Abstract
A high-order compact difference scheme for solving the two-dimensional (2D) elliptic problems is proposed by including compact approximations to the leading truncation error terms of the central difference scheme. A multigrid method is employed to overcome the difficulties caused by conventional iterative methods when they are used to solve the linear algebraic system arising from the high-order compact scheme. Numerical experiments are conducted to test the accuracy and efficiency of the present method. The computed results indicate that the present scheme achieves the fourth-order accuracy and the effect of the multigrid method for accelerating the convergence speed is significant.
Highlights
In this paper, we consider the 2D elliptic equation with a mixed derivative term in the form of − ε ( ∂2u ∂x2 + ∂2u ∂y2 ) β (x, y) ∂2u ∂x∂y p ∂u ∂x (1)+ q (x, y) ∂u + r (x, y) u = f (x, y)
Since Brant published his pioneering work [26], multigrid method has been widely used in numerical solutions of various kinds of differential equations discretized by finite difference methods [14, 17, 23, 27,28,29,30,31,32,33,34,35,36,37,38,39] and finite element methods [40,41,42,43]
The results reveal that the fourth-order compact difference scheme on coarse grid can achieve equivalent accuracy to the central difference scheme on fine grid
Summary
We consider the 2D elliptic equation with a mixed derivative term in the form of. Since Brant published his pioneering work [26], multigrid method has been widely used in numerical solutions of various kinds of differential equations discretized by finite difference methods [14, 17, 23, 27,28,29,30,31,32,33,34,35,36,37,38,39] and finite element methods [40,41,42,43] It has been proved a high efficient and effective iterative method for solving linear elliptic problems.
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