Finite Difference Method for The Burgers Equation

Burgers equation: u_t+ uu_x= λu_xx is a fundamental partial differential equation arising from fluid mechanics. It occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow. It was named after Johannes Martinus Burgers (1895–1981). J. M. Burgers did studies on the equation in 1940s principally as a model problem of the interaction between nonlinear and dissipative phenomena.

The equation arises in model studies of turbulence and shock wave theory. In physical application of shock waves in fluids, coefficient λ has the meaning of viscosity. For light fluids or gases the solution considers the inviscid limit as λ tends to zero.

The solution of the Burgers equation is classified into two categories: numerical solutions and analytic solutions. In both methods, the solutions have been valid for λ ɛ (0, 1).

In this paper we have solved the Burgers equation using finite difference methods where λ is not restricted to the interval (0, 1). In this work we have managed to solve the Burgers equation with λ ɛ (0, 10/3). The methods involved developing a finite difference scheme, analyzing the scheme for stability and solving the resulting system of equations using Mathcad 2000 professional. It is our hope that this will be of great contribution to the mathematical knowledge in the application of the Burgers equation.