Arbitrary Order Solutions for the Eikonal Equation using a Discontinuous Galerkin Method

TL;DR

This paper introduces a high-order discontinuous Galerkin method with artificial viscosity for solving the eikonal equation, achieving accurate solutions for complex geometries while satisfying entropy conditions.

Contribution

It presents a novel arbitrary-order polynomial space approach for the eikonal equation using a discontinuous Galerkin framework with artificial viscosity, ensuring entropy compliance.

Findings

Demonstrates optimal convergence order

Achieves stable solutions for shocks and rarefactions

Successfully models complex airfoil geometries

Abstract

We provide a method to compute the entropy-satisfying weak solution to the eikonal equation in an arbitrary-order polynomial space. The method uses an artificial viscosity approach and is demonstrated for the signed distance function, where exact solutions are available. The method is designed specifically for an existing high-order discontinuous-Galerkin framework, which uses standard convection, diffusion, and source terms. We show design order of accuracy and good behavior for both shocks and rarefaction type solutions. Finally the distance function around a complex multi-element airfoil is computed using a high-order-accurate representation.