Invited Lecture
Relaxation Runge-Kutta methods: fully-discrete entropy-stability for hyperbolic PDEs
King Abdullah University of Science and Technology, Saudi Arabia
Abstract
Recent advances have enabled the development of efficient high-order entropy-stable discretizations for the Euler and Navier-Stokes equations. However, the strict entropy-stability property is destroyed by standard explicit time discretizations. I will present a class of Runge–Kutta-like methods, related to projection methods, that guarantee conservation or stability with respect to any inner-product norm, and thus provide fully-discrete entropy stability for symmetric hyperbolic systems at the same cost as standard explicit Runge-Kutta time stepping. Because of the methods’ special form, they retain many desirable properties (including order of accuracy, approximate linear stability, and strong stability preservation) of the original Runge–Kutta method. I will show several numerical examples, including an extension to preservation of stability for arbitrary convex entropies such as the standard entropy for the Euler equations.
This is joint work with H. Ranocha (TU Braunschweig, Germany), M. Alsayyari, M. Parsani, and L. Dalcin (KAUST, Saudi Arabia).
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