A First-Order System Approach for Burgers’ Equation

International Journal of Research and Scientific Innovation (IJRSI) | Volume V, Issue VII, July 2018 | ISSN 2321–2705

A First-Order System Approach for Burgers’ Equation

Rajashekhar P L

Assistant Professor, Department of Mathematics, RVCE, Bengaluru, Karnataka, India

Abstract: In this paper, we try to use a first-order system approach for Burgers’ equation. A non-linear partial differential equation is converted into a system of linear equations with non-linear source terms, known as a ‘relaxation system’. The new strategy is to solve an equivalent first order hyperbolic system instead of the second-order non-linear partial differential equation. The numerical solutions obtained through this approach have same order accuracy in space and time.

Key words: Relaxation system, operator splitting method, Strang splitting, Discrete Boltzmann equation

I. INTRODUCTION

The non-linear parabolic partial differential equation

∂u/∂t+u ∂u/∂x=k((∂^2 u)/(∂x^2 )) … (1),

where k > 0 is the constant is known as Burgers’ equation. It arises in applications modelling traffic flow, fluid flow in certain conditions, atmospheric behaviour, and many other physical systems. This equation appears often as a simplification of a more sophisticated model. It is one of the simplest partial differential equation that combining both non-linear propagation effects and diffusive effects.

II. FIRST-ORDER SYSTEM APPROACH FOR BURGERS’ EQUATION

Burgers’ equation can be solved by ‘operator splitting method’. Operator splitting is a powerful method for numerical investigation of complex models. The basic idea of the operator splitting methods is based on splitting of complex problem into a sequence of simpler split sub-problems. The sub operators are usually chosen with regard to different physical process. The technique is generally used in one of the two ways: It is used in methods in which one splits the differential operator such that each split system only involves derivatives along on of the coordinate axes. Alternatively, it is used as a means to split the differential operator into several parts, where each part represents a particular physical phenomenon, such as convection, diffusion, etc. In either case, the corresponding numerical method is defined as a sequence of solver of each of the split problems. This can lead to very efficient methods, since one can treat each part of the original operator independently.

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