Hyperbolic Approximation of High-Order PDEs

I will present a general technique for approximating high-order PDEs by first-order systems of PDEs, through the introduction of auxiliary variables with relaxed constraints. Such approximations have been pioneered by a number of authors in recent years, as first-order hyperbolic PDEs possess some advantages over higher-order PDEs: they respect causality and they tend to yield non-stiff numerical integration problems. I will show how some of these approaches can be placed in a single framework that can then be generalized in order to approximate, in principle, arbitrary systems of nonlinear PDEs by those of first-order. I will review recent progress in proving the convergence of these relaxed systems, as well as exploring their efficient numerical discretization. I will showcase the application of this idea to the nonlinear Schrodinger and Korteweg-de Vries equations, as well as to tsunami modeling.