Philipp Birken: Towards awesome iterative solvers for accurate numerical weather prediction: Preserving linear and nonlinear invariants

Abstract:

During the past decade, significant effort has been put into high order discontinuous Galerkin (DG) methods for compressible flows. These methods are both highly accurate, and well suited for High Performance Computing due to their high arithmetic intensity. At Environment and Climate Change Canada (ECCC), a redesign of their dynamical core to simulate atmospheric flow towards a DG method is underway, in collaboration with a consortium of partners including Lund University.

Such flows are characterized by low Mach numbers and a gravity source term, leading to stiff problems requiring implicit discretizations. The design goal is thus a suitable highly parallel stable DG implementation. Additionally, to solve the arising nonlinear equation systems, iterative methods are needed that are fast, parallel and use little memory.

A successful design principle has been to find methods that preserve properties of the differential equation, such as the underlying conservation law or the second law of thermodynamics in the form of an entropy inequality. The question thus arises if these approximate solutions in turn preserve the properties of the discretization.

In this talk, we will discuss preservation of linear and nonlinear invariants, namely global and local conservation, and of entropy. As it turns out, many commonly used methods preserve the local conservation of an underlying implicit scheme. We present extensions of the Lax-Wendroff theorem for a fixed, finite number of iterations each time step. Regarding entropy, iterative methods do not preserve this. We present various ways to fix this problem by adjusting Newton's method or adding a relaxation step.

This is joint work with Viktor Linders (Carrier AB), Stéphane Gaudreault, Vincent Magnoux and Shoyon Panday (all ECCC).