area weighted and control volume approaches studied

a framework for developing a mimetic tensor artificial viscosity for lagrangian hydrocodes on arbitrary polygonal meshes we construct a new mimetic tensor artificial viscosity on general polygonal meshes the tensor artificial viscosity is based on discretization of coordinate invariant operators divergence of a tensor and gradient of a vector the focus of this paper is on the non symmetric form div email protected u of the tensor artificial viscosity the discretizations of this operator is derived for the case of a full tensor coefficient m however in the numerical experiments we only use scalar m we prove that the new tensor viscosity preserves spatial symmetry on special meshes we demonstrate performance of the new viscosity for the noh implosion sedov explosion and saltzman piston problems on a set of various polygonal meshes in both cartesian and axisymmetric coordinate systems

variational methods in relativistic quantum mechanics this review is devoted to the study of stationary solutions of lin ear and nonlinear equations from relativistic quantum mechanics involving the dirac operator the solutions are found as critical points of an energy func tional contrary to the laplacian appearing in the equations of nonrelativistic quantum mechanics the dirac operator has a negative continuous spectrum which is not bounded from below this has two main consequences first the energy functional is strongly indefinite second the euler lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap between the negative and positive continuous spectra moreover since we work in the space domain r 3 the palais smale condition is not satisfied for these reasons the problems discussed in this review pose a challenge in the calculus of variations the existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems in the first part we consider the fixed eigenvalue problem for models of a free self interacting relativistic particle they allow to describe the localized state of a spin 1 2 particle a fermion which propagates without changing its shape this includes the soler models and the maxwell dirac or klein gordon dirac equations the second part is devoted to the presentation of min max principles al lowing to characterize and compute the eigenvalues of linear dirac operators with an external potential in the gap of their essential spectrum many con sequences of these min max characterizations are presented among them a new kind of hardy like inequalities and a stable algorithm to compute the eigenvalues in the third part we look for normalized solutions of nonlinear eigenvalue problems the eigenvalues are lagrange multipliers lying in a spectral gap we review the results that have been obtained on the dirac fock model which is a nonlinear theory describing the behavior of n interacting electrons in an external electrostatic field in particular we focus on the problematic definition of the ground state and its nonrelativistic limit in the last part we present a more involved relativistic model from quan tum electrodynamics in which the behavior of the vacuum is taken into ac count it being coupled to the real particles the main interesting feature of this model is that the energy functional is now bounded from below providing us with a good definition of a ground state

the construction of compatible hydrodynamics algorithms utilizing conservation of total energy the principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought however in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost or obeyed only to an approximation that often becomes worse with time this is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system although a finite truncation error is inherent to all discretization methods it is possible to satisfy certain global properties such as conservation of mass momentum and total energy to numerical roundoff error the purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties in particular it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in lagrangian form and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system for staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy this allows for the specification of forces that can be of an arbitrary complexity such as those derived from an artificial viscosity or subzonal pressures these forces originate only in discrete form nonetheless the change in internal energy caused by them is still completely determined the procedure given here is compared to the method of support operators to which it is closely related difficulties with conservation of momentum volume and entropy are also discussed the proper treatment of boundary conditions and differencing with respect to time are detailed

the internal consistency stability and accuracy of the discrete compatible formulation of lagrangian hydrodynamics this work explores the somewhat subtle meaning and consequences of the salient properties of the discrete compatible formulation of lagrangian hydrodynamics in particular since this formulation preserves total energy to roundoff error the amount of error in the conservation of total energy cannot be used to gauge the internal consistency of calculations as is often done with the older forms of this algorithm however the compatible formulation utilizes two definitions of zone volume the first is the usual definition whereby the volume of a zone is defined as some prescribed function of the coordinates of the points that define it the second is given as the integration in time of the continuity equation for zone volume as expressed in lagrangian form it is the use of this latter volume in the specific internal energy equation that enables total energy to be exactly conserved these two volume definitions are generally not precisely equal it is the analysis of this difference that forms the first part of this study it is shown that this difference in zone volumes can be used to construct a practical internal consistency measure that not only takes the place of the lack of total energy conservation of the older forms of lagrangian hydrodynamics but is more general in that it can be defined on a single zone basis it can also be used to ascertain the underlying spatial and temporal order of accuracy of any given set of calculations the difference in these two definitions of zone volume may be interpreted as a type of entropy error however this entropy error is found to be significant only when a given calculation becomes numerically unstable otherwise it remains at or far beneath truncation error levels in fact it can be utilized to provide an upper bound on the size of the spatial truncation error for a stable computation it is also shown how this volume difference can be used as an indicator of numerical difficulties since exact local conservation of total energy does not guarantee numerical stability or the quality of any numerical calculation the discrete compatible formulation of lagrangian hydrodynamics utilizes a two level predictor corrector type of time integration scheme a stability analysis both analytical and numerical is given this analysis reveals a novel stability diagram that has not been heretofore published and gives definitive information as to how the stabilizing corrector step should be centered in time

