Finite Difference Methods
Overview
Finite difference methods is one of the primary numerical techniques used to solve differential equations.
The method usually involves breaking up the domain of the equation instead a grid of points, separated
from each other by a small finite difference. At each point, the derivatives in the differential equation
are approximated by numerical techniques.
(see
differentiation for examples)
Mesh
The finite difference method works by paritioning up the input space into a grid of equally spaced points. The value of the function is known on the
points on the boundary.
The following grid shows a grid for a function of two variables, typically {% x %} and {% t %}.
Approximations
The calculation proceeds by replacing the derivatives in the equation of interest by a numeric difference equation that servers as an
approximation. (see
Numeric Differentiation).
In the case of a function of two variables {% x %} and {% t %}, we would have the follwoing approximations.
{% \frac{\partial{u}}{\partial{t}} \approx \frac{u(x_i,t_{j+1}) - u(x_i,t_j)}{k} %}
{% \frac{\partial^2 u}{\partial x^2} \approx \frac{u(x_{i+1},t_j) -2u(x_i,t_j) + u(x_{i-1}, t_j) }{h^2} %}
Topics
- The Euler method is used to solve first order differential equations of a single variable,
with an equation such as
{% y'(x) = f(x) %}
which says that the derivative of y is a function of x.
- Two Point Boundeary Value
problems are second order ordinary differential equations of a single variable that specify the
values of the function at two (boundary) points. Two point boundary value problems are a fairly simple problem
that demonstrates the principles of finite difference methods below.
- Diffusion Problems
are partial differential equations of the form
{% a(x,t) \frac{\partial^2 u}{\partial x^2} + b(x,t) \frac{\partial u}{\partial x} + c(x,t) u = \frac{\partial u}{\partial t} + f(x,t) %}
