3. The finite difference method

In order to solve the Landau-Lifshitz equation () with the effective field () numerically, we have to convert it into a form, which can be translated into an algorithm for a digital computer with finite speed and memory.

We have to reduce the problem of finding a continuous solution to one with finite dimensionality [19]. In the finite difference method (as later in the finite element method) we replace the continuous solution domain by a discrete set of lattice points. In each lattice point we replace any differential operators by finite difference operators. The conditions on the boundary of the domain have to be replaced by their discrete counterparts.

For some differential equations, such as the wave equation in one dimension, it is even possible to construct exact algorithms by nonstandard finite difference schemes [20]. However, this is rarely the case, and so the finite difference method gives only an approximate solution.