High-order compact difference method for two-dimension elliptic and parabolic equations with mixed derivatives

Tingfu Ma, Yongbin Ge

In this article, firstly, based on Taylor series expansion and truncation error correction technology, combined with the fourth-order Pad\'e schemes of the first-order derivatives, a new fourth-order compact difference (CD) scheme is constructed to solve the two-dimensional (2D) linear elliptic equation with a mixed derivative. In this new scheme, unknown function and its first-order derivatives are regarded as the unknown variables in calculation. Then, the method is extended to solve the 2D parabolic equation with a mixed derivative. To match the spatial fourth-order accuracy, The backward differentiation formula (BDF) is employed to gain the fourth-order accuracy for the temporal discretization. Truncation error is analyzed to display that the present scheme is fourth-order accuracy in space. In order to solve the resulting large-scale linear equations, a multigrid method is employed to accelerate the convergence speed of the conventional relaxation methods. Finally, numerical results indicate that the present schemes obtain fourth-order convergence and are more accurate than those in the literature. 

Tbilisi Mathematical Journal, Vol. 13(4) (2020), pp. 141-167