Three discontinuous Galerkin schemes for the anisotropic heat conduction equation on non-aligned grids

TL;DR

This paper introduces and compares three discontinuous Galerkin schemes for anisotropic heat conduction on non-aligned grids, focusing on energy conservation, convergence, and numerical fluxes, with applications to axisymmetric magnetic fields.

Contribution

It presents a novel self-adjoint LDG scheme and two aligned schemes, analyzing their properties and performance for anisotropic heat conduction on non-aligned grids.

Findings

The LDG scheme conserves energy and converges with arbitrary order.

Aligned schemes have low perpendicular heat fluxes but face issues with energy conservation and convergence.

Numerical experiments demonstrate the trade-offs between schemes in different scenarios.

Abstract

We present and discuss three discontinuous Galerkin (dG) discretizations for the anisotropic heat conduction equation on non-aligned cylindrical grids. Our most favourable scheme relies on a self-adjoint local dG (LDG) discretization of the elliptic operator. It conserves the energy exactly and converges with arbitrary order. The pollution by numerical perpendicular heat fluxes degrades with superconvergence rates. We compare this scheme with aligned schemes that are based on the flux-coordinate independent approach for the discretization of parallel derivatives. Here, the dG method provides the necessary interpolation. The first aligned discretization can be used in an explicit time-integrator. However, the scheme violates conservation of energy and shows up stagnating convergence rates for very high resolutions. We overcome this partly by using the adjoint of the parallel derivative operator to construct a second self-adjoint aligned scheme. This scheme preserves energy, but reveals unphysical oscillations in the numerical tests, which result in a decreased order of convergence. Both aligned schemes exhibit low numerical heat fluxes into the perpendicular direction. We build our argumentation on various numerical experiments on all three schemes for a general axisymmetric magnetic field, which is closed by a comparison to the aligned finite difference (FD) schemes of References [1,2]