B-convergence Of Additive Runge-Kutta Methods For Stiff Problems

The stiff problems is a class of special initial-value problems for differential equations and have the widespread application background. But Runge-Kutta methods are used usually to solve the initial-value problems for differential equations. We often solve the stiff problems with the implicit Runge-Kutta methods. Although we can obtain the high-accuracy numerical solution, we pay large computational cost. In order to reduce the computational cost, we often use the diagonally-implicit Runge-Kutta methods to solve the initial-value problems. In recent years, much interest has been devoted to numerical integration of nonlinear stiff problems defined by operator that may be decomposed into a sum of two or more parts. To solve this class of stiff problems, we use additive Runge-Kutta methods. It is common to combine an implicit Runge-Kutta methods for stiff parts with explicit Runge-Kutta methods for nonstiff parts.In this article, we use the additive Runge-Kutta methods to solve the above stiff problems. We show that the order of optimal B-convergence of algebraically-stable and diagonally-stable additive Runge-Kutta methods is equal at least to the stage order for K_0,0 class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We also show that the order of optimal B-convergence of algebraically-stable and diagonally-stable(ANS-stable) additive Runge-Kutta methods is equal at least to the stage order for K_0,ω class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We show the optimal B-convergence of fractional step Runge-Kutta methods for K_0,0,K_0,ω class initial value problems. At last, we show that the order of optimal B-convergence of (θ, (p|-), (q|-))-algebraically-stable and diagonally-stable(ANS-stable) additive Runge-Kutta methods is equal at least to the stage order for K_σ,τ class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We show that the monotonicity of (θ, (p|-), (q|-))-algebraically-stable and the optimal B-convergence of fractional step Runge-Kutta methods for K_Φ,φ class initial value problems.