The numerical solution of extended initial value problems

In typical initial value problems for partial differential equations (e.g., the diffusion equation, wave equation, weather prediction, etc.) it is required to integrate ahead to find the solution at later times from data given at the initial time t = 0. While this is mathematically a well-posed, classifical formulation (the Cauchy Problem), in practice it is usually possible to continue to collect new data at later times. The question is answered of how best to use this data in the prediction process and numerical procedures are given to accomplish this together with some general theorems. These procedures are particularly valuable in problems where the data must be collected experimentally at a limited number of stations, since increased accuracy is achieved by collecting more data in time rather than by increasing the number of data stations as required by the usual difference methods.