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
A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin(DG)method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions.Th...A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin(DG)method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions.The staggered DG scheme defines the discrete pressure on the primal triangular mesh,while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh.In this paper,a new pair of equal-order-interpolation velocity-pressure finite elements is proposed.On the primary triangular mesh(the pressure elements),the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle.On the dual mesh instead(the velocity elements),the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries,while they are continuous inside.In other words,the basis functions on the dual mesh arc built by continuous finite elements on the subtriangles.This choice allows the construction of an efficient,quadrature-free and memory saving algorithm.In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations,the arbitrary high order of accuracy in time is achieved through tire use of time-dependent test and basis functions,in combination with simple and efficient Picard iterations.Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes,not only from a computer memory point of view,but also concerning the computational time.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funded by the research project STiMulUs,ERC Grant agreement no.278267Financial support has also been provided by 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)the PRIN2017 project.The authors have also received funding from the University of Trento via the Strategic Initiative Modeling and Simulation.
文摘A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin(DG)method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space dimensions.The staggered DG scheme defines the discrete pressure on the primal triangular mesh,while the discrete velocity is defined on a staggered edge-based dual quadrilateral mesh.In this paper,a new pair of equal-order-interpolation velocity-pressure finite elements is proposed.On the primary triangular mesh(the pressure elements),the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle.On the dual mesh instead(the velocity elements),the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries,while they are continuous inside.In other words,the basis functions on the dual mesh arc built by continuous finite elements on the subtriangles.This choice allows the construction of an efficient,quadrature-free and memory saving algorithm.In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations,the arbitrary high order of accuracy in time is achieved through tire use of time-dependent test and basis functions,in combination with simple and efficient Picard iterations.Several numerical tests on classical benchmarks confirm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes,not only from a computer memory point of view,but also concerning the computational time.
