A high-order monotonicity-preserving scheme for hyperbolic conservation laws

Abstract

This work aims at designing a numerical model for discretizing non-linear hyperbolic systems of conservation laws, with high-order accuracy, while verifying a positivity property of the discretization. From a finite-volume discretization of the PDEs, we identify three positivity constraints for the resulting numerical scheme to be a convex combination of the stencil supporting the discretization. These constraints are then introduced into the MUSCL-like least-square interpolation procedure by using limiters conveniently sized. Lastly, a high-order stability-preserving time integration method (SSPRK) enables to generate the final algorithm. Extensive numerical tests on scalar problems help to assess the benefits and potentialities of the resulting algorithm. Then, upon using a local diagonalization, we extend the scope of the algorithm to non linear hyperbolic systems. To this aim, we propose two strategies for identifying local directions that promotes this diagonalization. Numerical tests on standard problems of gas dynamics, help to appreciate the advantages and drawbacks of each strategy. The resulting numerical scheme is a fourth-order least-square positive scheme (LSQP4) and it is designed and formulated for discretizing unsteady multi-dimensional hyperbolic conservation laws, over non-Cartesian meshes.