In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamicall...In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamically compatiblefirst-order hyperbolic systems.By construction,the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm.A very peculiar feature of our approach is that entropy is discretized directly,while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization.The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs,including both,conservative and non-conservative products,as well as potentially stiff algebraic relaxation source terms,provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law,such as the conservation of total energy density.The proposed family offinite volume schemes is based on the seminal work of Abgrall[1],where for thefirst time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented.We apply our new approach to three particular thermodynamically compatible systems:the equations of ideal magnetohydrodynamics(MHD)with thermodynamically compatible generalized Lagrangian multiplier(GLM)divergence cleaning,the unifiedfirst-order hyperbolic model of continuum mechanics proposed by Godunov,Peshkov,and Romenski(GPR model)and thefirst-order hyperbolic model for turbulent shallow waterflows of Gavrilyuk et al.In addition to formal mathematical proofs of the properties of our newfinite volume schemes,we also present a large set of numerical results in order to show their potential,efficiency,and practical applicability.展开更多
ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lew...ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed CCFL=0.95for their experiments. Nonetheless, we noted that the experiments conducted in this study with CCFL=0.95produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.展开更多
In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory(WENO)schemes in two dimensional Cartesian meshes.We compare them in terms of accuracy,performance for smooth an...In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory(WENO)schemes in two dimensional Cartesian meshes.We compare them in terms of accuracy,performance for smooth and shocked solutions,and efficiency in CPU timing.For linear systems both schemes are high order accurate,however for nonlinear systems,analysis and numerical simulation results verify that one of them(Class A)is only second order accurate,while the other(Class B)is high order accurate.The WENO scheme in Class A is easier to implement and costs less than that in Class B.Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes,despite of the difference in their formal order of accuracy.The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.展开更多
We propose an a-posteriori error/smoothness indicator for standard semidiscrete finite volume schemes for systems of conservation laws,based on the numerical production of entropy.This idea extends previous work by th...We propose an a-posteriori error/smoothness indicator for standard semidiscrete finite volume schemes for systems of conservation laws,based on the numerical production of entropy.This idea extends previous work by the first author limited to central finite volume schemes on staggered grids.We prove that the indicator converges to zero with the same rate of the error of the underlying numerical scheme on smooth flows under grid refinement.We construct and test an adaptive scheme for systems of equations in which the mesh is driven by the entropy indicator.The adaptive scheme uses a single nonuniform grid with a variable timestep.We show how to implement a second order scheme on such a space-time non uniform grid,preserving accuracy and conservation properties.We also give an example of a p-adaptive strategy.展开更多
This paper presents and analyzes a Discrete Duality Finite Volume(DDFV)method to solve 2D diffusion problems under prescribed Robin boundary conditions.The derivation of a symmetric discrete problem is established.The...This paper presents and analyzes a Discrete Duality Finite Volume(DDFV)method to solve 2D diffusion problems under prescribed Robin boundary conditions.The derivation of a symmetric discrete problem is established.The existence and uniqueness of a solution to this discrete problem are shown via the positive definiteness of its associated matrix.We show that the discrete scheme meets the Neumann problem when the parameterα→0(and,in a sense,whenα→∞the Dirichlet problem).This work is a continuation of our work regarding the development of DDFV methods.The main innovation here is taking into account Robin’s boundary conditions.We provide a few steps of Matlab implementation and numerical tests to confirm the effectiveness of the method.展开更多
We have developed approximate Riemann solvers for ideal MHD equations based on a relaxation approach in [4], [5]. These lead to entropy consistent solutions with good properties like guaranteed positive density. We de...We have developed approximate Riemann solvers for ideal MHD equations based on a relaxation approach in [4], [5]. These lead to entropy consistent solutions with good properties like guaranteed positive density. We describe the extension to higher order and multiple space dimensions. Finally we show our implementation of all this into the astrophysics code FLASH.展开更多
We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes,focusing in particular on the cells close to the boundaries of the domain.In fact,most techniq...We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes,focusing in particular on the cells close to the boundaries of the domain.In fact,most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that,taking into account the boundary conditions,fills the ghost cells with appropriate values,so that a standard reconstruction can be applied also in the boundary cells.In Naumann et al.(Appl.Math.Comput.325:252–270.https://doi.org/10.1016/j.amc.2017.12.041,2018),motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network,a different technique was explored that avoids the use of ghost cells,but instead employs for the boundary cells a different stencil,biased towards the interior of the domain.In this paper,extending that approach,which does not make use of ghost cells,we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids.In several numerical tests,we compare the novel reconstruction with the standard approach using ghost cells.展开更多
