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. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 366–378, 2007