A variable mesh C-SPLAGE method of accuracy $O(k^2{h}_l^{-1}+{\mathit{kh}}_l+h_l^3)$ for 1D nonlinear parabolic equations

Abstract

A new high accuracy two-level implicit cubic spline formulation for the solution of nonlinear parabolic equation u_xx = ϕ(x, t, u, u_x, u_t) is presented. The strategy of the cubic spline alternating group explicit (C-SPLAGE) method is developed. The proposed C-SPLAGE method requires only 3-spatial grid points and applicable to singular problems. The convergence analysis of the proposed C-SPLAGE method is given. The obtained C-SPLAGE method is compared to the corresponding successive over relaxation (SOR) method both in terms of stability and performance.

Introduction

High accuracy numerical approximation of scalar nonlinear parabolic initial boundary value problems has received a lot of attention during the last three decades. It is well known that the main difficulty in these problems is the presence of singular terms in the differential equation, whose high order approximation on non-uniform meshes have been the main focus of numerous research endeavors. Problems of this type arise in numerous applications from science and engineering, such as fluid flow at high Reynolds number, heat transfer with small diffusion parameters, just to name a few. In the context of the cubic spline method, the robust approximation of differential equation requires the use of the non-uniform meshes in the solution space. Jain and Aziz [1] have first used cubic spline technique for the solution of two point nonlinear boundary value problems. Later, Mohanty et al. [2] have extended their method to a variable mesh. The numerical solution of two point nonlinear boundary value problem using a variable mesh has been studied by Jain et al. [3], Mohanty [4], [5] and Bieniasz [6]. Recently, Mohanty and Singh [7] and Mohanty [8] have developed new two-level implicit variable mesh schemes for the solution of nonlinear parabolic equations. First, Papamichael and Whiteman [9] have used cubic spline technique for the solution of one-dimensional heat conduction equation. Later, using second order accurate cubic spline approximations, Rubin and Graves [10], and Jain and Lohar [11] have discussed numerical solution of nonlinear viscous flow problems. Recently, using three spatial grid points, Mohanty and Jain [12] have developed a cubic spline alternating group explicit (C-SPLAGE) method of order 2 in time and 4 in space for the solution of one-dimensional nonlinear parabolic equation on a uniform mesh. In this paper, using a variable mesh, we discuss a new two-level implicit highly accurate C-SPLAGE method for the solution of one-dimensional nonlinear parabolic equation. However, for constant mesh case, the proposed method reduces to the method discussed by Mohanty and Jain [12]. Further, recently Mohanty [13] has discussed a two-level implicit variable mesh cubic spline method for singularly perturbed non-linear parabolic initial boundary value problems. Different numerical methods both for the initial value and the initial boundary value problems for ordinary and partial differential equations have been already discussed in the literature [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Difficulties were experienced in the past for the numerical solution of parabolic equations with singular coefficients using cubic spline approach, especially on a non-uniform mesh. The solutions become unbounded in the vicinity of the singular point. We modify our approximation in such a way that the solutions become bounded in the vicinity of the singularity.

In Section 2, we discuss the derivation procedure of the proposed cubic spline method. In Section 3, we discuss the application of the proposed cubic spline method to heat equation with singular coefficients and present C-SPLAGE and Newton-C-SPLAGE iterative methods for the solution of one-space dimensional linear and nonlinear parabolic equations. Since these methods are coupled compactly and explicit in nature, they are suitable for the use on parallel computers. Formation of variable mesh grid points is discussed briefly and comparative results are given in Section 4. Finally, concluding remarks and scope for further research work are given in Section 5.