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A finite difference scheme for two-dimensional singularly perturbed convection-diffusion problem with discontinuous source term

Ram Shiromani, Niall Madden, V. Shanthi

TL;DR

This work addresses a two-dimensional singularly perturbed convection-diffusion equation on the unit square with discontinuous forcing, where $0<\epsilon<1$ multiplies the diffusion terms. It develops a fitted, piecewise-uniform Shishkin mesh and a modified upwind finite difference scheme to resolve both exponential and characteristic interior layers, achieving parameter-robust, nearly first-order convergence. A rigorous stability analysis and a solution decomposition into regular and singular components underpin an epsilon-uniform error bound of the form $\|U-u\|_{\bar{\Omega}^{N,N}} \le C N^{-1}\ln^2 N$, validated by numerical experiments with constant and variable coefficients. The results demonstrate that the method accurately resolves interacting interior layers and discontinuities, providing a reliable tool for simulations of SPDEs in engineering contexts.

Abstract

We propose a finite difference scheme for the numerical solution of a two-dimensional singularly perturbed convection-diffusion partial differential equation whose solution features interacting boundary and interior layers, the latter due to discontinuities in source term. The problem is posed on the unit square. The second derivative is multiplied by a singular perturbation parameter, $ε$, while the nature of the first derivative term is such that flow is aligned with a boundary. These two facts mean that solutions tend to exhibit layers of both exponential and characteristic type. We solve the problem using a finite difference method, specially adapted to the discontinuities, and applied on a piecewise-uniform (Shishkin). We prove that that the computed solution converges to the true one at a rate that is independent of the perturbation parameter, and is nearly first-order. We present numerical results that verify that these results are sharp.

A finite difference scheme for two-dimensional singularly perturbed convection-diffusion problem with discontinuous source term

TL;DR

This work addresses a two-dimensional singularly perturbed convection-diffusion equation on the unit square with discontinuous forcing, where multiplies the diffusion terms. It develops a fitted, piecewise-uniform Shishkin mesh and a modified upwind finite difference scheme to resolve both exponential and characteristic interior layers, achieving parameter-robust, nearly first-order convergence. A rigorous stability analysis and a solution decomposition into regular and singular components underpin an epsilon-uniform error bound of the form , validated by numerical experiments with constant and variable coefficients. The results demonstrate that the method accurately resolves interacting interior layers and discontinuities, providing a reliable tool for simulations of SPDEs in engineering contexts.

Abstract

We propose a finite difference scheme for the numerical solution of a two-dimensional singularly perturbed convection-diffusion partial differential equation whose solution features interacting boundary and interior layers, the latter due to discontinuities in source term. The problem is posed on the unit square. The second derivative is multiplied by a singular perturbation parameter, , while the nature of the first derivative term is such that flow is aligned with a boundary. These two facts mean that solutions tend to exhibit layers of both exponential and characteristic type. We solve the problem using a finite difference method, specially adapted to the discontinuities, and applied on a piecewise-uniform (Shishkin). We prove that that the computed solution converges to the true one at a rate that is independent of the perturbation parameter, and is nearly first-order. We present numerical results that verify that these results are sharp.
Paper Structure (10 sections, 17 theorems, 163 equations, 4 figures, 2 tables)

This paper contains 10 sections, 17 theorems, 163 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The continuous problem
  3. Stability of the differential operator
  4. Solution decomposition
  5. Numerical method
  6. A fitted mesh
  7. The finite difference method
  8. Error analysis
  9. Numerical Experiments
  10. Conclusions

Key Result

Theorem 2.1

(Maximum principle) Let $L_{\epsilon}$ be the differential operator given in (eqncon). If $\phi(x,y)\geq 0$ on $\partial \Omega$, $L_{\epsilon}\phi(x,y)\geq 0$ for all $(x,y)\in \Omega$, $[\phi](d_1,y)=[\phi](x,d_2)=0$, $[\frac{\partial \phi}{\partial x}](d_1,y)\leq 0$ and $[\frac{\partial \phi}{\p

Figures (4)

  • Figure 1: Notation for subregions and boundaries the domains
  • Figure 2: Surface and contour plots of numerical solution to \ref{['ex1']} with $\epsilon = 10^{-2}$ and $N=128$
  • Figure 3: Surface graph and contour plots of numerical solution to \ref{['ex2']} with $\epsilon = 10^{-2}$ and $N=128$
  • Figure 4: Error graphs of numerical solution with $\epsilon = 10^{-2}$ and $N=128$

Theorems & Definitions (32)

  • Theorem 2.1
  • proof
  • Lemma 2.2: Stability
  • proof
  • Lemma 2.3
  • Theorem 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 22 more