An Overview of Structure Preserving Approximations of Nonlinear Conservation Equations

Realistic physical models of conservation equations are highly nonlinear systems of partial differential equations (PDE). Proving existence and uniqueness of solutions to these PDE systems is very often beyond reach, and in some cases the models are known to be ill-posed. In absence of a strong mathematical framework guaranteeing some form of compactness and well-posedness, one is lead to construct approximation techniques that (in addition to being consistent, of course) preserve key structures of the PDE system. For instance if the solution map to a PDE system is known to map pointwise to a convex subset of $\mathbb^m$, then a structure preserving approximation method would be one that ensures that the approximate solution map does the same. This concept is a multi-dimensional generalization of the maximum principle. Techniques satisfying this type of property are called invariant-domain preserving. It also often desirable to make sure that approximate solutions exactly solve a problem in some canonical situations. For instance, if a fluid flow is in hydrostatic equilibrium, one would like the approximate solution to satisfy this equilibrium as well. If the free surface of a lake is a rest, one would like the approximate model to deliver a solution that is also at rest. Approximation techniques that satisfy this type of property are said to be well-balanced. Many other forms of structures can be identified and preserved: involutions (think of Gauss laws of magnetism); asymptotic limits with respect to some model parameter; etc. In this talk I will give a brief overview of the state of the art on structure preserving approximation techniques for nonlinear conservation equations.