Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations
KAUST DepartmentKing Abdullah University of Science and Technology (KAUST), Extreme Computing Research Center (ECRC), Computer Electrical and Mathematical Science and Engineering Division (CEMSE), Thuwal, 23955-6900, Saudi Arabia
Extreme Computing Research Center
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program
Permanent link to this recordhttp://hdl.handle.net/10754/659959
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AbstractThe framework of inner product norm preserving relaxation Runge-Kutta methods [D. I. Ketcheson, SIAM J. Numer. Anal., 57 (2019), pp. 2850-2870] is extended to general convex quantities. Conservation, dissipation, or other solution properties with respect to any convex functional are enforced by the addition of a relaxation parameter that multiplies the Runge-Kutta update at each step. Moreover, other desirable stability (such as strong stability preservation) and efficiency (such as low storage requirements) properties are preserved. The technique can be applied to both explicit and implicit Runge-Kutta methods and requires only a small modification to existing implementations. The computational cost at each step is the solution of one additional scalar algebraic equation for which a good initial guess is available. The effectiveness of this approach is proved analytically and demonstrated in several numerical examples, including applications to high order entropy-conservative and entropy-stable semidiscretizations on unstructured grids for the compressible Euler and Navier-Stokes equations.
CitationRanocha, H., Sayyari, M., Dalcin, L., Parsani, M., & Ketcheson, D. I. (2020). Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations. SIAM Journal on Scientific Computing, 42(2), A612–A638. doi:10.1137/19m1263480
SponsorsThis work was supported by the King Abdullah University of Science and Technology (KAUST).
We are thankful for the computing resources of the Supercomputing Laboratory and the Extreme Computing Research Center at King Abdullah University of Science and Technology