Finite difference method

Finite difference methods are a class of numerical methods for solving ordinary and partial differential equations.

First, the area for which the equation is to apply, discretized by a finite number of grid points. One-dimensional intervals to be broken down into equal subintervals, multi-dimensional areas in rectangular grid. The derivatives of the unknown function at the grid points are then approximated by difference quotients (see numerical differentiation ). The differential equation is approximated in this way by a system of difference equations that can be solved by various algorithms for the numerical solution of systems of equations.

Processes of this type find widespread application among others, fluid dynamic simulations, for example in meteorology and astrophysics. A special finite difference method for the numerical solution of the heat equation is the Crank- Nicolson method.

Among the pioneers of the finite-difference method for partial differential equations include Lewis Fry Richardson, Richard Southwell, Richard Courant, Kurt Friedrichs, Hans Lewy, Peter Lax and John von Neumann.

Example

Consider the boundary value problem

The solution function can be calculated exactly to here.

To solve the difference method, the interval is discretized by the grid points for the mesh size. The discretization of the second derivative is carried out with the central difference quotient of the second derivative

This results in the inner grid points, the difference equations

For the numerical approximations of the solution values ​​. Using the given boundary conditions and this is a system of linear equations with the equations for the unknown.

In matrix form the system to be solved is here

Since in each row occur a maximum of only three unknowns, it is a system with a sparse coefficient matrix, more precisely, a system with tridiagonal Toeplitz matrix.