Topic 6: Partial differential equations (I - Laplace's equation)

Introduction

A common problem in scientific computing is to solve a time-dependent partial differential equation (PDE) in 1, 2 or 3 spatial dimensions.

Scientific computing have learned (often the hard way) that PDEs can be a mine-field of computational difficulty, manifested through persistent instabilities, difficulaties with boundary values, or very slow convergence.

Standard forms

The general form of a PDE in two independent dimensions (x,y) is

$$A(x,y)\frac{\partial^2 U}{\partial x^2} + B(x,y)\frac{\partia... ...frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}),$$

If B² = AC the equation is parabolic (e.g, 1-dimensional heat flow equation), if B² < AC it is elliptic (e.g. Laplace's equation) and if B² > AC it is hyperbolic (e.g. 1-dimensional wave equation).

These different classes of PDE exhibit different kinds of instabilities and error propagation, which actually reflects the underlying character of the natural phenomena they represent.

Laplace's equation

The harmonic, potential or Laplace equation in two rectangular coordinates is

$$\frac{\partial^2 U(x,y)}{\partial x^2} = \frac{\partial^2 U(x,y)}{\partial y^2}$$

The problem is to find U(x,y) within a boundary, given its values on the boundary.

There is a fairly obvious representation of the PDE in terms of finite differences. The solution of this set of equations will often involve intricate iteration to deal with the specified boundary contitions.

Tutorial goals

Solve the 2-D Laplace equation in a square region with U = 1 along one edge, and U=0 along the other three edges.