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

TL;DR

This work addresses efficient wall-distance computations by solving DE-based formulations (Eikonal, Hamilton-Jacobi, Poisson) with high-order central difference schemes. It demonstrates that high-order central schemes, particularly for the HJ equation, yield substantial speedups (approximately $1.4$–$2.8$ times) with marginal accuracy gains over upwind baselines, while a Localized Artificial Diffusivity (LAD) based HJ formulation can match Eikonal accuracy and run about $1.5\times$ faster than the baseline HJ solver. The study also shows that Eikonal solutions with high-order schemes incur some dispersion-related errors due to the lack of diffusion, whereas the LAD/HJ approach provides robust, accurate results across steady and unsteady cases, including dendrite grain burnback in solid rocket motors. Curvature corrections based on Nakanishi further improve wall-distance accuracy near highly curved walls, making the methods attractive for CAD, mesh generation, and overset-grid applications where accurate wall-distance away from walls is critical.

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.

Figures (17)

• Figure 1: Algorithm flow chart for (a) Baseline UW solver and (b) Enhanced high-order solver
• Figure 2: Wall distance field (a,b,c) and the corresponding error field (d,e,f) for 3D box case (case-1) obtained by solving Eikonal, Hamilton-Jacobi equation and Poisson equation with the baseline solver (g) Comparison of wall distance along the mid-vertical line with the exact solution.
• Figure 3: Wall distance field obtained using (a) Eikonal, (b) Hamilton-Jacobi and (c) Poisson equations for complex geometries case (case-2).
• Figure 4: (a) Contours of wall distance field and absolute wall distance error. Variation of (b) wall distance and (c) absolute wall distance error along mid vertical line (marked with dashed lines in contour plots) for flat plate, channel and 2D box cases.
• Figure 5: (a) $L^2$ norm error with varying grid resolutions and (b) CPU time taken per 100 iterations with different schemes for channel case.