In this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-con...In this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source terms.This scheme is constructed with a single stencil polynomial reconstruction operator,a one-step space-time ADER integration which is suitably designed for dealing even with stiff sources,a nodal solver with relaxation to determine the mesh motion,a path-conservative integration technique for the treatment of non-conservative products and an a posteriori stabilization procedure derived from the so-called Multidimensional Optimal Order Detection(MOOD)paradigm.In this work we consider the seven equation Baer-Nunziato model of compressible multi-phase flows as a representative model involving non-conservative products as well as relaxation source terms which are allowed to become stiff.The new scheme is validated against a set of test cases on 2D/3D unstructured moving meshes on parallel machines and the high order of accuracy achieved by the method is demonstrated by performing a numerical convergence study.Classical Riemann problems and explosion problems with exact solutions are simulated in 2D and 3D.The overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm.展开更多
In this paper, we present two semi-implicit-type second-order compact approximate Tay-lor(CAT2) numerical schemes and blend them with a local a posteriori multi-dimensionaloptimal order detection (MOOD) paradigm to so...In this paper, we present two semi-implicit-type second-order compact approximate Tay-lor(CAT2) numerical schemes and blend them with a local a posteriori multi-dimensionaloptimal order detection (MOOD) paradigm to solve hyperbolic systems of balance lawswith relaxed source terms. The resulting scheme presents the high accuracy when applied tosmooth solutions, essentially non-oscillatory behavior for irregular ones, and offers a nearlyfail-safe property in terms of ensuring the positivity. The numerical results obtained from avariety of test cases, including smooth and non-smooth well-prepared and unprepared initialconditions, assessing the appropriate behavior of the semi-implicit-type second order CAT-MOODschemes. These results have been compared in the accuracy and the efficiency witha second-order semi-implicit Runge-Kutta (RK) method.展开更多
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
In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws.High order accuracy in space is obtain...In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws.High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkinmethod recently proposed in[20].In the Lagrangian framework considered here,the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element.For the spacetime basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points.The moving space-time elements are mapped to a reference element using an isoparametric approach,i.e.the spacetime mapping is defined by the same basis functions as the weak solution of the PDE.We show some computational examples in one space-dimension for non-stiff and for stiff balance laws,in particular for the Euler equations of compressible gas dynamics,for the resistive relativistic MHD equations,and for the relativistic radiation hydrodynamics equations.Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.展开更多
Higher order finite difference Weighted Essentially Non-oscillatory(WENO)schemes for conservation laws represent a technology that has been reasonably consolidated.They are extremely popular because,when applied to mu...Higher order finite difference Weighted Essentially Non-oscillatory(WENO)schemes for conservation laws represent a technology that has been reasonably consolidated.They are extremely popular because,when applied to multidimensional problems,they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes.They come in two flavors.There is the classical finite difference WENO(FD-WENO)method(Shu and Osher in J.Comput.Phys.83:32–78,1989).However,in recent years there is also an alternative finite difference WENO(AFD-WENO)method which has recently been formalized into a very useful general-purpose algorithm for conservation laws(Balsara et al.in Efficient alternative finite difference WENO schemes for hyperbolic conservation laws,submitted to CAMC,2023).However,the FD-WENO algorithm has only very recently been formulated for hyperbolic systems with non-conservative products(Balsara et al.in Efficient finite difference WENO scheme for hyperbolic systems with non-conservative products,to appear CAMC,2023).In this paper,we show that there are substantial advantages in obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative products.Such an algorithm is documented in this paper.We present an AFD-WENO formulation in a fluctuation form that is carefully engineered to retrieve the flux form when that is warranted and nevertheless extends to non-conservative products.The method is flexible because it allows any Riemann solver to be used.The formulation we arrive at is such that when non-conservative products are absent it reverts exactly to the formulation in the second citation above which is in the exact flux conservation form.The ability to transition to a precise conservation form when