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A hybrid approach for singularly perturbed parabolic problem with discontinuous data

Nirmali Roy, Anuradha Jha

Abstract

In this article, we study a two-dimensional singularly perturbed parabolic equation of the convection-diffusion type, characterized by discontinuities in the source term and convection coefficient at a specific point in the domain. These discontinuities lead to the development of interior layers. To address these layers and ensure uniform convergence, we propose a hybrid monotone difference scheme that combines the central difference and midpoint upwind schemes for spatial discretization, applied on a piecewise-uniform Shishkin mesh. For temporal discretization, we employ the Crank-Nicolson method on a uniform mesh. The resulting scheme is proven to be uniformly convergent, order achieving almost two in space and two in time. Numerical experiments validate the theoretical error estimates, demonstrating superior accuracy and convergence when compared to existing methods.

A hybrid approach for singularly perturbed parabolic problem with discontinuous data

Abstract

In this article, we study a two-dimensional singularly perturbed parabolic equation of the convection-diffusion type, characterized by discontinuities in the source term and convection coefficient at a specific point in the domain. These discontinuities lead to the development of interior layers. To address these layers and ensure uniform convergence, we propose a hybrid monotone difference scheme that combines the central difference and midpoint upwind schemes for spatial discretization, applied on a piecewise-uniform Shishkin mesh. For temporal discretization, we employ the Crank-Nicolson method on a uniform mesh. The resulting scheme is proven to be uniformly convergent, order achieving almost two in space and two in time. Numerical experiments validate the theoretical error estimates, demonstrating superior accuracy and convergence when compared to existing methods.

Paper Structure

This paper contains 7 sections, 14 theorems, 107 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Analytical aspects of the solution
  3. Semi discretization
  4. Discretization in space
  5. Truncation error analysis
  6. Numerical examples
  7. Conclusion

Key Result

Lemma 2.1

Suppose that a function $y(x,t)\in C^{0}(\bar{\Omega})\cap C^{2}(\Omega^{-}\cup \Omega^{+})$ satisfies $y(x,t)\le0,~(x,t)\in\Gamma_{c}, ~[\frac{\delta y}{\delta x}](d,t)\ge0,t>0,~~\mathcal{L}y(x,t)\ge0,~(x,t)\in \Omega^{-}\cup\Omega^{+},$ then $y(x,t)\ge0,~\forall(x,t)\in\bar{\Omega}.$

Figures (4)

  • Figure 1: Numerical solution for $\epsilon=2^{-16}$when $N=64$ for Example \ref{['ex-a']}.
  • Figure 2: Error for $\epsilon=2^{-16}$ when $N=64$ for Example \ref{['ex-a']}.
  • Figure 3: Numerical solution for $\epsilon=2^{-22}$ when $N=64$ for Example \ref{['ex-b']}.
  • Figure 4: Error for $\epsilon=2^{-22}$ when $N=64$ for Example \ref{['ex-b']}.

Theorems & Definitions (29)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 19 more