Analysis of linear and nonlinear stiff problems using the RK‐Butcher algorithm

Abstract

I present a numerical solution of linear and nonlinear stiff problems using the RK-Butcher algorithm. The obtained discrete solutions using the RK-Butcher algorithm are found to be very accurate and are compared with the exact solutions of the linear and nonlinear stiff problems and also with the Runge-Kutta method based on arithmetic mean (RKAM). A topic of stability for the RK-Butcher algorithm is discussed in detail. Error graphs for discrete and exact solutions are presented in a graphical form to show the efficiency of the RK-Butcher algorithm. The results obtained show that RK-Butcher algorithm is more useful for solving linear and nonlinear stiff problems and the solution can be obtained for any length of time.

Parareal Algorithms Implemented with IMEX Runge-Kutta Methods

“…For this type of RK methods, the stiff part is assigned with the implicit component of the IMEX RK method to satisfy stability requirement and the nonstiff or mildly stiff part is assigned with the explicit component to reduce computational cost. For more details about the RK methods and the IMEX RK methods, the interested reader may refer to [23][24][25][26][27][28][29][30][31][32][33] and references therein.…”