Introduction [ edit | edit source] There exist very good books (see ([#References|references])) about numerical integration of PDE evolution problem. Note that all methods (finite difference, finite element, spectral methods) contains as a final step the time integration of a ODE system. In this section this time integration is treated. Problem to be solved is the following probmathevovec Problem: (Cauchy problem) Find functions obeying the ODE system where is known. We will not present here details about stability and precision calculation of the numerical integration schemes, however the reader should always keep in mind this crucial problem. Generally speaking, knowledge of mathematical properties of solutions should always be used to verify numerical results. For instance, if the solution of problem probmathevovec is known to be bounded, and that the solution obtained numerically diverges, then numerical solution will be obviously considered as bad. This problem of stability is the first that the numerician meet.
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