a maximum principle preserving finite element method for scalar conservation equations this paper introduces a first order viscosity method for the explicit approximation of scalar conservation equations with lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension provided the lumped mass matrix is positive definite the method is shown to satisfy the local maximum principle under a usual cfl condition the method is independent of the cell type for instance the mesh can be a combination of tetrahedra hexahedra and prisms in three space dimensions

short note volume consistency in a staggered grid lagrangian hydrodynamics scheme staggered grid lagrangian schemes for compressible hydrodynamics involve a choice of how internal energy is advanced in time the options depend on two ways of defining cell volumes an indirect one that guarantees total energy conservation and a direct one that computes the volume from its definition as a function of the cell vertices it is shown that the motion of the vertices can be defined so that the two volume definitions are identical a so modified total energy conserving staggered scheme is applied to the coggeshall adiabatic compression problem and now also entropy is basically exactly conserved for each lagrangian cell and there is increased accuracy for internal energy the overall improvement as the grid is refined is less than what might be expected

staggered grids for three dimensional convection of a multicomponent fluid in a porous medium convection in a porous medium may produce strong nonuniqueness of patterns we study this phenomena for the case of a multicomponent fluid and develop a mimetic finite difference scheme for the three dimensional problem discretization of the darcy equations in the primitive variables is based on staggered grids with five types of nodes and on a special approximation of nonlinear terms this scheme is applied to the computer study of flows in a porous parallelepiped filled by a two component fluid and with two adiabatic lateral planes we found that the continuous family of steady stable states exists in the case of a rather thin enclosure when the depth is increased only isolated convective regimes may be stable we demonstrate that the non mimetic approximation of nonlinear terms leads to the destruction of the continuous family of steady states

the mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes we study the mimetic finite difference discretization of diffusion type problems on unstructured polyhedral meshes we demonstrate high accuracy of the approximate solutions for general diffusion tensors the second order convergence rate for the scalar unknown and the first order convergence rate for the vector unknown on smooth or slightly distorted meshes on non matching meshes and even on meshes with irregular shaped polyhedra with flat faces we show that in general the meshes with non flat faces require more than one flux unknown per mesh face to get optimal convergence rates

entropy viscosity method for the single material euler equations in lagrangian frame a new finite element method for solving the euler equations in lagrangian coordinates is proposed the method is stabilized by adding artificial diffusion terms compatible with positivity of mass and internal energy a minimum principle on the specific entropy and all generalized entropy inequalities two options of first order artificial diffusion are considered one is in the spirit of the eulerian based method guermond et al 2011 23 22 zingan et al 2013 and the other is similar to existing viscosity stabilizations in lagrangian frame e g dobrev et al 2012 the method is verified to be high order for smooth solutions even with active viscosity terms this is achieved by using high order finite element spaces and an entropy based viscosity stabilization that degenerates the first order viscous terms this stabilization automatically distinguishes smooth and singular regions the formal accuracy and convergence properties of the proposed methods are tested on a series of benchmark problems this is the first result extending the entropy viscosity methodology to the lagrangian hydrodynamics

spectral methods for the euler equations i fourier methods and shock capturing spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals current spectral collociation methods are put into historical context the basic concepts of fourier spectral collocation methods are provided filtering strategies for shock capturing approaches are also presented fourier shock capturing techniques are evaluated using a one dimensional periodic astrophysical nozzle problem

energy preservation and entropy in lagrangian space and time staggered hydrodynamic schemes usual space and time staggered sts leap frog lagrangian hydrodynamic schemes such as von neumann richtmyer s 1950 wilkins 1964 and their variants are widely used for their simplicity and robustness despite their known lack of exact energy conservation since the seminal work of trulio and trigger 1950 and despite the later corrections of burton 1991 it is generally accepted that these schemes cannot be modified to exactly conserve energy while retaining all of the following properties sts stencil with velocities half time centered with respect to positions explicit second order algorithm locally implicit for internal energy and definite positive kinetic energy it is shown here that it is actually possible to modify the usual sts hydrodynamic schemes in order to be exactly energy preserving regardless of the evenness of their time centering assumptions and retaining their simple algorithmic structure burton s conservative scheme 1991 is found as a special case of time centering which cancels the term here designated as incompatible displacements residue in contrast von neumann richtmyer s original centering can be preserved provided this residue is properly corrected these two schemes are the only special cases able to capture isentropic flow with a third order entropy error instead of second order in general the momentum equation is presently obtained by application of a variational principle to an action integral discretized in both space and time the internal energy equation follows from the discrete conservation of total energy entropy production by artificial dissipation is obtained to second order by a prediction correction step on the momentum equation the overall structure of the equations explicit for momentum locally implicit for internal energy remains identical to that of usual sts leap frog schemes though complementary terms are required to correct the effects of time step changes and artificial viscosity updates in deriving these schemes an apparently novel approach of flux in time was introduced to correct numerical residues and ensure energy conservation this method can be applied to essentially any numerical scheme whenever required or desired provided space and time numerical consistency is preserved numerical test cases are presented confirming the conservative character of the new csts schemes down to computer round off errors and showing various improvements compared to the standard von neumann richtmyer and wilkins sts schemes mostly on shock levels shock velocities singularity induced distortions and cfl stability limits

