MA584
: Numerical solutions of PDEs – Finite Difference
Methods
Zhilin
Li
1. Introduction
1.1 The Problems
We want to solve ordinary / partial differential
equations (ODE/PDE), especially linear second order ODE/PDEs, and first order
systems, numerically.
What is a differential equation? It is an equation
whose unknown is a function, say u (x), or u (x,y), or u (t,x), or u (t,x,y,z)
etc. The equation involves the derivatives or partial derivatives of the unknown
function.
Differential equations have been used intensively to
model many physical problem including fluid / solid mechanics, biology,
material sciences, economics, ecology, sports and computer sciences, for
example, the navier stokes equations in fluid dynamics, biharmonic equations
for stress, the Maxwell equations in electro-magnetics, and many others.
Unfortunately, while differential equations can
describe many physical problems, only very small portion of them can be solved
exactly in terms of elementary function (polynomials, sin x, cos x, log x, ex,
ax etc.) and their combinations (composite functions). Quite often, even if a differential equation
can be solved analytically, great efforts and sound mathematical theories are
needed. The closed form of the solution may be too complicated to be actually
useful.
If the analytic solution is not available, we want
to find an approximate solution to the differential equation. Typically, there
are two approaches:
- · Semi-analytic methods. Sometime we can appoximate the solution using series, integral equations, perturbation techniques, asymptotic approximations etc. The solution is often expressed as some simpler function.
- · Numerical approximate solutions in which some numbers are obtained. Nowadays, those numbers are obtained from computers. The development of modern computers make it possible to solve many problems that were impossible just a few decades, or years ago. ...........