non-conservative products are absent ensures,via the Lax-Wendroff theorem,that shock locations will be exactly captured by the method.We present two formulations of AFD-WENO that can be used with hyperbolic systems with non-conservative products and stiff source terms with slightly differing computational complexities.The speeds of our new AFD-WENO schemes are compared to the speed of the classical FD-WENO algorithm from the first of the above-cited papers.At all orders,AFD-WENO outperforms FD-WENO.We also show a very desirable result that higher order variants of AFD-WENO schemes do not cost that much more than their lower order variants.This is because the larger number of floating point operations associated with larger stencils is almost very efficiently amortized by the CPU when the AFD-WENO code is designed to be cache friendly.This should have great,and very beneficial,implications for the role of our AFD-WENO schemes in the Peta-and Exascale computing.We apply the method to several stringent test problems drawn from the Baer-Nunziato system,two-layer shallow water equations,and the multicomponent debris flow.The method meets its design accuracy for the smooth flow and can handle stringent problems in one and multiple dimensions.Because of the pointwise nature of its update,AFD-WENO for hyperbolic systems with non-conservative products is also shown to be a very efficient performer on problems with stiff source terms.展开更多
基金W.B.has been financed by the European Research Council(ERC)under the European Union’s Seventh Framework Programme(FP7/2007-2013)with the research project STiMulUs,ERC Grant agreement no.278267R.L.has been partially funded by the ANR under the JCJC project“ALE INC(ubator)3D”JS01-012-01the“International Centre for Mathematics and Computer Science in Toulouse”(CIMI)partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02.The authors would like to acknowledge PRACE for awarding access to the SuperMUC supercomputer based in Munich,Germany at the Leibniz Rechenzentrum(LRZ).Parts of thematerial contained in this work have been elaborated,gathered and tested while W.B.visited the Mathematical Institute of Toulouse for three months and R.L.visited the Dipartimento di Ingegneria Civile Ambientale e Meccanica in Trento for three months.
文摘In this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source terms.This scheme is constructed with a single stencil polynomial reconstruction operator,a one-step space-time ADER integration which is suitably designed for dealing even with stiff sources,a nodal solver with relaxation to determine the mesh motion,a path-conservative integration technique for the treatment of non-conservative products and an a posteriori stabilization procedure derived from the so-called Multidimensional Optimal Order Detection(MOOD)paradigm.In this work we consider the seven equation Baer-Nunziato model of compressible multi-phase flows as a representative model involving non-conservative products as well as relaxation source terms which are allowed to become stiff.The new scheme is validated against a set of test cases on 2D/3D unstructured moving meshes on parallel machines and the high order of accuracy achieved by the method is demonstrated by performing a numerical convergence study.Classical Riemann problems and explosion problems with exact solutions are simulated in 2D and 3D.The overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm.
基金the European Union’s NextGenerationUE-Project:Centro Nazionale HPC,Big Data e Quantum Computing,“Spoke 1”(No.CUP E63C22001000006)E.Macca was partially supported by GNCS No.CUP E53C22001930001 Research Project“Metodi numericiper problemi differenziali multiscala:schemi di alto ordine,ottimizzazione,controllo”+1 种基金E.Macca and S.Boscarino would like to thank the Italian Ministry of Instruction,University and Research(MIUR)to supportthis research with funds coming from PRIN Project 2022(2022KA3JBA,entitled“Advanced numericalmethods for time dependent parametric partial differential equations and applications”)Sebastiano Boscarinohas been supported for this work from Italian Ministerial grant PRIN 2022 PNRR“FIN4GEO:forward andinverse numerical modeling of hydrothermalsystemsin volcanic regions with application to geothermal energyexploitation”(No.P2022BNB97).E.Macca and S.Boscarino are members of the INdAM Research groupGNCS.
文摘In this paper, we present two semi-implicit-type second-order compact approximate Tay-lor(CAT2) numerical schemes and blend them with a local a posteriori multi-dimensionaloptimal order detection (MOOD) paradigm to solve hyperbolic systems of balance lawswith relaxed source terms. The resulting scheme presents the high accuracy when applied tosmooth solutions, essentially non-oscillatory behavior for irregular ones, and offers a nearlyfail-safe property in terms of ensuring the positivity. The numerical results obtained from avariety of test cases, including smooth and non-smooth well-prepared and unprepared initialconditions, assessing the appropriate behavior of the semi-implicit-type second order CAT-MOODschemes. These results have been compared in the accuracy and the efficiency witha second-order semi-implicit Runge-Kutta (RK) method.