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

compatible energy conserving bounds preserving remap of hydrodynamic fields for an extended ale scheme abstract from the very origins of numerical hydrodynamics in the lagrangian work of von neumann and richtmyer 83 the issue of total energy conservation as well as entropy production has been problematic because of well known problems with mesh deformation lagrangian schemes have evolved into arbitrary lagrangian eulerian ale methods 39 that combine the best properties of lagrangian and eulerian methods energy issues have persisted for this class of methods we believe that fundamental issues of energy conservation and entropy production in ale require further examination the context of the paper is an ale scheme that is extended in the sense that it permits cyclic or periodic remap of data between grids of the same or differing connectivity the principal design goals for a remap method then consist of total energy conservation bounded internal energy and compatibility of kinetic energy and momentum we also have secondary objectives of limiting velocity and stress in a non directional manner keeping primitive variables monotone and providing a higher than second order reconstruction of remapped variables in particular the new contributions fall into three categories associated with energy conservation and entropy production reconstruction and bounds preservation of scalar and tensor fields and conservative remap of nonlinear fields the paper presents a derivation of the methods details of implementation and numerical results for a number of test problems the methods requires volume integration of polynomial functions in polytopal cells with planar facets and the requisite expressions are derived for arbitrary order

elimination of artificial grid distortion and hourglass type motions by means of lagrangian subzonal masses and pressures the bane of lagrangian hydrodynamics calculations is the premature breakdown of grid topology that results in severe degradation of accuracy and run termination often long before the assumption of a lagrangian zonal mass has ceased to be valid at short spatial grid scales this is usually referred to by the terms hourglass mode or keystone motion associated in particular with underconstrained grids such as quadrilaterals and hexahedrons in two and three dimensions respectively at longer spatial lengths relative to the grid spacing there is what is referred to ubiquitously as spurious vorticity or the long thin zone problem in both cases the result is anomalous grid distortion and tangling that has nothing to do with the actual solution as would be the case for turbulent flow in this work we show how such motions can be eliminated by the proper use of subzonal lagrangian masses and associated densities and pressures these subzonal pressures give rise to forces that resist these spurious motions the pressure is no longer a constant in a zone it now accurately reflects the density gradients that can occur within a zone due to its differential distortion subzonal lagrangian masses can be choosen in more than one manner to obtain subzonal density and pressure variation however these masses arise in a natural way from the intersection of the lagrangian contours through which no mass flows that are associated with both the lagrangian zonal and nodal masses in a staggered spatial grid hydrodynamics formulation this is an extension of the usual lagrangian assumption that is often applied to only the zonal mass we show that with a proper discretization of the subzonal forces resulting from subzonal pressures hourglass motion and spurious vorticity can be eliminated for a very large range of problems

isogeometric analysis of lagrangian hydrodynamics axisymmetric formulation in the rz cylindrical coordinates a recent isogeometric analysis iga formulation of lagrangian shock hydrodynamics 4 is extended to the 3d axisymmetric case the euler equations of compressible hydrodynamics are formulated using the rz cylindrical coordinates and are discretized in the weak form using nurbs based iga artificial shock viscosity and internal energy projection are added to stabilize the formulation the resulting discretization exhibits good accuracy and robustness properties it also gives exact symmetry preservation on the appropriately constructed meshes several benchmark examples are computed to examine the performance of the proposed formulation

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

a second order maximum principle preserving lagrange finite element technique for nonlinear scalar conservation equations this paper proposes an explicit at least second order maximum principle sat isfying lagrange finite element method for solving nonlinear scalar conservation equations the technique is based on a new viscous bilinear form introduced in guermond and nazarov com put methods appl mech engrg 272 2014 pp 198 213 a high order entropy viscosity method and the boris book zalesak flux correction technique the algorithm works for arbitrary meshes in any space dimension and for all lipschitz fluxes the formal second order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems

a high order cell centered lagrangian scheme for compressible fluid flows in two dimensional cylindrical geometry the goal of this paper is to present high order cell centered schemes for solving the equations of lagrangian gas dynamics written in cylindrical geometry a node based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume it allows to derive finite volume numerical schemes that are compatible with the geometric conservation law gcl two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator the first one corresponds to the control volume scheme while the second one corresponds to the so called area weighted scheme both formulations share the same discretization for the total energy equation in both schemes fluxes are computed using the same nodal solver which can be viewed as a two dimensional extension of an approximate riemann solver the control volume scheme is conservative for momentum total energy and satisfies a local entropy inequality in its first order semi discrete form however it does not preserve spherical symmetry on the other hand the area weighted scheme is conservative for total energy and preserves spherical symmetry for one dimensional spherical flow on equi angular polar grid the two dimensional high order extensions of these two schemes are constructed employing the generalized riemann problem grp in the acoustic approximation many numerical tests are presented in order to assess these new schemes the results obtained for various representative configurations of one and two dimensional compressible fluid flows show the robustness and the accuracy of our new schemes

