Abstract
Based on the locally one-dimensional strategy, we propose two high order finite difference schemes for solving two-dimensional linear parabolic equations. In the first method, fourth order approximation in space and (2,2) Padé formula in time are considered. These lead to a fourth order finite difference scheme in both space and time. For the second method, we employ sixth order approximation in space and (3,3) Padé formula in time. This yields a novel sixth order scheme in both space and time. The methods are proved to be unconditionally stable, and the Sheng–Suzuki barrier is successfully avoided. Numerical experiments are given to illustrate our conclusions as well as computational effectiveness.
Highlights
Among them, splitting strategies including alternating direction implicit (ADI) and locally one-dimensional (LOD) methods have been extensively explored for high order difference schemes [4, 6,7,8,9, 11,12,13,14,15,16,17]
5 Conclusion In this paper, we propose two types of high order splitting finite difference schemes for solving the two-dimensional parabolic equation
Based on the LOD strategy, we separate the two-dimensional equation into two one-dimensional equations to construct the new schemes
Summary
Consider the following two-dimensional linear parabolic equation:. together with the initial condition u(x, y, 0) = φ0(x, y) and boundary conditions u(0, y, t) = g0, u(1, y, t) = g1, u(x, 0, t) = d0, u(x, 1, t) = d1, where = [0, l] × [0, l] is a spatial domain and l is a positive real number. φ0 is a sufficiently smooth function and g0, g1, d0, d1 are constants. For the one-dimensional hyperbolic equation, Gao and Chi [20] used semi-discrete method to transform it into a system consisting of ordinary differential equations with respect to time, whose exact solution containing an infinite matrix series was approximated by (1, 1) and (2, 2) Padé approximations They obtained two schemes with third and fifth order accuracy in time, respectively. Substituting Eq (15) into Eq (3) and dropping the truncated errors O(h4), we obtain the following semi-discrete scheme of fourth order accuracy in space: 24 ∂t i,j+1 12 ∂t i,j 24 ∂t i,j–1 a. Combining Eq (14) with Eq (21), we obtain a fourth-order difference scheme in both time and space for solving Eq (1) as follows:.
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