On a general method for investigation of finite difference schemes

Abstract

Recently sufficiently deep knowledge has been accumulated about the numerical solution of special types of nonlinear partial differential equations (PDE) and also convergence analysis methods of discrete algorithms have been developed [1, 2]. An important problem is to work out general methods for the investigation of finite difference and variational algorithms for the solution of nonlinear PDE. Construction of such methodology is based on generalization of stability and convergence results for the solution of nonlinear ordinary differential equations (ODE) and the linear PDE. In the case of ODE we will mention the books [3, 4], in which a deep analysis of the basic numerical analysis concepts of approximation, stability, and convergence is given. Recent results of stability theory are comprehensively surveyed in [5]. A fairly complete theory has been already constructed for the numerical solution of the linear PDE. The theorems on necessary and sufficient stability conditions of finite-difference schemes are proved and efficient methods for verification these conditions are developed [6, 7]. Therefore attempts to relate investigation of the convergence of nonlinear finite difference schemes to the convergence analysis of some classes of linear problems seems to be fruitful. Usually we take the equation linearized around a given solution of the nonlinear PDE as an example of such a linear problem. Let us mention the most important works related to this trend. The results of [3] are generalized in [8, 9] and a quite general convergence theory for the nonlinear finite difference schemes is obtained. Similar results are also proved in [10], whose ideas are based on the analysis of nonlinear iterative methods developed in [11]. We point out also a rather general methodology for investigation of twoand threestep finite difference schemes developed in [12, 13]. The results given in [14] for the solution of the nonlinear operator equations are used substantially in this methodology. In this article we generalize the methodology for investigation of nonlinear difference schemes, which was proposed in our papers [15-19] for the solution of diffusion-reaction and nonlinear Schrdinger type problems. Our aim is to obtain sufficient stability conditions of finite difference schemes which will enable us to use the well-known investigation scheme "approXimation + stability = convergence" in the case of nonlinear difference schemes and to prove the existence and uniqueness of the finite difference scheme solution at the same time. First we consider thoroughly the case when the neighborhood of a solution of a differential problem is independent of discrete mesh parameters. The efficiency of the proposed methodology is demonstrated for a concrete finite difference scheme, which approximates the well-known Kuramoto-Tsuzuki problem. After this the given investigation scheme is generalized for the case when radius of the region, in which stability of nonlinear difference scheme is proved, depends on the discrete mesh parameters r, h, and also when, generally speaking,