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the European Research Council under the European Union’s Seventh Framework Programme(FP7/2007-2013)under the research project STiMulUs,ERC Grant agreement no.278267.
文摘In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws.High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkinmethod recently proposed in[20].In the Lagrangian framework considered here,the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element.For the spacetime basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points.The moving space-time elements are mapped to a reference element using an isoparametric approach,i.e.the spacetime mapping is defined by the same basis functions as the weak solution of the PDE.We show some computational examples in one space-dimension for non-stiff and for stiff balance laws,in particular for the Euler equations of compressible gas dynamics,for the resistive relativistic MHD equations,and for the relativistic radiation hydrodynamics equations.Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.
基金support via NSF grant NSF-AST-2009776,NASA grant NASA-2020-1241 and NASA grant 80NSSC22K0628support from a Vajra award,VJR/2018/00129 and also a travel grant from Notre Dame International.CWS acknowledges support via NSF grant DMS-2309249+2 种基金support via the NSF Grants NSF-19-04774,NSF-AST-2009776,NASA-2020-1241,and NASA-80NSSC22K0628support from a Vajra award,VJR/2018/00129support via AFOSR Grant FA9550-20-1-0055 and NSF Grant DMS-2010107.
文摘Higher order finite difference Weighted Essentially Non-oscillatory(WENO)schemes for conservation laws represent a technology that has been reasonably consolidated.They are extremely popular because,when applied to multidimensional problems,they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes.They come in two flavors.There is the classical finite difference WENO(FD-WENO)method(Shu and Osher in J.Comput.Phys.83:32–78,1989).However,in recent years there is also an alternative finite difference WENO(AFD-WENO)method which has recently been formalized into a very useful general-purpose algorithm for conservation laws(Balsara et al.in Efficient alternative finite difference WENO schemes for hyperbolic conservation laws,submitted to CAMC,2023).However,the FD-WENO algorithm has only very recently been formulated for hyperbolic systems with non-conservative products(Balsara et al.in Efficient finite difference WENO scheme for hyperbolic systems with non-conservative products,to appear CAMC,2023).In this paper,we show that there are substantial advantages in obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative products.Such an algorithm is documented in this paper.We present an AFD-WENO formulation in a fluctuation form that is carefully engineered to retrieve the flux form when that is warranted and nevertheless extends to non-conservative products.The method is flexible because it allows any Riemann solver to be used.The formulation we arrive at is such that when non-conservative products are absent it reverts exactly to the formulation in the second citation above which is in the exact flux conservation form.The ability to transition to a precise conservation form when non-conservative products are absent ensures,via the Lax-Wendroff theorem,that shock locations will be exactly captured by the method.We present two formulations of AFD-WENO that can be used with hyperbolic systems with non-conservative products and stiff source terms with slightly differing computational complexities.The speeds of our new AFD-WENO schemes are compared to the speed of the classical FD-WENO algorithm from the first of the above-cited papers.At all orders,AFD-WENO outperforms FD-WENO.We also show a very desirable result that higher order variants of AFD-WENO schemes do not cost that much more than their lower order variants.This is because the larger number of floating point operations associated with larger stencils is almost very efficiently amortized by the CPU when the AFD-WENO code is designed to be cache friendly.This should have great,and very beneficial,implications for the role of our AFD-WENO schemes in the Peta-and Exascale computing.We apply the method to several stringent test problems drawn from the Baer-Nunziato system,two-layer shallow water equations,and the multicomponent debris flow.The method meets its design accuracy for the smooth flow and can handle stringent problems in one and multiple dimensions.Because of the pointwise nature of its update,AFD-WENO for hyperbolic systems with non-conservative products is also shown to be a very efficient performer on problems with stiff source terms.
