Fast algorithms for coupled Burgers' equations by sharing characteristic curves within BSLM

Abstract

This paper introduces a new perspective of the traditional view on the velocity of each physical particle in the coupled Burgers' equation in the backward semi-Lagrangian method (BSLM). The proposed methods reduce the number of Cauchy problems to be solved by observing a single virtual characteristic curve with a velocity. This can drastically reduce the computational cost of determining the departure point. And then, we solve the derived system reflected by the single virtual characteristic curve. Moreover, an efficient strategy for the derived linear system of equations is provided. Four examples are tested to demonstrate the adaptability and efficiency of the proposed method. The test results show that the proposed method has third-order accuracy in time and space. In addition, compared with the existing method of solving the problem along two particles with different velocities, we confirm that the proposed method significantly reduces computational cost while maintaining accuracy well.