We consider constraint preserving multidimensional evolution equations.A prototypical example is provided by the magnetic induction equation of plasma physics.The constraint of interest is the divergence of the magnet...We consider constraint preserving multidimensional evolution equations.A prototypical example is provided by the magnetic induction equation of plasma physics.The constraint of interest is the divergence of the magnetic field.We design finite volume schemes which approximate these equations in a stable manner and preserve a discrete version of the constraint.The schemes are based on reformulating standard edge centered finite volume fluxes in terms of vertex centered potentials.The potential-based approach provides a general framework for faithful discretizations of constraint transport and we apply it to both divergence preserving as well as curl preserving equations.We present benchmark numerical tests which confirm that our potential-based schemes achieve high resolution,while being constraint preserving.展开更多
This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix.The resulting scheme is much simpler than the original on...This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix.The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws,while retaining the attractive features of the original solver:the method is entropy-satisfying,differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field,in particular to the intermediate ones,since the full eigenstructure of the underlying hyperbolic system is used.To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws:Euler equations of compressible gasdynamics with ideal gas and real gas equation of state,classical and relativistic MHD equations as well as the equations of nonlinear elasticity.To the knowledge of the authors,apart from the Euler equations with ideal gas,an Osher-type scheme has never been devised before for any of these complicated PDE systems.Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of PNPM schemes on unstructured meshes recently proposed in[9].展开更多
We extend the weighted essentially non-oscillatory(WENO)schemes on two dimensional triangular meshes developed in[7]to three dimensions,and construct a third order finite volume WENO scheme on three dimensional tetrah...We extend the weighted essentially non-oscillatory(WENO)schemes on two dimensional triangular meshes developed in[7]to three dimensions,and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes.We use the Lax-Friedrichs monotone flux as building blocks,third order reconstructions made from combinations of linear polynomials which are constructed on diversified small stencils of a tetrahedral mesh,and non-linear weights using smoothness indicators based on the derivatives of these linear polynomials.Numerical examples are given to demonstrate stability and accuracy of the scheme.展开更多
We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations.By all regime,we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolve...We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations.By all regime,we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M,i.e.such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1.The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M.This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy.A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism.A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed.Then a simple and efficient semi implicit scheme is also proposed.The resulting scheme is stable under a CFL condition driven by the(slow)material waves and not by the(fast)acoustic waves and so verifies the all regime property.Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.展开更多
文摘In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamically compatiblefirst-order hyperbolic systems.By construction,the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm.A very peculiar feature of our approach is that entropy is discretized directly,while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization.The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs,including both,conservative and non-conservative products,as well as potentially stiff algebraic relaxation source terms,provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law,such as the conservation of total energy density.The proposed family offinite volume schemes is based on the seminal work of Abgrall[1],where for thefirst time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented.We apply our new approach to three particular thermodynamically compatible systems:the equations of ideal magnetohydrodynamics(MHD)with thermodynamically compatible generalized Lagrangian multiplier(GLM)divergence cleaning,the unifiedfirst-order hyperbolic model of continuum mechanics proposed by Godunov,Peshkov,and Romenski(GPR model)and thefirst-order hyperbolic model for turbulent shallow waterflows of Gavrilyuk et al.In addition to formal mathematical proofs of the properties of our newfinite volume schemes,we also present a large set of numerical results in order to show their potential,efficiency,and practical applicability.
文摘ADER-WAF methods were first introduced by researchers E.F. Toro and V.A. Titarev. The linear stability criterion for the model equation for the ADER-WAF schemes is CCFL≤1, where CCFLdenotes the Courant-Friedrichs-Lewy (CFL) coefficient. Toro and Titarev employed CCFL=0.95for their experiments. Nonetheless, we noted that the experiments conducted in this study with CCFL=0.95produced solutions exhibiting spurious oscillations, particularly in the high-order ADER-WAF schemes. The homogeneous one-dimensional (1D) non-linear Shallow Water Equations (SWEs) are the subject of these experiments, specifically the solution of the Riemann Problem (RP) associated with the SWEs. The investigation was conducted on four test problems to evaluate the ADER-WAF schemes of second, third, fourth, and fifth order of accuracy. Each test problem constitutes a RP characterized by different wave patterns in its solution. This research has two primary objectives. We begin by illustrating the procedure for implementing the ADER-WAF schemes for the SWEs, providing the required relations. Afterward, following comprehensive testing, we present the range for the CFL coefficient for each test that yields solutions with diminished or eliminated spurious oscillations.
基金The research of R.Zhang is supported in part by NSFC grant 10871190The research of M.Zhang is supported in part by NSFC grant 10671190 and the research of C.-W+1 种基金Shu is supported in part by ARO grant W911NF-08-1-0520NSF grant DMS-0809086.
文摘In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory(WENO)schemes in two dimensional Cartesian meshes.We compare them in terms of accuracy,performance for smooth and shocked solutions,and efficiency in CPU timing.For linear systems both schemes are high order accurate,however for nonlinear systems,analysis and numerical simulation results verify that one of them(Class A)is only second order accurate,while the other(Class B)is high order accurate.The WENO scheme in Class A is easier to implement and costs less than that in Class B.Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes,despite of the difference in their formal order of accuracy.The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.
