Mathematical Physics - Volume II - Numerical Methods

1. Finite difference method and Finite element method

1.1 Finite difference method

The basic idea behind the finite difference method is to replace the derivatives of a given function with their approximate values. In order to achieve this goal, points which form a mesh of nodes are introduced, and the solution is determined for them. The basic concept of finite different method and its realization will be shown us ing examples which involve parabolic, hyperbolic and elliptical partial differential equations. 1.1.1 Finite difference method for parabolic partial differential equations A mesh in plane xt is a set of points ( x n , t j ) = ( x 0 + nh , t 0 + jk ) , where n and j are integers, ( x 0 , t 0 ) is the referent point, and ( x n , t j ) are called mesh points or nodes . Positive numbers h and k are mesh steps along the x and t directions, respectively. If h and k are constants, the mesh is uniform, and if they are equal, the mesh is quadratic. If we use the compact designation

u n j = u ( x n , t j ) .

(1.1)