An error predict-correction formula of the load vector in the BSLM for solving three-dimensional Burgers’ equations

Abstract

This paper aims to develop an algorithm reducing the computational cost of the backward semi-Lagrangian method for solving nonlinear convection–diffusion equations. For this goal, we introduce an error predict-correction formula (EPCF) of the load term for the Helmholtz system. The EPCF is built up to involve the same values as solving the perturbed Cauchy problem, which allows the reuse of the values. These reuses dramatically reduce the computational cost by eliminating the interpolation processes to evaluate the load term in most computational domains. The algorithm also produces a stable solution, even where the original solution has a sharp gradient, by introducing an indicator to distinguish the sharpness of the solution. Furthermore, the proposed algorithm discusses fast solvers of the discrete systems for the perturbed Cauchy problem, for which we utilize an eigenvalue decomposition theory. To show the adaptability and efficiency of the proposed algorithm, we apply it to the convection–diffusion equations such as two and three-dimensional viscous Burgers’ equations. Through several simulations, we confirm that the proposed algorithm for the load term radically reduces the computational cost, and the indicator for stiff components guarantees precision.