On the use of high order central difference schemes for differential equation based wall distance computations

TL;DR

This paper introduces high-order central difference schemes for efficient wall distance computation via PDEs, demonstrating faster performance and comparable accuracy to traditional methods, with applications in rocket motor simulations.

Contribution

It develops a high-order central difference solver for wall distances, introduces a pseudo HJ formulation with LAD, and incorporates curvature correction for improved accuracy.

Findings

High-order schemes are 1.4-2.8 times faster than upwind schemes.

The pseudo HJ formulation predicts results with accuracy comparable to Eikonal equations.

The methods effectively handle steady and unsteady cases, including rocket motor simulations.

Abstract

A computationally efficient high-order solver is developed to compute the wall distances by solving the relevant partial differential equations, namely: Eikonal, Hamilton-Jacobi (HJ) and Poisson equations. In contrast to the upwind schemes widely used in the literature, we explore the suitability of high-order central difference schemes (explicit/compact) for the wall-distance computation. While solving the Hamilton-Jacobi equation, the high-order central difference schemes performed approximately $1.4-2.8$ times faster than the upwind schemes with a marginal improvement in the solution accuracy. A new pseudo HJ formulation based on the localized artificial diffusivity (LAD) approach has been proposed. It is demonstrated to predict results with an accuracy comparable to that of the Eikonal equation and the simulations are $\approx$ 1.5 times faster than the baseline HJ solver using upwind schemes. A curvature correction is also incorporated in the HJ equation to correct for the near-wall errors due to concave/convex wall curvatures. We demonstrate the efficacy of the proposed methods on both the steady and unsteady test cases and exploit the unsteady wall-distance solver to estimate the instantaneous shape and burning surface area of a dendrite propellant grain in a solid propellant rocket motor.