A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations
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A Novel Staggered Semi‑implicit Space‑Time Discontinuous Galerkin Method for the Incompressible Navier‑Stokes Equations F. L. Romeo1 · M. Dumbser1 · M. Tavelli1 Received: 31 December 2019 / Revised: 14 March 2020 / Accepted: 18 May 2020 © Shanghai University 2020
Abstract 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 are built by continuous finite elements on the subtriangles. This choice allows the construction of an efficient, 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 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. Keywords 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 Mathematics Subject Classification 65M60 * M. Dumbser [email protected] F. L. Romeo francesco.romeo‑[email protected] M. Tavelli [email protected] 1
Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123 Trento, Italy
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Vol.:(0123456789)
Communications on Applied Mathematics and Computation
1 Introduction In this article, we propose a new pair of velocity-pressure elements in the framework of the family of arbitrary high-order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) methods for the two-dimensional incompressible Navier-Stokes equations on unstructured meshes, extending the works in [116, 117] by Tavelli and Dumbser. The discretization of the incompressible Navier-Stokes equations was mainly carried out in the past using finite difference methods [78, 102, 103, 127] or continuous finite
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