High-order difference schemes for 2D elliptic and parabolic problems with mixed derivatives

We propose a 9-point fourth-order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two-level high-order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth-order accurate in space and second- or lower-order accurate in time depending on the choice of a weighted average parameter μ. Unconditional stability is proved for 0.5 ≤ μ, ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high-order accuracy of the schemes are presented.