A truncation error analysis of third-order MUSCL scheme for nonlinear conservation laws

This article is a rebuttal to the claim found in the literature that the monotonic upstream-centered scheme for conservation laws (MUSCL) cannot be third-order accurate for nonlinear conservation laws. We provide a rigorous proof for third-order accuracy of the MUSCL scheme based on a careful and detailed truncation error analysis. Throughout the analysis, the distinction between the cell average and the point value will be strictly made for the numerical solution as well as for the target operator. It is shown that the average of the solutions reconstructed at a face by Van Leer's kappa-scheme recovers a cubic solution exactly with kappa=1/3, the same is true for the average of the nonlinear fluxes evaluated by the reconstructed solutions, and a dissipation term is already sufficiently small with a third-order truncation error. Finally, noting that the target spatial operator is a cell-averaged flux derivative, we prove that the leading truncation error of the MUSCL finite-volume scheme is third-order with kappa=1/3. The importance of the diffusion scheme is also discussed: third-order accuracy will be lost when the third-order MUSLC scheme is used with an incompatible fourth-order diffusion scheme for convection-diffusion problems. Third-order accuracy is verified by thorough numerical experiments for both steady and unsteady problems. This article is intended to serve as a reference to clarify confusions about third-order accuracy of the MUSCL scheme, as a guide to correctly analyze and verify the MUSCL scheme for nonlinear equations, and eventually as the basis for clarifying high-order unstructured-grid schemes in a subsequent article.

Automated summary

This article provides a rigorous proof for the third-order accuracy of the MUSCL scheme for nonlinear conservation laws through detailed truncation error analysis. It also highlights the importance of compatible diffusion schemes to maintain third-order accuracy, as using incompatible schemes could lead to a loss of accuracy.