Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of ...Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.展开更多
This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms.Such PDEs are referred to as...This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms.Such PDEs are referred to as curl-free or curl-preserving,respectively.They arise very frequently in equations for hyperelasticity and compressible multiphase flow,in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation.Experience has shown that if nothing special is done to account for the curl-preserving vector field,it can blow up in a finite amount of simulation time.In this paper,we catalogue a class of DG-like schemes for such PDEs.To retain the globally curl-free or curl-preserving constraints,the components of the vector field,as well as their higher moments,must be collocated at the edges of the mesh.They are updated using potentials collocated at the vertices of the mesh.The resulting schemes:(i)do not blow up even after very long integration times,(ii)do not need any special cleaning treatment,(iii)can oper-ate with large explicit timesteps,(iv)do not require the solution of an elliptic system and(v)can be extended to higher orders using DG-like methods.The methods rely on a spe-cial curl-preserving reconstruction and they also rely on multidimensional upwinding.The Galerkin projection,highly crucial to the design of a DG method,is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the verti-ces of the mesh with the help of a multidimensional Riemann solver.A von Neumann sta-bility analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work.The stability analysis confirms that with the increasing order of accuracy,our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation.We also show that PNPM-like methods,which only evolve the lower moments while reconstructing the higher moments,retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity.The quadratic energy preservation of these methods is also shown to be excellent,especially at higher orders.The methods are also shown to be curl-preserving over long integration times.展开更多
Several important PDE systems,like magnetohydrodynamics and computational electrodynamics,are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-pre...Several important PDE systems,like magnetohydrodynamics and computational electrodynamics,are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion.Recently,new classes of PDE systems have emerged for hyperelasticity,compressible multiphase flows,so-called firstorder reductions of the Einstein field equations,or a novel first-order hyperbolic reformulation of Schrödinger’s equation,to name a few,where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field.We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume(FV)WENO-like schemes for PDEs that support a curl-preserving involution.(Some insights into discontinuous Galerkin(DG)schemes are also drawn,though that is not the prime focus of this paper.)This is done for two-and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction.The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented.In two dimensions,a von Neumann analysis of structure-preserving WENOlike schemes that mimetically satisfy the curl constraints,is also presented.It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems.Numerical results are also presented to show that the edge-centered curl-preserving(ECCP)schemes meet their design accuracy.This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy.By its very design,this paper is,therefore,intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.展开更多
GPU computing is expected to play an integral part in all modern Exascale supercomputers.It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such s...GPU computing is expected to play an integral part in all modern Exascale supercomputers.It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers.It is,therefore,very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity.Not every algorithm that is used in the space-time update of the solution of hyperbolic PDEs will take well to GPUs.However,we identify a small core of algorithms that take exceptionally well to GPU computing.Based on an analysis of available options,we have been able to identify weighted essentially non-oscillatory(WENO)algorithms for spatial reconstruction along with arbitrary derivative(ADER)algorithms for time extension followed by a corrector step as the winning three-part algorithmic combination.Even when a winning subset of algorithms has been identified,it is not clear that they will port seamlessly to GPUs.The low data throughput between CPU and GPU,as well as the very small cache sizes on modern GPUs,implies that we have to think through all aspects of the task of porting an application to GPUs.For that reason,this paper identifies the techniques and tricks needed for making a successful port of this very useful class of higher order algorithms to GPUs.Application codes face a further challenge—the GPU results need to be practically indistinguishable from the CPU results—in order for the legacy knowledge bases embedded in these applications codes to be preserved during the port of GPUs.This requirement often makes a complete code rewrite impossible.For that reason,it is safest to use an approach based on OpenACC directives,so that most of the code remains intact(as long as it was originally well-written).This paper is intended to be a one-stop shop for anyone seeking to make an OpenACC-based port of a higher order Godunov scheme to GPUs.We focus on three broad and high-impact areas where higher order Godunov schemes are used.The first area is computational fluid dynamics(CFD).The second is computational magnetohydrodynamics(MHD)which has an involution constraint that has to be mimetically preserved.The third is computational electrodynamics(CED)which has involution constraints and also extremely stiff source terms.Together,these three diverse uses of higher order Godunov methodology,cover many of the most important applications areas.In all three cases,we show that the optimal use of algorithms,techniques,and tricks,along with the use of OpenACC,yields superlative speedups on GPUs.As a bonus,we find a most remarkable and desirable result:some higher order schemes,with their larger operations count per zone,show better speedup than lower order schemes on GPUs.In other words,the GPU is an optimal stratagem for overcoming the higher computational complexities of higher order schemes.Several avenues for future improvement have also been identified.A scalability study is presented for a real-world application using GPUs and comparable numbers of high-end multicore CPUs.It is found that GPUs offer a substantial performance benefit over comparable number of CPUs,especially when all the methods designed in this paper are used.展开更多
基金support via NSF grants NSF-19-04774,NSF-AST-2009776,NASA-2020-1241NASA grant 80NSSC22K0628.DSB+3 种基金HK acknowledge support from a Vajra award,VJR/2018/00129a travel grant from Notre Dame Internationalsupport via AFOSR grant FA9550-20-1-0055NSF grant DMS-2010107.
文摘Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.
