Monotonicity-Preserving Lax–Wendroff Scheme for Solving Scalar Hyperbolic Conservation Laws

Abstract

In this paper, we construct the monotonicity-preserving Lax–Wendroff (MP-Lax–Wendroff) scheme based on the MP scheme as proposed by Suresh and Huynh [11], which is a high-order and high-resolution method for hyperbolic conservation laws. It is well known that the total-variation-diminishing (TVD) methods possess the order reduction at smooth extremum points. However, the MP scheme not only preserves the non-oscillatory behavior of the procedure, but also prevents the order reduction of the method. We provide an analysis of the MP procedure using the numerical Lax–Wendroff flux sophistically in detail. Due to lack of robustness of the MP scheme, a rigorous and efficient MP-Lax–Wendroff scheme is introduced to enhance the robustness of the MP process. This means that a TVD numerical flux is applied to damp the oscillations at disturbed discontinuity points to get the high-resolution property. Therefore, we apply this idea to construct the improved MP-Lax–Wendroff scheme (we call it MP-R-Lax–Wendroff scheme). The computational performance of the MP-R-Lax–Wendroff scheme for linear and nonlinear hyperbolic conservation laws is conducted through some prototype examples which in turn validate our theoretical results.