Time-point relaxation Runge-Kutta methods for ordinary differential equations

Abstract

We investigate convergence, order, and stability properties of time-point relaxation Runge-Kutta methods for systems of ordinary differential equations. These methods can be implemented in Gauss-Jacobi, or Gauss-Seidel modes denoted by TRGJRK( k) or TRGSRK( k), respectively, where k stands for the number of Picard-Lindelöf iterations. As k → ∞, these modes tend to the same one-step method for ordinary differential equations called diagonal split Runge-Kutta (DSRK) method. It is proved that these methods are convergent with order min{ k, p}, where p is the order of the underlying Runge-Kutta method. Recurrence relations resulting from application of TRGJRK( k), TRGSRK( k) and DSRK methods to the two-dimensional test system u′ = λu - μν, ν′ = μu + λν, t ⩾ 0, where λ and μ are real parameters, are derived and stability regions in the (λ, μ)-plane are plotted for some methods using a variant of the boundary locus method. In most cases stability regions increase as the number of Picard-Lindelöf iterations k becomes larger.