A Weighted Compact Finite Volume Scheme for Hyperbolic Conservation Laws

Abstract

In this paper, we introduce a novel fifth-order weighted compact finite volume scheme that integrates the concept of adaptive order weighted essentially non-oscillatory (WENO) schemes into existing linear compact schemes. Our proposed method employs a new stencil selection criterion for the smoothness indicator, resulting in a convex combination of one fifth-order linear compact reconstruction and two third-order linear reconstructions. This innovative combination preserves the high-resolution characteristics of compact schemes while effectively capturing discontinuities with the accuracy of WENO methods. To demonstrate the efficiency of our scheme, we applied it to solve a range of nonlinear scalar equations and systems of Euler equations in both one- and two-dimensional spaces. The results show that the intended fifth-order accuracy is maintained in smooth regions, and the resolution near shocks or discontinuities is significantly enhanced without any numerical oscillations. Additionally, the computational cost associated with using the same stencils is reduced.

Keywords

Hyperbolic conservation laws / Compact schemes / Weighted essentially non-oscillatory (WENO) / High resolution / Adaptive order / 65M08 / 35L65

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