[Paper Review] A Narrow-stencil finite difference method for approximating viscosity solutions of fully nonlinear elliptic partial differential equations with applications to Hamilton-Jacobi-Bellman equations

Xiaobing Feng, Thomas Lewis|arXiv (Cornell University)|Jul 24, 2019

Advanced Mathematical Modeling in Engineering27 references4 citations

TL;DR

This paper introduces a novel narrow-stencil finite difference method for approximating viscosity solutions of fully nonlinear second-order elliptic PDEs, including Hamilton-Jacobi-Bellman (HJB) equations. By incorporating a numerical moment—defined as the difference of two central Hessian approximations—the scheme stabilizes low-regularity solutions, ensuring well-posedness, stability in ℓ² and ℓ∞ norms, and convergence via a discrete comparison principle, even without monotonicity.

ABSTRACT

This paper presents a new narrow-stencil finite difference method for approximating the viscosity solution of second order fully nonlinear elliptic partial differential equations including Hamilton-Jacobi-Bellman equations. The proposed finite difference method naturally extends the Lax-Friedrichs method for first order problems to second order problems by introducing a key stabilization and guiding term called a "numerical moment". The numerical moment uses the difference of two (central) Hessian approximations to resolve the potential low-regularity of viscosity solutions. It is proved that the proposed scheme is well posed (i.e, it has a unique solution) and stable in both the l-2 norm and the l-infinity norm. The highlight of the paper is to prove the convergence of the proposed scheme to the viscosity solution of the underlying fully nonlinear second order problem using a novel discrete comparison argument. This paper extends the one-dimensional analogous method of Feng, Kao, and Lewis to the higher-dimensional setting. Numerical tests are presented to gauge the performance of the proposed finite difference methods and to validate the convergence result of the paper.

Motivation & Objective

• To develop a convergent narrow-stencil finite difference method for fully nonlinear second-order elliptic PDEs, including HJB equations, in dimensions d ≥ 2.
• To overcome the limitation of wide-stencil schemes by enabling narrow-stencil methods without relying on monotonicity.
• To extend the one-dimensional Lax-Friedrichs-like method from [14] to higher dimensions, addressing challenges from off-diagonal Hessian terms.
• To establish convergence for non-monotone schemes using a novel discrete comparison argument, bypassing the Barles-Souganidis framework.
• To validate the method numerically on problems with low-regularity solutions and degenerate or non-diagonally dominant coefficient matrices.

Proposed method

• The method introduces a numerical moment as a stabilization and guiding term, defined as the difference between two central Hessian approximations.
• The numerical moment acts as a high-order linear perturbation that enhances resolution of low-regularity solutions on narrow stencils.
• Multiple discrete approximations of ∇u and D²u are used locally to capture function behavior on a fixed grid, compensating for narrow-stencil limitations.
• The scheme is constructed to be well-posed and stable in both ℓ² and ℓ∞ norms, ensuring robust numerical behavior.
• A new discrete comparison principle is developed to prove convergence, replacing the need for monotonicity in the Barles-Souganidis framework.
• The method is extended to parabolic problems via the method of lines, enabling application to time-dependent HJB equations.

Experimental results

Research questions

• RQ1 Can a narrow-stencil finite difference scheme converge to the viscosity solution of a fully nonlinear second-order elliptic PDE without requiring monotonicity?
• RQ2 How can a numerical moment be designed to stabilize low-regularity solutions in higher-dimensional finite difference schemes?
• RQ3 Can the discrete comparison principle be generalized to non-monotone schemes to ensure convergence?
• RQ4 What is the convergence rate of the proposed scheme for problems with non-smooth or degenerate solutions?
• RQ5 Can the method be applied to HJB equations with non-diagonally dominant or degenerate diffusion matrices?

Key findings

• The proposed scheme is well-posed, with a unique solution, and stable in both ℓ² and ℓ∞ norms.
• The scheme achieves convergence to the viscosity solution via a novel discrete comparison argument, even without monotonicity.
• Numerical tests on a problem with a non-diagonally dominant diffusion matrix show nearly second-order convergence (order ≈ 1.97) for h = 2.89e-02.
• For a degenerate elliptic problem with a solution in W^{4/3,p} but not in C², the method achieves convergence order ≈ 0.40, consistent with the solution’s low regularity.
• The method successfully stabilizes numerical artifacts in low-regularity regimes, as demonstrated in tests involving the infinite Laplacian and non-divergence form linear PDEs.
• The numerical moment acts as a low-regularity indicator and enables robust performance across a wide class of fully nonlinear and linear non-divergence form PDEs.