A high order correction to the Lax-Friedrich's method for approximating stationary Hamilton-Jacobi equations

TL;DR

This work addresses the challenge of solving nondegenerate stationary Hamilton-Jacobi equations with Dirichlet data beyond the first-order accuracy limit of monotone schemes. It introduces two finite difference schemes that incorporate a high-order correction via a numerical moment to the Lax-Friedrichs method, complemented by a cutoff mechanism to preserve admissibility and stability. The authors develop a Schauder-fixed-point framework to prove admissibility, stability, and convergence for the modified scheme, and demonstrate through extensive 1D and 2D numerical experiments that the non-monotone methods can achieve up to second-order accuracy for smooth solutions and robustly approximate low-regularity viscosity solutions. The work also connects the high-order corrections to vanishing viscosity concepts and outlines a practical path to extending these ideas to nonuniform grids and to dynamic Hamilton-Jacobi problems. Overall, the paper provides a theoretical and computational template for leveraging high-order non-monotone stabilizers to surpass the Godunov barrier in stationary Hamilton-Jacobi computations, with broad implications for applications in control, optics, and imaging.

Abstract

A new class of non-monotone finite difference (FD) approximation methods for approximating solutions to non-degenerate stationary Hamilton-Jacobi problems with Dirichlet boundary conditions is proposed and analyzed. The new FD methods add a high order correction to the Lax-Friedrich's method while utilizing a novel cutoff to preserve the convergence properties of the Lax-Friedrich's approximation. Since monotone methods are limited to first order accuracy by the Godunov barrier, the proposed approach provides a template for boosting the accuracy of a monotone method using a modified numerical moment stabilizer with a high-order auxiliary boundary condition. Numerical tests are provided to test the utility of the approach while a novel admissibility and stability analysis technique lays a foundation for analyzing non-monotone methods.

Figures (3)

• Figure 3.1: A two-dimensional example of the mesh $\mathcal{T}_{\mathbf{h}}$ and enforcement of the auxiliary boundary condition \ref{['bc2a']} or \ref{['bc2b']}. The solid line corresponds to $\partial \Omega$. The black nodes are the unknown values in $\mathcal{T}_{\mathbf{h}} \cap \Omega$. The red nodes are the Dirichlet boundary data in $\mathcal{T}_{\mathbf{h}} \cap \partial \Omega$. The blue nodes represent ghost points and are uniquely determined by the choice of the auxiliary boundary condition.
• Figure 4.1: Sample plots locally bounding the values of $U_{\alpha \pm \mathbf{e}_i}$ to ensure $U$ solves both \ref{['FD_scheme']} and \ref{['FD_filtered']}. The values for $U_{\alpha \pm 2 \mathbf{e}_i}$ are fixed. The value for $U_\alpha$ can range over the entire interval $[\underline{U}_\alpha , \overline{U}_\alpha]$ which is shaded in cyan. The value for $U_{\alpha \pm \mathbf{e}_i}$ can range over the blue subintervals that are determined by the values of $U_{\alpha \pm 2\mathbf{e}_i}$, $\overline{U}_\alpha$, and $\underline{U}_\alpha$.
• Figure 5.1: Approximations for Example 2 in one dimension using the Lax-Friedrich's method on the left, the modified high-order corrector scheme \ref{['FD_filtered']} with $c=2$ in the middle, and the modified high-order corrector scheme \ref{['FD_filtered']} with $c=10$ on the right. The approximation with $c=10$ agrees with the solution to the proposed high order scheme \ref{['FD_scheme']}. All approximations correspond to $h$=2.53e-02. Similar results are observed for both boundary conditions \ref{['bc2a']} and \ref{['bc2b']}.

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Remark 3.1
• Lemma 3.1
• proof
• Corollary 3.2
• Theorem 4.1
• proof
• Theorem 4.2
• proof