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

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