Numerical cubature for fully symmetric regions and adaptive methods for PDEs

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This research programme has two principal thrusts. One direction continues work carried out in thelast five years on the numerical solution of time dependent partial differential equations in onespace variable. Differential equations occur in many physical models, and often the solutionexhibits narrow layers with peaks or sharp variations. In many cases these peaks or areas of rapidchange move in time. For an approximate method of solution to be able to catch these features, thediscrete model must adapt to the solution. Software to approximate to the solutions with reliable errorestimates will be developed. We will investigate the use of high order interpolants in a method oflines context, in order to obtain inexpensive and reliable error estimates. Using anequi-distribution algorithm we will ensure that the error globally satisfies a user-defined errortolerance.The second thrust, which also is a continuation of a long standing research programme, isthe construction of methods for the numerical evaluation of integrals over multidimensionalregions having specified symmetries.  Examples of such regions are the n dimensional cube,sphere, simplex, octahedron, spherical surface, and the whole of n-space with a symmetricweight function. As shown in an earlier paper there is a strong connection between formulasfor the surface of the sphere and for the simplex. It is intended to investigate connectionsbetween formulas for the octahedron and the simplex. A long term plan is to set up softwareto accept as input the symmetries of the region, and a requested polynomial degree, and thento compute consistent structures for the formulas, set up the defining equations, and solvethem.