A new WENO based Chebyshev Spectral Volume method for solving one- and two-dimensional conservation laws

Abstract

High-order methods have attracted considerable attention in the CFD fields because of their potential in achieving higher accuracy with a lower cost than low order methods. In this paper, a new class of high order spectral volume method based on Chebyshev polynomials as an approximation function called Chebyshev Spectral Volume (CSV) method is presented for hyperbolic conservation laws in quadrilateral meshes. As we know, the hyperbolic conservation law may contain discontinuities even if the initial conditions are smooth. To capture these discontinuities, we propose a new limiter to reconstruct the numerical approximation on the target cell, which is marked as a troubled cell. It is based on the Chebyshev polynomials in the target and its immediate neighboring cells to achieve a high order of accuracy and non-oscillatory properties. Also, a new indicator is introduced to detect troubled cells. This indicator depends neither on the numerical order of accuracy nor on a problem-dependent parameter. A series of 1D and 2D scalar and system conservation laws are presented to evaluate the performance of the CSV method coupled with the new limiter and indicator.