An alternating-direction implicit scheme for parabolic equations with mixed derivatives

Abstract

An alternating-direction implicit method for N-dimensional parabolic equations with mixed derivatives is considered. The method requires the solution of N tridiagonal matrix equations per time-step and combines computational simplicity with the possibility of unconditional stability for any N. The regimes of conditional stability for N ⩽ 6 show that the scheme is less effective for higher dimensional problems, owing to the proliferation of mixed derivatives. An alternative scheme (requiring 2 N tridiagonal operations) which involves a single iteration to time-centre the mixed derivatives is shown to improve accuracy and stability. In particular the iterative scheme allows second-order accuracy and unconditional stability in the important special cases of two and three space dimensions.