基金Open Access funding provided by ETH Zurich.The funding has been acknowledged.DSB acknowledges support via NSF grants NSF-19-04774,NSF-AST-2009776 and NASA-2020-1241.
文摘This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms.Such PDEs are referred to as curl-free or curl-preserving,respectively.They arise very frequently in equations for hyperelasticity and compressible multiphase flow,in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation.Experience has shown that if nothing special is done to account for the curl-preserving vector field,it can blow up in a finite amount of simulation time.In this paper,we catalogue a class of DG-like schemes for such PDEs.To retain the globally curl-free or curl-preserving constraints,the components of the vector field,as well as their higher moments,must be collocated at the edges of the mesh.They are updated using potentials collocated at the vertices of the mesh.The resulting schemes:(i)do not blow up even after very long integration times,(ii)do not need any special cleaning treatment,(iii)can oper-ate with large explicit timesteps,(iv)do not require the solution of an elliptic system and(v)can be extended to higher orders using DG-like methods.The methods rely on a spe-cial curl-preserving reconstruction and they also rely on multidimensional upwinding.The Galerkin projection,highly crucial to the design of a DG method,is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the verti-ces of the mesh with the help of a multidimensional Riemann solver.A von Neumann sta-bility analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work.The stability analysis confirms that with the increasing order of accuracy,our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation.We also show that PNPM-like methods,which only evolve the lower moments while reconstructing the higher moments,retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity.The quadratic energy preservation of these methods is also shown to be excellent,especially at higher orders.The methods are also shown to be curl-preserving over long integration times.
基金Dinshaw S.Balsara acknowledges support via NSF grants NSF-19-04774,NSFAST-2009776 and NASA-2020-1241Michael Dumbser acknowledges the financial support received from the Italian Ministry of Education,University and Research(MIUR)in the frame of the Departments of Excellence Initiative 2018-2022 attributed to DICAM of the University of Trento(grant L.232/2016)and in the frame of the PRIN 2017 project Innovative numerical methods for evolutionary partial differential equations and applications.
文摘Several important PDE systems,like magnetohydrodynamics and computational electrodynamics,are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion.Recently,new classes of PDE systems have emerged for hyperelasticity,compressible multiphase flows,so-called firstorder reductions of the Einstein field equations,or a novel first-order hyperbolic reformulation of Schrödinger’s equation,to name a few,where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field.We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume(FV)WENO-like schemes for PDEs that support a curl-preserving involution.(Some insights into discontinuous Galerkin(DG)schemes are also drawn,though that is not the prime focus of this paper.)This is done for two-and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction.The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented.In two dimensions,a von Neumann analysis of structure-preserving WENOlike schemes that mimetically satisfy the curl constraints,is also presented.It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems.Numerical results are also presented to show that the edge-centered curl-preserving(ECCP)schemes meet their design accuracy.This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy.By its very design,this paper is,therefore,intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.
基金support via the NSF grants NSF-19-04774,NSF-AST-2009776,NASA-2020-1241the NASA grant 80NSSC22K0628。
文摘GPU computing is expected to play an integral part in all modern Exascale supercomputers.It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers.It is,therefore,very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity.Not every algorithm that is used in the space-time update of the solution of hyperbolic PDEs will take well to GPUs.However,we identify a small core of algorithms that take exceptionally well to GPU computing.Based on an analysis of available options,we have been able to identify weighted essentially non-oscillatory(WENO)algorithms for spatial reconstruction along with arbitrary derivative(ADER)algorithms for time extension followed by a corrector step as the winning three-part algorithmic combination.Even when a winning subset of algorithms has been identified,it is not clear that they will port seamlessly to GPUs.The low data throughput between CPU and GPU,as well as the very small cache sizes on modern GPUs,implies that we have to think through all aspects of the task of porting an application to GPUs.For that reason,this paper identifies the techniques and tricks needed for making a successful port of this very useful class of higher order algorithms to GPUs.Application codes face a further challenge—the GPU results need to be practically indistinguishable from the CPU results—in order for the legacy knowledge bases embedded in these applications codes to be preserved during the port of GPUs.This requirement often makes a complete code rewrite impossible.For that reason,it is safest to use an approach based on OpenACC directives,so that most of the code remains intact(as long as it was originally well-written).This paper is intended to be a one-stop shop for anyone seeking to make an OpenACC-based port of a higher order Godunov scheme to GPUs.We focus on three broad and high-impact areas where higher order Godunov schemes are used.The first area is computational fluid dynamics(CFD).The second is computational magnetohydrodynamics(MHD)which has an involution constraint that has to be mimetically preserved.The third is computational electrodynamics(CED)which has involution constraints and also extremely stiff source terms.Together,these three diverse uses of higher order Godunov methodology,cover many of the most important applications areas.In all three cases,we show that the optimal use of algorithms,techniques,and tricks,along with the use of OpenACC,yields superlative speedups on GPUs.As a bonus,we find a most remarkable and desirable result:some higher order schemes,with their larger operations count per zone,show better speedup than lower order schemes on GPUs.In other words,the GPU is an optimal stratagem for overcoming the higher computational complexities of higher order schemes.Several avenues for future improvement have also been identified.A scalability study is presented for a real-world application using GPUs and comparable numbers of high-end multicore CPUs.It is found that GPUs offer a substantial performance benefit over comparable number of CPUs,especially when all the methods designed in this paper are used.
