A Robust Second-Order Godunov-Type Method for Burgers’ Equation

Details are given of the development of a finite-volume method for solving one-dimensional Burgers’ equation with second-order accuracy in space. Reconstructed with a TVD piecewise linear function, the equation is solved through Godunov’s Reconstruct-Evolve-Average algorithm. For a piecewise linear profile, the Riemann problem raised at cell boundaries does not have an exact solution and is always solved approximately. A novel method for the approximate solution of the piecewise linear profile is proposed. In this method, the conservation law inside the cells, by a physically meaningful assumption, is reduced to a simple ODE, which is solved analytically. From these ODEs, a robust but straightforward numerical flux function is derived, which is exponential with respect to the cell slope. Numerical experiments show the good performance of the method. The results are also compared with other second-order finite volume methods. It is shown that, in the solutions associated with shockwaves, the proposed method yields smaller L∞ error than other methods, which implies that the method is competently able to capture shockwaves. © 2022, The Author(s), under exclusive licence to Springer Nature India Private Limited.