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A Novel Three-Level Time-Split MacCormack Method for Solving Two-Dimensional Viscous Coupled Burgers Equations

Eric Ngondiep

TL;DR

The three-level explicit time-split MacCormack procedure in the numerical solutions of two-dimensional viscous coupled Burgers' equations subject to initial and boundary conditions shows the efficiency and effectiveness of the considered method compared to a large set of numerical schemes widely studied in the literature.

Abstract

In this paper, we analyze the three-level explicit time-split MacCormack procedure in the numerical solutions of two-dimensional viscous coupled Burgers' equations subject to initial and boundary conditions. The differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second order accurate in time and fourth convergent in space, while it is second order convergent in both time and space for high Reynolds numbers problems. This shows the efficiency and effectiveness of the considered method compared to a large set of numerical schemes widely studied in the literature for solving the two-dimensional time dependent nonlinear coupled Burgers' equations. Numerical examples which confirm the theoretical results are presented and discussed.

A Novel Three-Level Time-Split MacCormack Method for Solving Two-Dimensional Viscous Coupled Burgers Equations

TL;DR

The three-level explicit time-split MacCormack procedure in the numerical solutions of two-dimensional viscous coupled Burgers' equations subject to initial and boundary conditions shows the efficiency and effectiveness of the considered method compared to a large set of numerical schemes widely studied in the literature.

Abstract

In this paper, we analyze the three-level explicit time-split MacCormack procedure in the numerical solutions of two-dimensional viscous coupled Burgers' equations subject to initial and boundary conditions. The differential operators split the two-dimensional problem into two pieces so that the two-step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second order accurate in time and fourth convergent in space, while it is second order convergent in both time and space for high Reynolds numbers problems. This shows the efficiency and effectiveness of the considered method compared to a large set of numerical schemes widely studied in the literature for solving the two-dimensional time dependent nonlinear coupled Burgers' equations. Numerical examples which confirm the theoretical results are presented and discussed.

Paper Structure

This paper contains 4 sections, 1 theorem, 100 equations, 4 figures.

Table of Contents

  1. Introduction and motivation
  2. Detailed description of the three-level time-split scheme
  3. Numerical experiments and Convergence rate
  4. General conclusion and future works

Key Result

Lemma 2.1

Let $u_{ij}^{n}=u(x_{i},y_{j},t^{n})$ and $v_{ij}^{n}=v(x_{i},y_{j},t^{n})$ be the solutions satisfying equations $()$ and $()$, respectively. Thus, it holds: and where the linear operators $\delta_{x},$$\delta^{x}$ and $\delta^{2}_{x},$ are given by equation $().$

Figures (4)

  • Figure 1: Graphs corresponding to a three-level time-split MacCormack method
  • Figure 2: Graphs corresponding to a three-level time-split MacCormack method
  • Figure 3: Graphs corresponding to a three-level time-split MacCormack method
  • Figure 4: Graphs corresponding to a three-level time-split MacCormack method

Theorems & Definitions (2)

  • Lemma 2.1
  • proof