A Fully Implicit Jacobian-Free High-Order Discontinuous Galerkin Mesoscale Flow Solver

Abstract

In this work it is shown how to discretize the compressible Euler equations around a vertically stratified base state using the discontinuous Galerkin approach on collocated Gauss type grids. A stiffly stable Rosenbrock W-method is combined with an approximate evaluation of the Jacobian to integrate in time the resulting system of ODEs. Simulations with fully compressible equations for a rising thermal bubble are performed. Also included are simulations of an inertia gravity wave in a periodic channel. The proposed time-stepping method accelerates the simulation times with respect to explicit Runge-Kutta time stepping procedures having the same number of stages.

Parallel multilevel methods for implicit solution of shallow water equations with nonsmooth topography on the cubed-sphere

“…We plan to extend the approach to other climate models, such as the nonhydrostatic model based on the compressible Euler or Navier-Stokes equations [30,37,41].…”

A Scalable Fully Implicit Compressible Euler Solver for Mesoscale Nonhydrostatic Simulation of Atmospheric Flows

“…Therefore, in order to reduce the approximation error of the hydrostatic state, it is preferable [15,40] to introduce in the equations the following perturbed values…”

“…Despite of some successes on computing fully implicit solutions of the global shallow water equations [10,11,48,49], it is rare to see the application of fully implicit methods in nonhydrostatic atmospheric simulations. One of the contributions is [40], in which a discontinuous Galerkin solver for mesoscale flows using a fully implicit Rosenbrock scheme is proposed and shown to be superior to an explicit Runge-Kutta method. However, in their Jacobian-free Newton-Krylov framework for solving the nonlinear system, the preconditioner, which is critical for the performance of the solver, is not studied.…”

Implementation of a Semi-implicit Time Integration Scheme in Non-Hydrostatic Euler Equations

“…To overcome these issues, a time-split integration scheme was proposed [12,[13][14][15][16][17][18][19][20]. In this scheme, fast acoustic waves use a smaller time step while slower waves use a larger time step [21][22][23][24][25]. The scheme is based on the Runge-Kutta methods.…”

“…Nevertheless, the time step used in implicit schemes is greater than that used in explicit schemes. St-Cyr and Neckels [21], the authors discuss fully implicit time integration schemes for discontinuous Galerkin methods applied to atmospheric flow. In particular, they proposed Jacobian-free Newton-Krylov solvers without the computation of the Jacobian matrix.…”