Splitting-based time integration approaches such as fractional steps, alternating direction implicit, operator splitting, and locally one-dimensional methods partition the system of interest into components solve individual implicitly in a cost-effective way. This work proposes unified formulation splitting schemes framework general-structure additive Runge-Kutta (GARK) methods. Specifically, w...

Approximately calculating integrals over spherical surfaces in R3 can be done by simple extensions of one dimensional quadrature rules. This, however, does not make use of the symmetry or structure of the integration domain and potentially better schemes can be devised by directly using the integration surface in R3. We investigate several quadrature schemes for integration over a spherical sur...

We study a two-time-scale system of jump-diffusion stochastic differential equations. We analyze a class of multiscale integration methods for these systems, which, in the spirit of [1], consist of a hybridization between a standard solver for the slow components and short runs for the fast dynamics, which are used to estimate the effect that the fast components have on the slow ones. We obtain...

We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte-Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at finite noise and to gain higher order a...

A simple accelerated third-order Runge-Kutta-type, fixed time step, integration scheme that uses just two function evaluations per step is developed. Because of the lower number of function evaluations, the scheme proposed herein has a lower computational cost than the standard third-order Runge-Kutta scheme while maintaining the same order of local accuracy. Numerical examples illustrating the...