A new type of simplified inverse Lax-Wendroff boundary treatment I: hyperbolic conservation laws

TL;DR

This work tackles the challenge of stable, high-order boundary treatments for hyperbolic conservation laws on fixed Cartesian grids with non-aligned boundaries. It introduces a new high-order simplified ILW (SILW) boundary method that splits ghost-point construction into interpolation of interior auxiliary points and Hermite-style extrapolation using boundary derivatives, controlled by a tunable α to optimize stability. The approach extends from 1D scalars to 1D Euler and 2D Euler equations, with stability analyses showing that fewer ILW terms are needed for stability at a given order, improving efficiency. Numerical tests on 1D/2D benchmarks, including non body-fitted grids and complex flows, validate robustness, accuracy, and practical computational gains.

Abstract

In this paper, we design a new kind of high order inverse Lax-Wendroff (ILW) boundary treatment for solving hyperbolic conservation laws with finite difference method on a Cartesian mesh. This new ILW method decomposes the construction of ghost point values near inflow boundary into two steps: interpolation and extrapolation. At first, we impose values of some artificial auxiliary points through a polynomial interpolating the interior points near the boundary. Then, we will construct a Hermite extrapolation based on those auxiliary point values and the spatial derivatives at boundary obtained via the ILW procedure. This polynomial will give us the approximation to the ghost point value. By an appropriate selection of those artificial auxiliary points, high-order accuracy and stable results can be achieved. Moreover, theoretical analysis indicates that comparing with the original ILW method, especially for higher order accuracy, the new proposed one would require fewer terms using the relatively complicated ILW procedure and thus improve computational efficiency on the premise of maintaining accuracy and stability. We perform numerical experiments on several benchmarks, including one- and two-dimensional scalar equations and systems. The robustness and efficiency of the proposed scheme is numerically verified.

Figures (14)

• Figure 2.1: The third order scheme with the new SILW procedure and $k_d=2$. The horizontal axis represents $C_a$ and the vertical axis represents the largest $|z(\mu)|$.
• Figure 2.2: Error contour of the linear problem with the third-order boundary treatment.
• Figure 2.3: Error contour of the linear problem with the fifth-order boundary treatment.
• Figure 3.1: The local coordinate rotation diagram.
• Figure 3.2: Two dimensional new SILW method diagram