Numerical moment stabilization of central difference approximations for linear stationary reaction-convection-diffusion equations with applications to stationary Hamilton-Jacobi equations

TL;DR

This paper introduces a numerical stabilization technique called a numerical moment for central difference schemes applied to reaction-convection-diffusion equations, improving convergence rates especially in convection-dominated scenarios and for Hamilton-Jacobi equations.

Contribution

The paper develops a high-order stabilized finite difference method using a numerical moment, enhancing stability and convergence for convection-dominated problems and Hamilton-Jacobi equations.

Findings

Higher-order local truncation errors achieved.

Demonstrated improved convergence rates over Lax-Friedrich's method.

Effective for both smooth and non-smooth viscosity solutions.

Abstract

Linear stationary reaction-convection-diffusion equations with Dirichlet boundary conditions are approximated using a simple finite difference method corresponding to central differences and the addition of a high-order stabilization term called a numerical moment. The focus is on convection-dominated equations, and the formulation for the method is motivated by various results for fully nonlinear problems. The method features higher-order local truncation errors than monotone methods consistent with the use of the central difference approximation for the gradient. Stability and rates of convergence are derived in the $\ell^2$ norm for the constant-coefficient case. Numerical tests are provided to compare the new methods to monotone methods. The methods are also tested for stationary Hamilton-Jacobi equations where they demonstrate higher rates of convergence than the Lax-Friedrich's method when the underlying viscosity solution is smooth and comparable performance when the underlying viscosity solution is not smooth.