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We propose entropy-preserving and entropy-stable partitioned Runge--Kutta(RK) methods. In particular, we extend the explicit relaxation Runge--Kuttamethods to IMEX--RK methods and a class of explicit second-order multiratemethods for stiff problems arising from scale-separable or grid-inducedstiffness in a system. The proposed approaches not only mitigate systemstiffness but also fully support entropy-preserving and entropy-stabilityproperties at a discrete level. The key idea of the relaxation approach is toadjust the step completion with a relaxation parameter so that thetime-adjusted solution satisfies the entropy condition at a discrete level. Therelaxation parameter is computed by solving a scalar nonlinear equation at eachtimestep in general; however, as for a quadratic entropy function, wetheoretically derive the explicit form of the relaxation parameter andnumerically confirm that the relaxation parameter works the Burgers equation.Several numerical results for ordinary differential equations and the Burgersequation are presented to demonstrate the entropy-conserving/stable behavior ofthese methods. We also compare the relaxation approach and the incrementaldirection technique for the Burgers equation with and without a limiter in thepresence of shocks.

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