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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 in practice this works well when the pairs of related ODEs come from the coupled cell networks formalism and, more generally, from fibrations of networks of manifolds.
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
This study computes the gradient of a function of numerical solutions of ordinary differential equations (ODEs) with respect to the initial condition. The adjoint method computes the gradient approximately by solving the corresponding adjoint system
It is well-known that a numerical method which is at the same time geometric structure-preserving and physical property-preserving cannot exist in general for Hamiltonian partial differential equations. In this paper, we present a novel class of para