Lecture Notes in Numerical Methods of Differential Equations

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This Ebook is designed for science and engineering students taking a course in numerical methods of differential equations. Most of the material in this Ebook has its origin based on lecture courses ...
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Chapter 5 Parabolic Equations

Pp. 100-124 (25)

Tadeusz Stys

Abstract

In this chapter, considerable attention is paid to construction of finite difference schemes with weight 0 ≤ σ ≤ 1 for diffusion equations in one and two space variables. Efficient algorithms are built for selected values of the weight σ, like implicit scheme (σ = 1), explicit scheme (σ = 0) and Crank Nicolson scheme (σ = 1/2). The average convergence of the scheme is proved in the norm of the Hilbert’s space H. The scheme with weight is implemented in the Mathematica module heatEqn and applied to the diffusion equations with initial boundary value conditions. In the last section, the heat equation with initial boundary value conditions is solved by the method of lines. The chapter ends with a set of questions.

Affiliation:

University of Warsaw.