Combined compact difference scheme for linear second-order partial differential equations with mixed derivative

A combined compact difference scheme is proposed for linear second-order partial differential equations with mixed derivative. The scheme is based on a nine-point stencil at the interior with sixth-order accurate local truncation error. Fourier analysis is used to analyze the spectral resolution of the proposed scheme. Numerical tests demonstrate at least sixth-order convergence rate with Dirichlet boundary condition and fifth-order with Robin boundary condition. A bonus is that high Reynolds numbers do not interfere with the order of accuracy.