High Resolution Finite Volume Scheme Based on the Quintic Spline Reconstruction on Non-uniform Grids

Abstract

This paper presents a compact quintic spline reconstruction for finite volume method on non-uniform structured grids. The primitive function of a dependent variable is reconstructed by a piece-wise quintic polynomial by requiring the derivatives up to fourth order being continuous at the cell interfaces. This procedure results in a block tridiagonal system of linear equations which can be solved efficiently by incorporating certain boundary closure relations. There are some distinct features in the quintic spline reconstruction. Firstly, the reconstruction stencil is compact; Secondly, the reconstruction can be applied on arbitrary non-uniform grids; and finally, the reconstruction is continuous at cell interface without the need of a Riemann solver. To stabilize the scheme, the sixth order artificial viscosity is introduced. The quintic spline reconstruction achieves sixth-order accuracy on uniform grids without artificial viscosity and fifth-order accuracy on both uniform and non-uniform grids when artificial viscosity is added. The splined scheme is blended with shock-capturing WENO scheme to suppress non-physical oscillations near discontinuities. Numerical results demonstrate the accuracy, robustness and efficiency of the proposed scheme.