High Order Spectral Volume Method for Category 1, Problem 1 and Category 2 Problems

A recently developed high-order finite volume (FV) method named the spectral volume (SV) method is employed to solve two categories of problems of the Fourth Computational Aeroacoustics (CAA) Workshop on Benchmark Problems: Problem 1 in Category 1 (C1P1) and Category 2 (C2) problems. In the SV method, a simplex grid element or cell is named a spectral volume, which is further partitioned into subcells called control volumes (CVs). The cell-averaged solutions on the CVs are the unknowns or degrees-of-freedom (DOFs). The DOFs are used to build high-order polynomials in the SV, which generate high-order updates for the unknowns. Therefore the resultant SV schemes are not optimized for any particular range of wave frequencies. In order to make a fair comparison with the finite difference method, the total number of DOFs is determined based on the requirement of an average 1 DOF per unit length for C1P1. SV schemes of as high as 100th order of accuracy are tested on C1P1. It was found that SV schemes of very high order are necessary for waves at 3.7 PPW (points per wave), and SV schemes showed a dissipative behavior for waves with less than 2 PPW. For C2 problems, a 4th order SV scheme was employed, and the numerical solutions agree well with the analytical solutions.