high order mimetic finite difference method for diffusion problems on polygonal meshes the mimetic finite difference mfd methods mimic important properties of physical and mathematical models as a result conservation laws solution symmetries and the fundamental identities of the vector and tensor calculus are held for discrete models the mfd methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non convex and degenerate elements the existing mfd methods for solving diffusion type problems are second order accurate for the conservative variable temperature pressure energy etc and only first order accurate for its flux we developed new high order mfd methods which are second order accurate for both scalar and vector variables the second order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes

a subcell remapping method on staggered polygonal grids for arbitrary lagrangian eulerian methods we describe a new remapping algorithm for use in arbitrary lagrangian eulerian ale simulations the new features of this remapper are designed to complement a staggered mesh lagrangian phase in which the cells may be general polygons in two dimensions and which uses subcell discretizations to control unphysical mesh distortion and hourglassing our new remapping algorithm consists of three stages a gathering stage in which we interpolate momentum internal energy and kinetic energy to the subcells in a conservative way a subcell remapping stage in which we conservatively remap mass momentum internal and kinetic energy from the subcells of the lagrangian mesh to the subcells of the new rezoned mesh a scattering stage in which we conservatively recover the primary variables subcell density nodal velocity and cell centered specific internal energy on the new rezoned mesh we prove that our new remapping algorithm is conservative reversible and satisfies the debar consistency condition we also demonstrate computationally that our new remapping method is robust and accurate for a series of test problems in one and two dimensions

algebraic limitations on two dimensional hydrodynamics simulations algebraic limitations imposed by the use of connected straightline segments to define meshes for hydrodynamics simulations in two dimensional cylindrical geometries are shown it is shown that in the simplest smooth isentropic flow of the spherical expansion of a gas with point symmetry commonly and currently used finite difference finite volume or finite element staggered grid hydrodynamics equations cannot simultaneously preserve energy entropy and sphericity on an equal angler mesh it is further shown why finite difference codes tend to preserve sphericity and entropy while finite element codes tend to preserve sphericity and energy exact difference representations of interface cell face pressures and work terms and of the elements of the strain rate tensor in a cell are shown

a finite element framework for some mimetic finite difference discretizations in this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified nedelec raviart thomas finite element methods for model problems in h curl and h div this provides a simple and transparent way to analyze such mimetic finite difference discretizations using the well known results from finite element theory the finite element framework that we develop is also crucial for the design of efficient multigrid methods for mimetic finite difference discretizations since it allows us to use canonical inter grid transfer operators arising from the finite element framework we provide special local fourier analysis and numerical results to demonstrate the efficiency of such multigrid methods

the force work differencing of exceptional points in the discrete compatible formulation of lagrangian hydrodynamics this study presents the force and mass discretization of exceptional points in the compatible formulation of lagrangian hydrodynamics it concludes a series of papers that develop various aspects of the theoretical exposition and the operational implementation of this numerical algorithm exceptional points are grid points at the termination of lines internal to the computational domain and where boundary conditions are therefore not applied these points occur naturally in most applications in order to ameliorate spatial grid anisotropy and the consequent timestep reduction that will otherwise arise for grids with highly tapered regions or a center of convergence they have their velocity enslaved to that of neighboring points in order to prevent large excursions of the numerical solution about them how this problem is treated is given herein for the aforementioned numerical algorithm such that its salient conservation properties are retained in doing so the subtle aspects of this algorithm that are due to the interleaving of spatial contours that occur with the use of a spatially staggered grid mesh are illuminated these contours are utilized to define both forces and the work done by them and are the central construct of this type of finite volume differencing additionally difficulties that occur due to uncertainties in the specification of the artificial viscosity are explored and point to the need for further research in this area

local flux mimetic finite difference methods we develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes to approximate the velocity vector variable the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n gonal mesh face in three dimensions to approximate the pressure scalar variable the method uses one degree of freedom per element a specially chosen quadrature rule for the l 2 product of vector functions allows for a local flux elimination and reduction of the method to a cell centered finite difference scheme for the pressure unknowns under certain assumptions first order convergence is proved for both variables and second order convergence is proved for the pressure the assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method for general polyhedral meshes non symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions numerical results confirm the theory

