Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

Kang, Shinhoo; Constantinescu, Emil M.

doi:10.1007/s10915-022-01982-w

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

Journal Article · Sun Sep 04 00:00:00 EDT 2022 · Journal of Scientific Computing

DOI:https://doi.org/10.1007/s10915-022-01982-w· OSTI ID:1885765

^[1]; Constantinescu, Emil M. ^[1]

Argonne National Lab. (ANL), Argonne, IL (United States)

In this work, we propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

View Accepted Manuscript (DOE)

Research Organization:: Argonne National Laboratory (ANL), Argonne, IL (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC)

Grant/Contract Number:: AC02-06CH11357

OSTI ID:: 1885765

Journal Information:: Journal of Scientific Computing, Journal Name: Journal of Scientific Computing Journal Issue: 1 Vol. 93; ISSN 0885-7474

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

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