Error analysis of variable stepsize Runge–Kutta methods for a class of multiply-stiff singular perturbation problems

In this paper, we present some results on the error behavior of variable stepsize stiffly-accurate Runge–Kutta methods applied to a class of multiply-stiff initial value problems of ordinary differential equations in singular perturbation form, under some weak assumptions on the coefficients of the considered methods. It is shown that the obtained convergence results hold for stiffly-accurate Runge–Kutta methods which are not algebraically stable or diagonally stable. Some results on the existence and uniqueness of the solution of Runge–Kutta equations are also presented.