With this award the principal investigator will continue his numerical analysis of scalar hyperbolic equations in one space dimension, in order to derive improved error bounds and to extend his techniques to systems of such equations in higher space dimensions. In particular, he will employ Besov space approximation theory to develop an improved regularity theory for scalar conservation laws that is applicable to moving mesh numerical methods. He will also attempt to extend moving mesh finite element methods to systems of equations by providing a theoretical justification for some practical moving mesh algorithms in current use. There are many methods available today for the numerical solution of a single partial differential equation in one space variable, but only a precious few of these methods can be used to solve systems of such equations in more than one space variable. One reason for this is that many methods lack a solid theoretical foundation. With this award the principal investigator will continue developing theoretical justifications for certain numerical schemes in the hope that better schemes for higher-dimensional systems can be found.
