Skew-Hermitian based iterations for nine-point approximations of convection-diffusion problems

We study skew-Hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high-order compact (HOC) approximations of two-dimensional convection-diffusion problems. Such discretisations lead to nine-point system matrices with block-tridiagonal structures. The nonsymmetric discretisation matrix for a constant-coefficient grid-aligned flow problem is shown to be positive definite and thus Hermitian and skew-Hermitian splitting (HSS) methods can be considered for the solution of the corresponding linear system. For the solution of this linear system using the HSS iteration, we derive an analytical expression for the optimal value for the upper bound of the contraction factor. Comparison of the HSS iteration with triangular and skew-Hermitian splitting (TSS) methods and their corresponding block variants (BTSS) are carried out for various test problems. Cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given. Our results show that the combination of high-order compact discretisations and skew-Hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two-dimensional convection-diffusion equations.