Novelly Fifth-Order HUS-WENO Scheme for Simulating Hyperbolic Conservation Laws on Structured Grids

Source abstract

In this study, we design a new fifth-order finite difference hybrid unequal-sized weighted essentially non-oscillatory (HUS-WENO) scheme for simulating hyperbolic conservation laws on 1D and 2D structured grids. By utilizing the information defined on one five-point stencil, we reestablish one quartic polynomial and its three derivative polynomials. We develop an enhanced troubled cell indicator by combining a hierarchically simplified Newton iteration method. This indicator accurately identifies all real extremum points of the reconstruction polynomial within the minimal interval in 1D space. This troubled cell indicator doesn’t have any manually adjustable parameters and offers a universal approach across different examples. The hybrid approach is bifurcated into two aspects: if the reconstruction polynomial exhibits no extremum points or those extremum points lie exterior to the minimal interval, the cell isn’t the troubled cell and the fifth-order linear upwind reconstruction is employed. Conversely, the cell is the troubled cell. In such instances, the US-WENO spatial reconstruction, renowned for its exceptional shock-capture capabilities, is employed. This ensures the preservation of the essentially non-oscillatory nature. Then, we propose the fifth-order HUS-WENO scheme by applying this newly developed troubled cell indicator. This scheme is designed with the flexibility to be readily extended to multidimensional applications. A principal advantage of the HUS-WENO scheme is its enhanced computational efficiency, which affords an approximate reduction of 23%-33% of CPU time relative to the conventional US-WENO scheme of the same order when simulating some benchmark tests.