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

a mimetic discretization of the reissner mindlin plate bending problem we present a mimetic approximation of the reissner mindlin plate bending problem which uses deflections and rotations as discrete variables the method applies to very general polygonal meshes even with non matching or non convex elements we prove linear convergence for the method uniformly in the plate thickness

a compatible energy and symmetry preserving lagrangian hydrodynamics algorithm in three dimensional cartesian geometry this work presents a numerical algorithm for the solution of fluid dynamics problems with moderate to high speed flow in three dimensions cartesian geometry is chosen owing to the fact that in this coordinate system no curvature terms are present that break the conservation law structure of the fluid equations written in lagrangian form these equations are discretized utilizing compatible control volume differencing with a staggered grid placement of the spatial variables the concept of compatibility means that the forces used in the momentum equation to advance velocity are also incorporated into the internal energy equation so that these equations together define the total energy as a quantity that is exactly conserved in time in discrete form multiple pressures are utilized in each zone they produce forces that resist spurious vorticity generation this difficulty can severely limit the utility of the lagrangian formulation in two dimensions and make this representation otherwise virtually useless in three dimensions an edge centered artificial viscosity whose magnitude is regulated by local velocity gradients is used to capture shocks the particular difficulty of exactly preserving one dimensional spherical symmetry in three dimensional geometry is solved this problem has both practical and pedagogical significance the algorithm is suitable for both structured and unstructured grids limitations that symmetry preservation imposes on the latter type of grids are delineated

stabilized shock hydrodynamics i a lagrangian method abstract a new supg stabilized formulation for lagrangian hydrodynamics of materials satisfying mie gruneisen equation of state is proposed it allows the use of simplex type triangular tetrahedral meshes as well as the more commonly used brick type quadrilateral hexahedral meshes the proposed method yields a globally conservative formulation in which equal order interpolation p1 or q1 isoparametric finite elements is applied to velocities displacements and pressure as a direct consequence and in contrast to traditional cell centered multidimensional hydrocode implementations the proposed formulation allows a natural representation of the pressure gradient on element interiors the supg stabilization involves additional design requirements specific to the lagrangian formulation a discontinuity capturing operator in the form of a noh type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions a set of challenging shock hydrodynamics benchmark tests for the euler equations of gas dynamics in one and two space dimensions is presented in the two dimensional case computations performed on quadrilateral and triangular grids are analyzed and compared these results indicate that the new formulation is a promising technology for hydrocode applications

short note a compact artificial viscosity equivalent to a tensor viscosity we present a compact artificial viscosity for staggered grid lagrangian hydrodynamics on polygonal cells in two cartesian dimensions and using a decomposition into triangles we show that this viscosity is equivalent to a tensor viscosity of campbell and shashkov for quadrilaterals

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 pseudospectral scheme for the numerical calculation of shocks abstract a pseudospectral numerical scheme is developed in order to calculate the propagation of a shock wave we use a two step time differencing method and a chebyshev transform method to compute the space derivative such a scheme has been used to reduce the oscillations due to gibb s phenomenon this numerical method is applied to the solution of the burgers equation without viscosity term the accuracy of the numerical solutions is compared to the one given by two finite difference methods

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

resolvent estimates and bounds on eigenvalues for dirac operators on the half line estimates for the eigenvalues of non self adjoint 1d dirac operators considered on the half line are obtained in terms of the l p norms of the potentials the proofs are based on the resolvent est

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

mimetic finite difference operators for second order tensors on unstructured grids abstract we use the support operators method to derive discrete approximations for the gradient of a vector and divergence of a tensor on unstructured grids in two dimensions these discrete operators satisfy discrete analogs of the integral identities of the differential operators on unstructured grids where vector functions are defined at the grid points and tensor functions are defined as tangential projections to the zone edges or as normal projections to the median mesh we evaluate the accuracy of the discrete operators by determining the order of convergence of the truncation error on structured and unstructured grids and show that the truncation error of the method is between first and second order depending on the smoothness of the grid in a test problem on a highly nonuniform grid we confirm that the convergence rate is between first and second order

a mortar mimetic finite difference method on non matching grids we consider mimetic finite difference approximations to second order elliptic problems on non matching multiblock grids mortar finite elements are employed on the non matching interfaces to impose weak flux continuity optimal convergence and in certain cases superconvergence is established for both the scalar variable and its flux the theory is confirmed by computational results

two new three dimensional contact algorithms for staggered lagrangian hydrodynamics abstract this paper presents two new three dimensional contact algorithms for staggered lagrangian hydrodynamics named discrete accurate matching method and discrete lagrangian multiplier method these new contact algorithms utilize the unified contact algorithm for contact searching contact segments from all potential contact surfaces are treated as a single set the basic idea of these new methods is to partition all contact segments into triangular facets firstly then for each pair of two triangular facets which correspond to a hitting node and a target node respectively these two triangular facets are projected to one plane and their intersection area is calculated the nodal masses and nodal forces of the hitting node and the target node are distributed to the intersection portions of these two triangular facets respectively according to the proportion of the intersection area to corresponding nodal area in the discrete accurate matching method the masses and forces of the intersection portions of these two triangular facets are added to corresponding contact nodes of each other and then the accelerations and velocities of the contact nodes are updated while in the discrete lagrangian multiplier method the intersection portions of these two triangular facets are considered as a 1d contact pair in the normal direction for which the contact force is explicitly calculated by the lagrangian multiplier method for a 1d contact pair of one hitting point and one target point then the contact force being added to corresponding contact nodes these new contact algorithms are assessed through several numerical tests performed on three dimensional structured and unstructured meshes the results of these tests show the accuracy and robustness of these new methods

