Some new additive Runge–Kutta methods and their applications

Abstract

We propose some new additive Runge-Kutta methods of orders ranging from 2 to 4 that may be used for solving some nonlinear system of ODEs, especially for the temporal discretization of some nonlinear systems of PDEs with constraints. Only linear ODEs or PDEs need to be solved at each time step with these new methods.

Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods

“…Accuracy results that support Theorem 3.5 and efficiency results are presented. Several different ARK methods of various orders were tested within the IDC prediction and correction loops: second order ARK2A1 [24] and ARK2ARS [2], third order ARK3KC [21] and ARK3BHR (Appendix 1. in [7]), and fourth order ARK4A2 [24] and ARK4KC [21]. For brevity, we select only a few of these to present.…”

“…Additive Runge-Kutta. One method of splitting an ODE is to partition the right hand side of an IVP into Λ parts [10,1,21,24]: …”

“…Under certain conditions on the RK coefficients, an ARK method of desired order can be obtained without splitting error [11,1,24,21]. Without loss of generality, we consider the case Λ = 2, where the IVP (2.1) simplifies to the IVP (1.1), where f S (t, y) is stiff and f N (t, y) is nonstiff.…”

“…In the literature, the definition of ARK for the case of Λ = 2 usually is presented in the form of (2.2) [1,21,10,24]. We reformulate the definition of ARK (Λ = 2) below to suit our analysis.…”

“…However, the stability measure for a mixed scheme, such as ARK, is much more complicated. In the literature, different measures for stability have been proposed for semi-implicit (including additive Runge-Kutta) methods [3,2,26,24]. We adopt the stability measure from [3,2], which is explained clearly in [25] via the following definitions.…”