New numerical methods for Burgers' equation based on semi-Lagrangian and modified equation approaches

In this paper, we develop a class of semi-Lagrangian finite difference schemes which are derived by a new algorithm based on the modified equation technique; and we apply those methods to the Burgers' equation. We show that the overall accuracy of the proposed semi-Lagrangian schemes depends on two factors: one is the global truncation error which can be obtained by the modified equation analysis, the other is a generic feature of semi-Lagrangian methods which characterizes their non-monotonic dependence on the time stepsize. The analytical results are confirmed by numerical tests.