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We explore higher-dimensional generalizations of the Runge-Kutta-Wentzel-Kramers-Brillouin method for integrating coupled systems of first-order ordinary differential equations with highly oscillatory solutions. Such methods could improve the performance and adaptability of the codes which are used to compute numerical solutions to the Einstein-Boltzmann equations. We test Magnus expansion-based methods on the Einstein-Boltzmann equations for a simple universe model dominated by photons with a small amount of cold dark matter. The Magnus expansion methods achieve an increase in run speed of about 50% compared to a standard Runge-Kutta integration method. A comparison of approximate solutions derived from the Magnus expansion and the Wentzel-Kramers-Brillouin (WKB) method implies the two are distinct mathematical approaches. Simple Magnus expansion solutions show inferior long range accuracy compared to WKB. However we also demonstrate how one can improve on the standard Magnus approach to obtain a new Jordan-Magnus method. This has a WKB-like performance on simple two-dimensional systems, although its higher-dimensional generalization remains elusive.
We categorify the RK family of numerical integration methods (explicit and implicit). Namely we prove that if a pair of ODEs are related by an affine map then the corresponding discrete time dynamical systems are also related by the map. We show that
Strong stability preserving (SSP) Runge-Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and implicit-expli
In this work we consider a mixed precision approach to accelerate the implemetation of multi-stage methods. We show that Runge-Kutta methods can be designed so that certain costly intermediate computations can be performed as a lower-precision comput
Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were proposed and analyzed in [8]. These specially designed methods use reduced precision or the implicit computations and full precision fo
We introduce a new class of Runge-Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge-Kutta methods, the new methods yield expected convergence properties when st