A new spectral difference method using hierarchical polynomial bases for hyperbolic conservation laws

To solve hyperbolic conservation laws, a new method is developed based on the spectral difference (SD) algorithm. The new scheme adopts hierarchical polynomials to represent the solution in each cell instead of Lagrange interpolation polynomials used by the original one. The degrees of freedom (DOFs) of the present scheme are the coefficients of these polynomials, which do not represent the states at the solution points like the original method. Therefore, the solution points defined in the original SD scheme are discarded, while the flux points are preserved to construct a Lagrange interpolation polynomial to approximate flux function in each cell. To update the DOFs, differential operators are applied to the governing equation as well as the Lagrange interpolation polynomial of flux function to evaluate first and higher order derivatives of both solution and flux at the centroid of the cell. The stability property of the current scheme is proved to be the same as the original SD method when the same solution space is adopted. One dimensional methods are always stable by the use of zeros of Legendre polynomials as inner flux points. For two dimensional problems, the introduction of Raviart-Thomas spaces for the interpolation of flux function proves stable schemes for triangles. Accuracy studies are performed with one- and two-dimensional problems. p-Multigrid algorithm is implemented with orthogonal hierarchical bases. The results verify the high efficiency and low memory requirements of implementation of p-multigrid algorithm with the proposed scheme.