A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier–Stokes Equations in Nonhydrostatic Mesoscale Modeling

Abstract

Abstract.A discontinuous Galerkin (DG) finite element formulation is proposed for the solution of the compressible Navier-Stokes equations for a vertically stratified fluid, which are of interest in mesoscale nonhydrostatic atmospheric modeling. The resulting scheme naturally ensures conservation of mass, momentum, and energy. A semi-implicit time-integration approach is adopted to improve the efficiency of the scheme with respect to the explicit Runge-Kutta time integration strategies usually employed in the context of DG formulations. A method is also presented to reformulate the resulting linear system as a pseudo-Helmholtz problem. In doing this, we obtain a DG discretization closely related to those proposed for the solution of elliptic problems, and we show how to take advantage of the numerical integration rules (required in all DG methods for the area and flux integrals) to increase the efficiency of the solution algorithm. The resulting numerical formulation is then validated on a collection of classical two-dimensional test cases, including density driven flows and mountain wave simulations. The performance analysis shows that the semi-implicit method is, indeed, superior to explicit methods and that the pseudo-Helmholtz formulation yields further efficiency improvements. 1. Introduction. In recent years, great attention has been devoted to the discontinuous Galerkin (DG) finite element method in the context of geophysical fluid dynamics applications. This is motivated by the fact that the DG framework simultaneously provides a high-order discretization, great flexibility in the choice of the computational grid, discrete balance relations, robustness with respect to unphysical oscillations, and compact computational stencils which are a key element in order to exploit distributed-memory parallel computers with up to tens of thousands of processors. Without attempting to provide a complete review of the literature, we mention here [47,2,30,39,34,32], where DG shallow water models are presented. The application of the DG method to compressible, nonhydrostatic atmospheric flows, using the Navier-Stokes equations or, when the flow is assumed to be inviscid, the Euler equations, is then considered in [31], where it is shown that the method represents a good candidate for the development of numerical climate and weather models. In the present paper, we continue the study initiated in [31] by focusing on the aspect of the time discretization which is, in fact, the most penalizing drawback of the DG method due to its high computational cost. This latter cost stems from the following

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

“…This could be improved, e.g. using semi-implicit time stepping to dampen the fast waves, see [10,32]. Because this qualitative characteristics of CPU time is observed also for all the following experiments, this analysis is presented for this test case only.…”

A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

“…This brief discussion on the cost of the SE and DG methods is completely independent from the time-integrators used. In this work we only discuss explicit methods but in a separate paper we present semi-implicit results [41].…”

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

“…Constructing the Schur complement for general DG polynomial spaces and boundary conditions remains an open problem. Thus far, we only know how to construct the Schur complement for collocated DG formulations (where the interpolation points coincide with the integration points) for a specific class of boundary conditions (see [15] and [16] for the solution to this problem for the Navier-Stokes equations). However, co-located interpolation and integration points currently exist for the quadrilateral (see, e.g., [15,29,30]) but they do not exist for the triangle.…”

“…Their SI DG formulation is based on low-order polynomial spaces (third order or less) and their approach is fundamentally different from ours in that they rely on a linearization of the nonlinear operators in conjunction with a special flux function that facilitates this linearization. Our approach [15,16] relies on extracting the linear operators containing the fastest wave speeds in the system and then discretizing them implicitly in time. While both approaches are very effective, our approach is more similar to the classical SI method first proposed by Robert et al [17].…”