Local maximum principles in high-order non-oscillatory schemes for non-linear hyperbolic equations

Abstract

In a preceding paper, (Capdeville, 2013), and following Linde's work (Linde, 2002 [2]), we constructed an approximate Rankine–Hugoniot Riemann solver (HLL-RH) for complex and non-linear hyperbolic problems.This solver rested upon a newly defined “smoothness-indicator” directly derived from polynomial considerations. The resulting scheme so generated was third-order accurate in time and fourth-order in space, for the smooth part of the solution, while remaining stable and accurate when computing shock waves.This paper presents an extension of this framework to construct a high-order scheme that satisfies a local maximum principle while preserving its initial non-oscillatory properties. By designing a more elaborated form of the “smoothness indicator,” we can present two new numerical strategies to enforce a local maximum principle property in a non-oscillatory multi-dimensional interpolation procedure.The first algorithm starts from a weighted essentially non-oscillatory combination of quadratic polynomials and integrates a maximum principle constraint to generate a fourth-order accurate interpolation method (WENO4); the second algorithm uses a low-order monotonicity-preserving polynomial and weights it with a cubic polynomial in a non-oscillatory way (WMP4).Upon employing a robust fourth-order monotonicity-preserving algorithm for the time-integration (SSPRK), the numerical method that combines a HLL-RH Riemann solver with one of those interpolation algorithms, is fully fourth-order accurate for smooth solutions and introduces a monotonicity-preserving constraint into a high-order non-oscillatory multi-dimensional interpolation procedure.Numerical tests and comparisons with existing methods give encouraging results for the computation of non-linear hyperbolic equations in problems that traditionally generate numerical difficulties.