one dimensional lagrangian implicit hydrodynamic algorithm for inertial confinement fusion applications a new one dimensional hydrodynamic algorithm specifically developed for inertial confinement fusion icf applications is presented the scheme uses a fully conservative lagrangian formulation in planar cylindrical and spherically symmetric geometries and supports arbitrary equations of state with separate ion and electron components fluid equations are discretized on a staggered grid and stabilized by means of an artificial viscosity formulation the space discretized equations are advanced in time using an implicit algorithm the method includes several numerical parameters that can be adjusted locally in regions with low courant friedrichs lewy cfl number where stability is not an issue they can be adjusted to optimize the accuracy in typical problems the truncation error can be reduced by a factor between 2 to 10 in comparison with conventional explicit algorithms on the other hand in regions with high cfl numbers the parameters can be set to guarantee unconditional stability the method can be integrated into complex icf codes this is demonstrated through several examples covering a wide range of situations from thermonuclear ignition physics where alpha particles are managed as an additional species to low intensity laser matter interaction where liquid vapor phase transitions occur

a new set of basis functions for the discrete geometric approach by exploiting the geometric structure behind maxwell s equations the so called discrete geometric approach allows to translate the physical laws of electromagnetism into discrete relations involving circulations and fluxes associated with the geometric elements of a pair of interlocked grids the primal grid and the dual grid to form a finite dimensional system of equations discrete counterparts of the constitutive relations must be introduced in addition they are referred to as constitutive matrices which must comply with precise properties symmetry positive definiteness consistency in order to guarantee the stability and consistency of the overall finite dimensional system of equations the aim of this work is to introduce a general and efficient set of vector functions associated with the edges and faces of a polyhedral primal grids or of a dual grid obtained from the barycentric subdivision of the boundary of the primal grid these vector functions comply with precise specifications which allow to construct stable and consistent discrete constitutive equations for the discrete geometric approach in the framework of an energetic method

some cell centered lagrangian laxwendroff hll hybrid schemes lagrangian hydrodynamics is treated by the laxwendroff method with the dissipative fluxes in the hll form including both artificial viscosity and artificial energy flux the symmetry of results for the noh problem on a cartesian mesh is very good on material interfaces we employ the eucclhyd nodal solver at the interface nodes with 1d acoustic approximate riemann solver on the interface edges the proposed method works reasonably well on several hydrodynamic tests including noh sedov saltzman sod woodwardcollela and triple point

a cell centered lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two dimensional cylindrical geometry we develop a new cell centered control volume lagrangian scheme for solving euler equations of compressible gas dynamics in cylindrical coordinates the scheme is designed to be able to preserve one dimensional spherical symmetry in a two dimensional cylindrical geometry when computed on an equal angle zoned initial grid unlike many previous area weighted schemes that possess the spherical symmetry property our scheme is discretized on the true volume and it can preserve the conservation property for all the conserved variables including density momentum and total energy several two dimensional numerical examples in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry accuracy and non oscillatory properties

a high order mimetic method on unstructured polyhedral meshes for the diffusion equation abstract we present a new family of mimetic finite difference schemes for solving elliptic partial differential equations in the primal form on unstructured polyhedral meshes these mimetic discretizations are built to satisfy local consistency and stability conditions the consistency condition is an exactness property i e the mimetic schemes are exact when the solution is a polynomial of an assigned degree the stability condition ensures the well posedness of the method the degrees of freedom are the solution moments on mesh faces and inside mesh cells higher order schemes are built using higher order moments the developed schemes are verified numerically on diffusion problems with constant and spatially variable possibly discontinuous tensorial coefficients

vorticity errors in multidimensional lagrangian codes abstract we investigate the apparent paradox as exemplified by the well known saltzman test problem of multidimensional lagrangian codes experiencing mesh tangling when computing one dimensional irrotational flows we demonstrate that the cause is the generation of spurious vorticity or vorticity error by a nonuniform mesh based on this we investigate two methods of constructing improved lagrangian vertex velocities by removing or filtering out this spurious vorticity rather than by the more common practice of introducing artificial viscosity the first method reconstructs the velocity from the known flow divergence and from the true vorticity computed by means of a transport equation the second method which is much simpler and more efficient subtracts a divergence free correction from the velocity such that the resulting velocity possesses the correct vorticity we then successfully apply this method to solve a two dimensional shock refraction problem a problem which exhibits nonzero intrinsic vorticity

an a posteriori error estimator for the mimetic finite difference approximation of elliptic problems we present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form we show that this estimator is reliable and efficient with respect to an energy type error comprising both flux and pressure its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes different permeability tensors and different types of computational meshes copyright 2008 john wiley sons ltd

analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems the maximum principle is one of the most important properties of solutions of partial differential equations its numerical analog the discrete maximum principle dmp is one of the most difficult properties to achieve in numerical methods especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic violation of the dmp may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one in this work we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference mfd method on two and three dimensional meshes these conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element efficient solutions are devised for meshes consisting of simplexes parallelograms and parallelepipeds and orthogonal locally refined elements as those used in the amr methodology on simplicial meshes it turns out that the mfd method coincides with the mixed hybrid finite element methods based on the low order raviart thomas vector space thus in this case we recover the well established conventional angle conditions of such approximations instead in the other cases a suitable design of the mfd method allows us to formulate a monotone discretization for which the existence of a dmp can be theoretically proved moreover on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the multi point flux approximation mpfa methods numerical experiments confirm the effectiveness of the considered monotonicity conditions

revisiting wall heating the results of computing a shock reflection are found to strongly depend on the chosen coordinate system lagrangian or eulerian this is seen by comparing the amount of wall heating associated with the solutions to noh s shock reflection problem computed in each reference frame here an explanation is offered that places a greater burden on the role of phase error or effective wave speed in producing this effect the reason for the greater difficulties in the lagrangian frame are the non uniformity of the mass coordinates in a converging geometry this produces a non steady state discrete shock profile

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

a conservative nodal variational multiscale method for lagrangian shock hydrodynamics abstract a new method based on a continuous piece wise linear approximation of the equations for lagrangian shock hydrodynamics is presented numerical instabilities are controlled by a stabilizing operator derived using the paradigm of the variational multiscale analysis encouraging numerical comparisons with existing methods in the case of quadrilateral and hexahedral elements indicate that the proposed method is capable of preventing hourglass patterns in the solution while maintaining accuracy in regions of smooth flow the proposed approach satisfies galilean invariance properties and hinges upon the interpretation of the lagrangian shock hydrodynamics equations as a system of nonlinear wave equations a specific implementation in terms of a predictor multi corrector version of the mid point time integrator guarantees global conservation of mass momentum and total energy for each iterate stability and formal order of accuracy are investigated applying the von neumann analysis to the linearized shock hydrodynamics equations in one dimension this approach yields tight bounds for stable time step estimation formal second order accuracy of the method in time and space and valuable indications on the choice of the most appropriate values for the stabilization parameters present in the formulation

numerical experiments in boundary layer stability numerical solution of the three dimensional incompressible navier stokes equations is used to study the instability of a flat plate boundary layer in a manner analogous to the vibrating ribbon experiments flow field structures are observed which are very similar to those found in the vibrating ribbon experiment to which computational initial conditions have been matched stream wise periodicity is assumed in the simulation so that the evolution occurs in time but the events that constitute the instability are so similar to the spatially occurring ones of the laboratory that it seems clear the physical processes involved are the same a spectral and finite difference numerical algorithm is employed in the simulation

tracking discontinuities in hyperbolic conservation laws with spectral accuracy it is well known that the spectral solutions of conservation laws have the attractive distinguishing property of infinite order convergence also called spectral accuracy when they are smooth e g c canuto m y hussaini a quarteroni t a zang spectral methods for fluid dynamics springer verlag heidelberg 1988 j p boyd chebyshev and fourier spectral methods second ed dover new york 2001 c canuto m y hussaini a quarteroni t a zang spectral methods fundamentals in single domains springer verlag berlin heidelberg 2006 if a discontinuity or a shock is present in the solution this advantage is lost there have been attempts to recover exponential convergence in such cases with rather limited success the aim of this paper is to propose a discontinuous spectral element method coupled with a level set procedure which tracks discontinuities in the solution of nonlinear hyperbolic conservation laws with spectral convergence in space spectral convergence is demonstrated in the case of the inviscid burgers equation and the one dimensional euler equations

flux reconstruction and solution post processing in mimetic finite difference methods abstract we present a post processing technique for the mimetic finite difference solution of diffusion problems in mixed form our post processing method yields a piecewise linear approximation of the scalar variable that is second order accurate in the l2 norm on quite general polyhedral meshes including non convex and non matching elements the post processing is based on the reconstruction of vector fields projected onto the mimetic space of vector variables this technique is exact on constant vector fields and is shown to be independent of the mimetic scalar product choice if a local consistency condition is satisfied the post processing method is computationally inexpensive optimal performance is confirmed by numerical experiments

a multi scale residual based anti hourglass control for compatible staggered lagrangian hydrodynamics abstract hourglassing is a well known pathological numerical artifact affecting the robustness and accuracy of lagrangian methods there exist a large number of hourglass control suppression strategies in the community of the staggered compatible lagrangian methods the approach of sub zonal pressure forces is among the most widely used however this approach is known to add numerical strength to the solution which can cause potential problems in certain types of simulations for instance in simulations of various instabilities to avoid this complication we have adapted the multi scale residual based stabilization typically used in the finite element approach for staggered compatible framework in this paper we describe two discretizations of the new approach and demonstrate their properties and compare with the method of sub zonal pressure forces on selected numerical problems

