Numerical Analysis and Scientific Computing Seminar | Jean-Luc Guermond, Invariant domain preserving explicit and implicit-explicit Runge Kutta techniques for systems of conservation equation

For Zoom Link please contact ddelreyfernandez@uwaterloo.ca  

Speaker

Jean-Luc Guermond, Department of Mathematics Texas A&M University

Title

Invariant domain preserving explicit and implicit-explicit Runge Kutta techniques for systems of conservation equation

Abstract

The first question addressed in the talk consists of constructing explicit and high-order accurate time stepping techniques that preserve the invariant-domain property of systems of first-order nonlinear conservation equations.  One well-known class of methods meeting this goal consists of the so-called strong stability preserving Runge-Kutta methods.  In this talk I will show how to go beyond the strong stability preserving paradigm and present a time stepping technique that makes every explicit Runge-Kutta method invariant-domain preserving.  The technique is space discretization agnostic and can be combined with continuous finite elements, discontinuous finite elements, and finite volumes. The key idea is that at each stage of the Runge Kutta scheme, one computes a low-order update, a high-order update, both defined from the same intermediate Runge Kutta stage, and then one applies the nonlinear, mass conservative limiting operator. The main advantage with respect to the strong stability preserving paradigm is more flexibility in the choice of the Runge Kutta scheme, thus allowing for less stringent restrictions on the time step. In the second part of the talk I will show how to extend the method to partial differential equations with second or higher-order derivatives in space. I will show that the above strategy can be adapted to implicit-explicit methods. One key step is to be able to construct a linearized first-order accurate method that is conservative and invariant-domain preserving.