Convergence of the compact finite difference method for second-order elliptic equations

Abstract

In this paper, we give a fourth-order compact finite difference scheme for the general forms of two point boundary value problems and two-dimensional elliptic partial differential equations (PDE’s). By decomposing the coefficient matrix into a sum of several matrixes after we discretize the original problems, we can obtain a lower bound for the smallest eigenvalue of the coefficient matrix. Thus we prove the compact finite difference scheme converges with the fourth order of accuracy. To solve the discretized block tri-diagonal matrix equations of the two-dimensional elliptic PDE’s, we develop an efficient iterative method: Full Multigrid Method. In our numerical experiments, we compare the compact finite difference method with the finite element method and Crank–Nicolson finite difference method. The results show that the compact finite difference scheme is a highly efficient and accurate method.

Introduction

In recent years, the compact finite difference method has been used widely in the large area of computational problems, for example, to compute hyperbolic equations. Generation of compact finite difference schemes are considered in [2], [17], [20]. Stability problems for the compact finite difference method on hyperbolic equations were discussed in [13], [14], [21]. There were also some works on applying the compact finite difference scheme for steady convection–diffusion problems [1], [3], [23], [24], Poisson equations [8], [9], integro-differential equations [10], [11], American option pricing problems [33], [34] and cardiac tissue models [31], [32].

In this paper, we derive fourth-order compact finite difference schemes for the general forms of two point boundary value problems and two-dimensional elliptic partial differential equations and give the convergence proof for them, respectively. In Section 2, we apply fourth-order compact finite difference schemes for the following general form of two point boundary value problems:

$$-u^{″}(x)+b(x)u^{\prime }(x)+c(x)u(x)=f(x)\text{,}x\in I\text{,}$$

$$u(a)=u_a\text{,}u(d)=u_d\text{,}$$

where $u^{\prime }(x)=\frac{\mathrm{d}u(x)}{\mathrm{d}x}$, $u^{″}(x)=\frac{{\mathrm{d}}^{2}u(x)}{\mathrm{d}{x}^2}$, and I = [a, d]. The coefficient functions b(x) and c(x) in Eq. (1.1), are given, and belong to the Hilbert space H²(I). And the source function also holds f(x) ∈ H²(I). Discretize equally the interval I = [a, d] into a grid T_h with nodes:

$$a=x_0<x_1<x_2<\cdots <x_{N-1}<x_N=d$$

and

$$x_i=x_0+\mathit{ih}\text{,}i=0\text{,}1\text{,}2\text{,}\dots \text{,}N\text{,}$$

h is the spacial step length. We obtain a tri-diagonal matrix equation by using the compact finite difference scheme to discretize Eq. (1.1). Then the following discrete norm error estimates are derived:

$$\Vert U-V\Vert ⩽{\mathit{Ch}}^4\text{,}$$

here C is a positive constant, V is the vector of approximate solutions, and U is the vector of accurate solutions. In Section 3, we discretize the following general form of the two-dimensional elliptic partial differential equations by the fourth-order compact finite difference scheme:

$$-(u_{\mathit{xx}}(x\text{,}y)+u_{\mathit{yy}}(x\text{,}y))+b_1(x\text{,}y)u_x(x\text{,}y)+b_2(x\text{,}y)u_y(x\text{,}y)+c(x\text{,}y)u(x\text{,}y)=f(x\text{,}y)\text{,}(x\text{,}y)\in \Omega \text{,}$$

$$u(x\text{,}y)=u_0(x\text{,}y)\text{,}(x\text{,}y)\text{on}\mathrm{\partial }\Omega \text{,}$$

where $u_z(x\text{,}y)=\frac{\mathrm{\partial }u(x\text{,}y)}{\mathrm{\partial }z}$, $u_{\mathit{zz}}(x\text{,}y)=\frac{{\mathrm{\partial }}^{2}u(x\text{,}y)}{\mathrm{\partial }{z}^2}$, and z stands for x or y. Coefficients functions b₁(x, y), b₂(x, y), c(x, y), and the source function f(x, y) are given functions, and they are all in the space H²(Ω). Boundary value function u₀(x, y) is already known. The coefficient matrix $\stackrel{\sim }{A}$ for Eq. (1.4) is block tri-diagonal. We obtain the same norm error estimates (1.3) for two-dimensional case. The results of numerical experiments in Section 4 show that the compact finite difference scheme is a highly efficient and accurate method by comparing the compact finite difference method with the finite element method and Crank–Nicolson finite difference method.

In the following sections, we always use u_i and v_i to stand for the accurate solution and approximate solution, respectively for second-order elliptic equations at partition point x_i.