An accurate approximation algorithm for Burgers' equation in the presence of small viscosity

Most of the existing numerical schemes developed to solve Burgers' equation cannot exhibit its correct physical behavior for very small values of viscosity. This difficulty can be overcome by using splitting methods derived for near-integrable system. This class of methods has positive real coefficients and can be used for non-reversible systems such as Burgers' equation. It also has the advantage of being able to account small viscosity in the accuracy. The algorithm is based on the combination of implicit-explicit finite difference schemes to solve each simplified problem and filtering technique to treat nonlinear instability. The resulting algorithm is accurate, efficient and easy to implement. The new numerical results are compared with numerical and exact solutions reported in the literature and found that they are very accurate for small values of the viscosity.