The Neumann boundary condition for the two-dimensional Lax-Wendroff scheme. II

TL;DR

This work extends the energy-method stability analysis for the 2D Lax-Wendroff scheme to second-order extrapolation boundary conditions in a quarter-plane. By introducing a boundary-aware modified energy and a carefully crafted norm, the authors derive an energy inequality that decays under a small-CFL regime, accounting for interior, boundary, and corner contributions. The main contribution is a rigorous stability estimate (under explicit CFL-type bounds and a smallness condition on $\alpha^2+\beta^2$) that preserves second-order accuracy near boundaries and at the corner, providing a prototype approach for high-order multi-dimensional boundary treatments. The results offer guidance for CFL selection and boundary design in transport problems with corners and motivate further refinements (e.g., stronger corner absorption) and GKS-style analyses.

Abstract

We study the stability of a two-dimensional Lax-Wendroff scheme in a quarter-plane. Following our previous work, we aim here at adapting the energy method in order to study second order extrapolation boundary conditions. We first show on the one-dimensional problem why modifying the energy is a necessity in order to obtain stability estimates. We then study the two-dimensional case and propose a modified energy as well as second order extrapolation boundary and corner conditions in order to maintain second order accuracy and stability of the whole scheme, including near the corner.

Figures (5)

• Figure 3.1: The spatial grid for the quarter-plane. Interior cells appear in blue, the boundary ghost cells appear in red and the corner ghost cell appears in green. The value $u_{j,k}^n$ corresponds to the approximation in the cell $[n\,\Delta t \, , \, (n+1)\, \Delta t) \times [j \Delta x,(j+1) \Delta x) \times [k \Delta y,(k+1) \Delta y)$.
• Figure 3.2: An illustration of admissible CFL parameters. The red area corresponds to the CFL parameters for which Theorem \ref{['thm1']} holds, the blue one to the optimal set of parameters (for which stability for the Cauchy problem holds).
• Figure 4.1: Negativity of the quadratic form associated with the corner contribution $\mathcal{C}$. In dark blue: the exterior of the ball. In yellow: the parameters $(\lambda \, |a|,\mu \, |b|)$ for which \ref{['CFLcauchy']} holds and the quadratic form is negative definite. In light blue: the parameters $(\lambda \, |a|,\mu \, |b|)$ for which \ref{['CFLcauchy']} holds and the quadratic form is not negative definite.
• Figure 4.2: Negativity of the quadratic form associated with the reduced corner contribution $\widetilde{\mathcal{C}}$. In dark blue: the exterior of the ball. In yellow: the parameters $(\lambda \, |a|,\mu \, |b|)$ for which \ref{['CFLcauchy']} holds and the quadratic form is negative definite. In light blue: the parameters $(\lambda \, |a|,\mu \, |b|)$ for which \ref{['CFLcauchy']} holds and the quadratic form is not negative definite.
• Figure 4.3: Negativity of the simplified boundary contribution $\widetilde{\mathcal{B}}$. In dark blue: the exterior of the ball. In yellow: the parameters $(\lambda \, |a|,\mu \, |b|)$ for which \ref{['CFLcauchy']} holds and the quadratic form is negative definite. In light blue: the parameters $(\lambda \, |a|,\mu \, |b|)$ for which \ref{['CFLcauchy']} holds and the quadratic form is not negative definite.

Theorems & Definitions (7)

• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof