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Efficient Finite Difference WENO Scheme for Hyperbolic Systems withnon-conservativeproducts

Communications on Applied Mathematics and Computation 2024年第2期6卷 907-962页

作者： Dinshaw S.Balsara Deepak Bhoriya Chi-Wang Shu Harish Kumar Physics Department University of Notre DameNotre DameUSA ACMS Department University of Notre DameNotre DameUSA Division of Applied Mathematics Brown UniversityProvidenceUSA Department of Mathematics Indian Institute of TechnologyDelhiIndia

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation *** 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 *** makes them quite attractive for several science and engineering ***,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative *** this paper,we perform such an extension which improves the domain of the applicability of such *** extension is carried out by writing the scheme in fluctuation *** use the HLLI Riemann solver of Dumbser and Balsara(***.304:275-319,2016)as a building block for carrying out this *** of the use of an HLL building block,the resulting scheme has a proper supersonic *** use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the *** 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 *** 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 *** 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 *** 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 *** some

关键词： Hyperbolic PDEs Numerical schemes non-conservative products Stiff source terms Finite difference WENO

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High Order Accurate Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD Finite Volume Schemes for non-conservative Hyperbolic Systems with Stiff Source Terms

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Communications in Computational Physics 2017年第1期21卷 271-312页

作者： Walter Boscheri Raphael Loubere Laboratory of AppliedMathematics Department of CivilEnvironmental and Mechanical EngineeringUniversity of TrentoI-38123 TrentoItaly CNRS and Institut de Mathematiques de Toulouse(IMT)Universite Paul-Sabatier ToulouseFrance

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 *** 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)*** 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 *** 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 *** Riemann problems and explosion problems with exact solutions are simulated in 2D and *** overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm.

关键词： Direct Arbitrary-Lagrangian-Eulerian a posteriori MOOD stabilization Baer-Nunziato model stiff source terms non-conservative products unstructured mesh ADER high order of accuracy in space and time high performance computing(HPC) hyperbolic conservation laws

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