numerical preservation of symmetry properties of continuum problems we investigate the problem of perfectly preserving a symmetry associated naturally with one coordinate system when calculated in a different coordinate system this allows a much wider range of problems that may be viewed as perturbations of the given symmetry to be investigated we study the problem of preserving cylindrical symmetry in two dimensional cartesian geometry and spherical symmetry in two dimensional cylindrical geometry we show that this can be achieved by a simple modification of the gradient operator used to compute the force in a staggered grid lagrangian hydrodynamics algorithm in the absence of the supposed symmetry we show that the new operator produces almost no change in the results because it is always close to the original gradient operator our technique thus results in a subtle manipulation of the spatial truncation error in favor of the assumed symmetry but only to the extent that it is naturally present in the physical situation this not only extends the range of previous algorithms and the use of new ones for these studies but for spherical or cylindrical calculations it reduces the sensitivity of the results to grid setup with equal angular zoning that has heretofore been necessary with these problems

spectral methods for the euler equations ii chebyshev methods and shock fitting on exprime chaque composante de la vitesse dans les equations d euler par une somme de polynomes de tchebycheff qui constitue une fonction d interpolation on utilise les expressions spectrales ainsi obtenues pour calculer plusieurs ecoulements gazeux bidimensionnels par la methode d ajustement d ondes de choc selon cette methode les conditions initiales correspondent a une onde de choc plane normale a un ecoulement supersonique et on determine la forme veritable de l onde a l aide des formules de rankine hugoniot qui expriment les conditions aux limites

group invariant solutions and optimal systems for multidimensional hydrodynamics the group properties of the three dimensional 3 d one temperature hydrodynamic equations including nonlinear conduction and a thermal source are presented a subgroup corresponding to axisymmetric geometry is chosen and the details of the construction of the one and two dimensional optimal systems are shown the two dimensional optimal system is used to generate 23 intrinsically different reductions of the 2 d partial differential equations to ordinary differential equations these ordinary differential equations can be solved to provide analytic solutions to the original partial differential equations two example analytic solutions are presented a 2 d axisymmetric flow with a p2 asymmetry and a 3 d spiraling flow

high order accurate finite difference schemes based on symmetry preservation abstract in this paper we present a mathematical approach that is based on modified equations and the method of equivariant moving frames for construction of high order accurate invariant finite difference schemes that preserve lie symmetry groups of underlying partial differential equations pdes in the proposed approach invariant or symmetry preserving numerical schemes with a desired or fixed order of accuracy are constructed from some non invariant base numerical schemes modified forms of pdes are used to improve the order of accuracy of existing schemes and these modified forms are obtained through addition of defect correction terms to the original forms of pdes these defect correction terms of modified pdes that are noted from truncation error analysis are either completely removed from schemes or their representation is significantly simplified by considering convenient moving frames this feature of the proposed method can especially be useful to avoid cumbersome numerical representations when high order schemes are developed from low order ones via the method of modified equations the proposed method is demonstrated via construction of invariant numerical schemes with fixed and higher order of accuracy for some common linear and nonlinear problems including the linear advection diffusion equation in 1d and 2d inviscid burgers equation and viscous burgers equation and the performance of these invariant numerical schemes is further evaluated our results indicate that such invariant numerical schemes obtained from existing base numerical schemes have the potential to significantly improve the quality of results not only in terms of desired higher order accuracy but also in the context of preservation of appropriate symmetry properties of underlying pdes

formulations of artificial viscosity for multi dimensional shock wave computations in this paper we present a new formulation of the artificial viscosity concept physical arguments for the origins of this term are given and a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated the first important property is that by definition a viscosity must always be dissipative transferring kinetic energy into internal energy and must never act as a false pressure the artificial viscous force should be galilean invariant and vary continuously as a function of the criterion used to determine compression and expansion and remain zero for the latter case these requirements significantly constrain the functional form that the artificial viscous force can have in addition an artificial viscosity should be able to distinguish between shock wave and adiabatic compression and not result in spurious entropy production when only the latter is present it must therefore turn off completely for self similiar motion where only a uniform stretching and or a rigid rotation occurs an additional important but more subtle condition where the artificial viscosity should produce no effect is along the direction tangential to a convergent shock front since the velocity is only discontinuous in the normal direction our principal result is the development of a new formulation of an edge centered artificial viscosity that is to be used in conjunction with a staggered spatial placement of variables that meets all of these standards and without the need for problem dependent numerical coefficients that have in the past made the artificial viscosity method appear somewhat arbitrary our formulation and numerical results are given with respect to two spatial dimensions but all of our arguments carry over directly to three dimensions a central feature of our development is the implementation of simple advection limiters in a straightforward manner in more than one dimension to turn off the artificial viscosity for the above mentioned conditions and to substantially reduce its effect when strong velocity gradients are absent

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity