A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations

doi:10.1007/s42967-020-00077-3

摘要/Abstract

摘要： A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible fows on unstructured triangular grids in two space dimensions. The staggered DG scheme defnes the discrete pressure on the primal triangular mesh, while the discrete velocity is defned on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure fnite 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 are built by continuous fnite elements on the subtriangles. This choice allows the construction of an efcient, quadraturefree 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 the use of time-dependent test and basis functions, in combination with simple and efcient Picard iterations. Several numerical tests on classical benchmarks confrm 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.

关键词: Incompressible Navier-Stokes equations, Semi-implicit space-time discontinuous Galerkin schemes, Staggered unstructured meshes, Space-time pressure correction method, High-order accuracy in space and time

Abstract: A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible fows on unstructured triangular grids in two space dimensions. The staggered DG scheme defnes the discrete pressure on the primal triangular mesh, while the discrete velocity is defned on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure fnite 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 are built by continuous fnite elements on the subtriangles. This choice allows the construction of an efcient, quadraturefree 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 the use of time-dependent test and basis functions, in combination with simple and efcient Picard iterations. Several numerical tests on classical benchmarks confrm 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.

Key words: Incompressible Navier-Stokes equations, Semi-implicit space-time discontinuous Galerkin schemes, Staggered unstructured meshes, Space-time pressure correction method, High-order accuracy in space and time

中图分类号:

65M60

F. L. Romeo, M. Dumbser, M. Tavelli. A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 607-647.

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