In this article, firstly, based on Taylor series expansion and truncation error correction technology, combined with the fourth-order Padé 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 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.
This work was supported in part by the National Natural Science Foundation of China under Grants 11772165 and 11961054, the National Natural Science Foundation of Ningxia under Grant 2018AAC02003 and 2020AAC03264, the National Key Research and Development Program of Ningxia under Grant 2018BEE03007, the Scientific Research Program in Higher Institution of Ningxia under Grant NGY2020110, Major Innovation Projects for Building First-class Universities in China's Western Region under Grant ZKZD2017009 and Western First-Class Subject Project of Ningxia Normal University under Grant NXYLXK2017B11.
The authors would like to thank the editors and the anonymous referees whose constructive comments are helpful to improve the quality of this paper.
"High-order compact difference method for two-dimension elliptic and parabolic equations with mixed derivatives." Tbilisi Math. J. 13 (4) 141 - 167, December 2020. https://doi.org/10.32513/tbilisi/1608606055