Higher-order accurate numerical solution of unsteady Burgers’ equation

摘要

Higher-order accurate finite-difference schemes for solving the unsteady Burgers’ equation which often arises in mathematical modeling used to solve problems in fluid dynamics are presented. The unsteady Burgers’ equation belongs to a few nonlinear partial differential equations which has an exact solution, and it allows one to compare the numerical solution with the exact one, and the properties of different numerical methods. We propose an explicit finite-difference scheme for a numerical solution of the heat equation with Robin boundary conditions. It has a sixth-order approximation in the space variable, and a third-order approximation in the time variable. As an application, we developed numerical schemes for solving a numerical solution of Burgers’ equation using the relationship between the heat and Burgers’ equations. This scheme has up to sixth-order approximation in the space variables. The main advantage of our approach is transition to one-dimensional equation which essentially reduces the computation costs compared to other direct methods for solving the unsteady Burgers’ equation. The numerical results of test examples are found in good agreement with exact solutions for a wide range of Reynolds number and confirm the approximation orders of the schemes proposed.