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

T. Lewis, X. Xue

TL;DR

This work introduces a numerical moment stabilization appended to a central-difference discretization of the convection-diffusion-reaction operator $L^{\epsilon}[u]$ to achieve higher-order accuracy in convection-dominated regimes while preserving stability in the vanishing-viscosity sense. By coupling a high-order stabilization term $M_{\mathbf{h}}^{p}$ with an auxiliary boundary operator, the authors obtain $\ell^{2}$-stable schemes that remain consistent for $\epsilon\ge0$ and extend naturally to the stationary Hamilton–Jacobi setting. They provide detailed consistency analyses, matrix-property observations, and a rigorous $\ell^{2}$-stability/convergence result for constant coefficients, along with extensive 1D and 2D numerical experiments showing improved accuracy on coarse meshes and robustness across smooth and nonsmooth viscosity solutions. The approach offers a practical, narrowly-stenciled alternative to monotone methods, with potential extensions to more advanced discretizations and fully nonlinear problems, including DG frameworks and rigorous admissibility results for HJ equations.

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.

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

TL;DR

This work introduces a numerical moment stabilization appended to a central-difference discretization of the convection-diffusion-reaction operator to achieve higher-order accuracy in convection-dominated regimes while preserving stability in the vanishing-viscosity sense. By coupling a high-order stabilization term with an auxiliary boundary operator, the authors obtain -stable schemes that remain consistent for and extend naturally to the stationary Hamilton–Jacobi setting. They provide detailed consistency analyses, matrix-property observations, and a rigorous -stability/convergence result for constant coefficients, along with extensive 1D and 2D numerical experiments showing improved accuracy on coarse meshes and robustness across smooth and nonsmooth viscosity solutions. The approach offers a practical, narrowly-stenciled alternative to monotone methods, with potential extensions to more advanced discretizations and fully nonlinear problems, including DG frameworks and rigorous admissibility results for HJ equations.

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 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.

Paper Structure

This paper contains 19 sections, 7 theorems, 63 equations, 2 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Difference operators
  4. The Lax-Friedrich's method and monotonicity
  5. Formulation
  6. Consistency results
  7. Matrix properties of the numerical moment
  8. Admissibility, stability, and convergence for constant coefficients
  9. Numerical Experiments
  10. One-Dimensional Tests
  11. Example 1: Problem \ref{['RCD']} with a smooth solution and $b$ positive
  12. Example 2: Problem \ref{['RCD']} with a smooth solution and $b$ sign-changing
  13. Example 3: Problem \ref{['RCD']} with a boundary layer as $\epsilon \to 0$
  14. Example 4: Non-smooth solution with Dirichlet data
  15. Two-Dimensional Tests
  16. ...and 4 more sections

Key Result

Lemma 4.1

The scheme HJeps_fd has a second order $\ell^\infty$ local truncation error over $\mathring{\mathcal{T}_{\mathbf{h}}}$. The boundary condition bc2a gives a $p$ order $\ell^\infty$ local truncation error over $( \mathcal{T}_{\mathbf{h}} \setminus \mathring{\mathcal{T}_{\mathbf{h}}} ) \cap \Omega$, an

Figures (2)

  • Figure 7.1: Plots for various approximation methods and $\epsilon$ values for Example 3 in one dimension using $\epsilon_h = h^q$ and $\gamma_h = h^p$. The plots for finer meshes restrict $x$ to be near $x=1$ in the right column.
  • Figure 7.2: Approximations for Example 4 in one dimension using the Lax-Friedrich's method ($\epsilon_h = 4h, \gamma_h =0$) on the left and the proposed non-monotone method with $\epsilon_h = 4h^2, \gamma_h = 1$ on the right. Both approximations correspond to $h$=2.02e-02.

Theorems & Definitions (14)

  • Remark 2.1
  • Lemma 4.1
  • Corollary 4.2
  • Theorem 4.3
  • proof
  • Remark 4.1
  • Lemma 6.1
  • proof
  • Theorem 6.2
  • Theorem 6.3
  • ...and 4 more