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
The purpose of this paper is to solve some of the trouble spots of the classical SPH method by proposing an alternative approach.First,we focus on the problem of the stability for two different SPH schemes,one is base...The purpose of this paper is to solve some of the trouble spots of the classical SPH method by proposing an alternative approach.First,we focus on the problem of the stability for two different SPH schemes,one is based on the approach of Vila[25]and another is proposed in this article which mimics the classical 1D LaxWendroff scheme.In both approaches the classical SPH artificial viscosity term is removed preserving nevertheless the linear stability of the methods,demonstrated via the von Neumann stability analysis.Moreover,the issue of the consistency for the equations of gas dynamics is analyzed.An alternative approach is proposed that consists of using Godunov-type SPH schemes in Lagrangian coordinates.This not only provides an improvement in accuracy of the numerical solutions,but also assures that the consistency condition on the gradient of the kernel function is satisfied using an equidistant distribution of particles in Lagrangian mass coordinates.Three different Riemann solvers are implemented for the first-order Godunov type SPH schemes in Lagrangian coordinates,namely the Godunov flux based on the exact Riemann solver,the Rusanov flux and a new modified Roe flux,following the work of Munz[17].Some well-known numerical 1D shock tube test cases[22]are solved,comparing the numerical solutions of the Godunov-type SPH schemes in Lagrangian coordinates with the first-order Godunov finite volume method in Eulerian coordinates and the standard SPH scheme with Monaghan’s viscosity term.展开更多
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.
文摘The purpose of this paper is to solve some of the trouble spots of the classical SPH method by proposing an alternative approach.First,we focus on the problem of the stability for two different SPH schemes,one is based on the approach of Vila[25]and another is proposed in this article which mimics the classical 1D LaxWendroff scheme.In both approaches the classical SPH artificial viscosity term is removed preserving nevertheless the linear stability of the methods,demonstrated via the von Neumann stability analysis.Moreover,the issue of the consistency for the equations of gas dynamics is analyzed.An alternative approach is proposed that consists of using Godunov-type SPH schemes in Lagrangian coordinates.This not only provides an improvement in accuracy of the numerical solutions,but also assures that the consistency condition on the gradient of the kernel function is satisfied using an equidistant distribution of particles in Lagrangian mass coordinates.Three different Riemann solvers are implemented for the first-order Godunov type SPH schemes in Lagrangian coordinates,namely the Godunov flux based on the exact Riemann solver,the Rusanov flux and a new modified Roe flux,following the work of Munz[17].Some well-known numerical 1D shock tube test cases[22]are solved,comparing the numerical solutions of the Godunov-type SPH schemes in Lagrangian coordinates with the first-order Godunov finite volume method in Eulerian coordinates and the standard SPH scheme with Monaghan’s viscosity term.