文摘We propose an a-posteriori error/smoothness indicator for standard semidiscrete finite volume schemes for systems of conservation laws,based on the numerical production of entropy.This idea extends previous work by the first author limited to central finite volume schemes on staggered grids.We prove that the indicator converges to zero with the same rate of the error of the underlying numerical scheme on smooth flows under grid refinement.We construct and test an adaptive scheme for systems of equations in which the mesh is driven by the entropy indicator.The adaptive scheme uses a single nonuniform grid with a variable timestep.We show how to implement a second order scheme on such a space-time non uniform grid,preserving accuracy and conservation properties.We also give an example of a p-adaptive strategy.
文摘This paper presents and analyzes a Discrete Duality Finite Volume(DDFV)method to solve 2D diffusion problems under prescribed Robin boundary conditions.The derivation of a symmetric discrete problem is established.The existence and uniqueness of a solution to this discrete problem are shown via the positive definiteness of its associated matrix.We show that the discrete scheme meets the Neumann problem when the parameterα→0(and,in a sense,whenα→∞the Dirichlet problem).This work is a continuation of our work regarding the development of DDFV methods.The main innovation here is taking into account Robin’s boundary conditions.We provide a few steps of Matlab implementation and numerical tests to confirm the effectiveness of the method.
文摘We have developed approximate Riemann solvers for ideal MHD equations based on a relaxation approach in [4], [5]. These lead to entropy consistent solutions with good properties like guaranteed positive density. We describe the extension to higher order and multiple space dimensions. Finally we show our implementation of all this into the astrophysics code FLASH.
基金MIUR-PRIN project 2017KKJP4X“Innovative numerical methods for evolutionary partial differential equations and applications”.Gabriella Puppo acknowledges also the support of 2019 Ateneo Sapienza research project no.RM11916B51CD40E1.
文摘We address the issue of point value reconstructions from cell averages in the context of third-order finite volume schemes,focusing in particular on the cells close to the boundaries of the domain.In fact,most techniques in the literature rely on the creation of ghost cells outside the boundary and on some form of extrapolation from the inside that,taking into account the boundary conditions,fills the ghost cells with appropriate values,so that a standard reconstruction can be applied also in the boundary cells.In Naumann et al.(Appl.Math.Comput.325:252–270.https://doi.org/10.1016/j.amc.2017.12.041,2018),motivated by the difficulty of choosing appropriate boundary conditions at the internal nodes of a network,a different technique was explored that avoids the use of ghost cells,but instead employs for the boundary cells a different stencil,biased towards the interior of the domain.In this paper,extending that approach,which does not make use of ghost cells,we propose a more accurate reconstruction for the one-dimensional case and a two-dimensional one for Cartesian grids.In several numerical tests,we compare the novel reconstruction with the standard approach using ghost cells.
基金E.Tadmor Research was supported in part by NSF grant 07-07949 and ONR grant N00014-091-0385.
文摘We consider constraint preserving multidimensional evolution equations.A prototypical example is provided by the magnetic induction equation of plasma physics.The constraint of interest is the divergence of the magnetic field.We design finite volume schemes which approximate these equations in a stable manner and preserve a discrete version of the constraint.The schemes are based on reformulating standard edge centered finite volume fluxes in terms of vertex centered potentials.The potential-based approach provides a general framework for faithful discretizations of constraint transport and we apply it to both divergence preserving as well as curl preserving equations.We present benchmark numerical tests which confirm that our potential-based schemes achieve high resolution,while being constraint preserving.
基金financed by the Italian Ministry of Research(MIUR)under the project PRIN 2007 and by MIUR and the British Council under the project British-Italian Partnership Programme for young researchers 2008-2009。
文摘This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix.The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws,while retaining the attractive features of the original solver:the method is entropy-satisfying,differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field,in particular to the intermediate ones,since the full eigenstructure of the underlying hyperbolic system is used.To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws:Euler equations of compressible gasdynamics with ideal gas and real gas equation of state,classical and relativistic MHD equations as well as the equations of nonlinear elasticity.To the knowledge of the authors,apart from the Euler equations with ideal gas,an Osher-type scheme has never been devised before for any of these complicated PDE systems.Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of PNPM schemes on unstructured meshes recently proposed in[9].
基金The research of the second author is supported by NSF grants AST-0506734 and DMS-0510345.
文摘We extend the weighted essentially non-oscillatory(WENO)schemes on two dimensional triangular meshes developed in[7]to three dimensions,and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes.We use the Lax-Friedrichs monotone flux as building blocks,third order reconstructions made from combinations of linear polynomials which are constructed on diversified small stencils of a tetrahedral mesh,and non-linear weights using smoothness indicators based on the derivatives of these linear polynomials.Numerical examples are given to demonstrate stability and accuracy of the scheme.
文摘We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations.By all regime,we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M,i.e.such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1.The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M.This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy.A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism.A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed.Then a simple and efficient semi implicit scheme is also proposed.The resulting scheme is stable under a CFL condition driven by the(slow)material waves and not by the(fast)acoustic waves and so verifies the all regime property.Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.
