Introduction
Second-order partial differential equations (PDEs) may be classified as parabolic, hyperbolic or elliptic. Parabolic and hyperbolic PDEs often model time dependent processes involving initial data.
In this Section we consider numerical solutions of parabolic problems.
Prerequisites
- review difference methods for first and second derivatives ( HELM booklet 31.3)
Learning Outcomes
- implement simple methods to obtain approximate solutions of the heat diffusion equation
Contents
1 Definitions2 Motivation
3 Approximating partial derivatives
4 An explicit numerical method for the heat equation
4.1 Implementation
5 Stability of the simple explicit scheme
5.1 Why is the stability constraint a problem?
6 The Crank-Nicolson method
6.1 In general
6.2 Stability of the Crank-Nicolson scheme
7 Cost -v- benefit