Derivation Of Order Conditions Research Articles

Extended multirate infinitesimal step methods: Derivation of order conditions

Multirate methods are specially designed for problems with multiple time scales. The multirate infinitesimal step method (MIS) was developed as a generalization of the so called split-explicit Runge–Kutta methods, where the integration of the fast part is conducted analytically. The MIS method was originally evolved for applications related to numerical weather prediction, i.e. the integration of the compressible Euler equation.In this work, an extension to MIS methods will be presented where an arbitrary Runge–Kutta method (RK) is applied for the integration of the fast component. Furthermore, the order convergence from the original MIS method will be reinvestigated including the derivation of conditions up to order four. Additionally will be presented how well-known methods such as recursive flux splitting multirate method, (Schlegel et al., 2012) partitioned Runge–Kutta method, (Jackiewicz and Vermiglio, 2000) and generalized additive Runge–Kutta method, (Sandu and Günther, 2015) are related to or can be cast as an extended MIS method. An exemplary MIS method of order four with five stages will show that the convergence behaviour not only depends on the applied method for the integration of the fast component. The method will further indicate that the used fast time step plays a significant role.

Automatic symbolic derivation of order conditions for two-stage complex Rosenbrock schemes

Rosenbrock methods are known to be good in numerical modeling of stiff problems. In this work, a new symbolic algorithm is proposed for automating the derivation of order conditions for two-stage Rosenbrock schemes with complex coefficients. Software implementation of the proposed algorithm is discussed in detail.

Order conditions for ARKN methods solving oscillatory systems

For the perturbed oscillators in one-dimensional case, J.M. Franco designed the so-called Adapted Runge–Kutta–Nyström (ARKN) methods and derived the sufficient order conditions as well as the necessary and sufficient order conditions for ARKN methods based on the B-series theory [J.M. Franco, Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770–787]. These methods integrate exactly the unperturbed oscillators and are highly efficient when the perturbing function is small. Unfortunately, some critical mistakes have been made in the derivation of order conditions in that paper. On the basis of the results from that paper, Franco extended directly the ARKN methods and the corresponding order conditions to multidimensional case where the perturbed function f does not depend on the first derivative y′ [J.M. Franco, New methods for oscillatory systems based on ARKN methods, Appl. Numer. Math. 56 (2006) 1040–1053]. In this paper, we present the order conditions for the ARKN methods for the general multidimensional perturbed oscillators where the perturbed function f may depend on only y or on both y and y′.

Order Conditions for General Two-Step Runge--Kutta Methods

The use of B-series for the derivation of order conditions for general two-step Runge--Kutta methods is illustrated.