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We construct eight implicit-explicit (IMEX) Runge-Kutta (RK) schemes up tothird order of the type in which all stages are implicit so that they can beused in the zero relaxation limit in a unified and convenient manner. Theseall-stages-implicit (ASI) schemes attain the strong-stability-preserving (SSP)property in the limiting case, and two are SSP for not only the explicit partbut also the implicit part and the entire IMEX scheme. Three schemes cancompletely recover to the designed accuracy order in two sides of therelaxation parameter for both equilibrium and non-equilibrium initialconditions. Two schemes converge nearly uniformly for equilibrium cases. TheseASI schemes can be used for hyperbolic systems with stiff relaxation terms ordifferential equations with some type constraints.

All-stages-implicit and strong-stability-preserving implicit-explicit  Runge-Kutta time discretization schemes for hyperbolic systems with stiff  relaxation terms