Abstract In this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for differential equations (SDEs). First, general order theory these established by the B-series and multicolored rooted tree. Then proposed CSSRK are applied to three special kinds SDEs corresponding conditions derived. particular, single integrand with additive noise, we construct some spe...

Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the is solved numerically. We propose to ensure positivity other by applying Runge-Kutta integration which method weights are adapted order enforce bounds. The chosen at each step after calculating stage derivatives, a way that also pre...

A connection between the algebra of rooted trees used in renormalization theory and Runge-Kutta methods is pointed out. Butcher’s group and B-series are shown to provide a suitable framework for renormalizing a toy model of field theory, following Kreimer’s approach. Finally B-series are used to solve a class of non-linear partial differential equations.

in this paper, a class of semi-implicit two-stage stochastic runge-kutta methods (srks) of strong global order one, with minimum principal error constants are given. these methods are applied to solve itô stochastic differential equations (sdes) with a wiener process. the efficiency of this method with respect to explicit two-stage itô runge-kutta methods (irks), it method, milstien method, sem...

We describe a fully explicit residual based construction to discretize the shallow water equations with friction on unstructured grids. The approach is by construction exactly well balanced for al the simple known steady equilibria, and it shows a super-covergent behavior for smooth non-trivial equilibria, as the implicit residual schemes considered in (Ricchiuto, J.Sci.Comp. 48,2011). Moreover...

We consider an abstract time-dependent, linear parabolic problem u′(t) = A(t)u(t), u(t0) = u0, where A(t) : D ⊂ X → X, t ∈ J , is a family of sectorial operators in a Banach space X with time-independent domain D. This problem is discretized in time by means of an A(θ) strongly stable Runge-Kutta method, 0 < θ < π/2. We prove that the resulting discretization is stable, under the assumption ‖(A...

In this article we consider partitioned Runge-Kutta (PRK) methods for Hamiltonian partial differential equations (PDEs) and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs.