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In this paper we define an efficient implementation of Runge-Kutta methods of Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems. The proposed implementation relies on an alternative low-rank formulation of the…
A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a…
Many time-dependent differential equations are equipped with invariants. Preserving such invariants under discretization can be important, e.g., to improve the qualitative and quantitative properties of numerical solutions. Recently,…
We provide a note on continuous-stage Runge-Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge-Kutta (RK)…
In this paper, we develop a higher order symmetric partitioned Runge-Kutta method for a coupled system of differential equations on Lie groups. We start with a discussion on partitioned Runge-Kutta methods on Lie groups of arbitrary order.…
Runge-Kutta methods are a popular class of numerical methods for solving ordinary differential equations. Every Runge-Kutta method is characterized by two basic parameters: its order, which measures the accuracy of the solution it produces,…
This paper presents an adaptive multiple-shooting method to solve stochastic multi-point boundary value problems. The heuristic to choose the shooting points is based on separating the effects of drift and diffusion terms and comparing the…
Multiphysics systems are driven by multiple processes acting simultaneously, and their simulation leads to partitioned systems of differential equations. This paper studies the solution of partitioned systems of differential equations using…
Stabilized Runge-Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized…
In the paper explicit functional continuous Runge-Kutta and Runge-Kutta-Nystr\"om methods for retarded functional differential equations are considered. New methods for first order equations as well as for second order equations of the…
Mixed precision Runge--Kutta methods have been recently developed and used for the time-evolution of partial differential equations. Two-derivative Runge--Kutta schemes may offer enhanced stability and accuracy properties compared to…
Recently, a new class of second order Runge-Kutta methods for It\^o stochastic differential equations with a multidimensional Wiener process was introduced by R\"o{\ss}ler. In contrast to second order methods earlier proposed by other…
In this paper the performance of a parallel iterated Runge-Kutta method is compared versus those of the serial fouth order Runge-Kutta and Dormand-Prince methods. It was found that, typically, the runtime for the parallel method is…
Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme…
This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carath\'eodory…
We show that existing Runge-Kutta methods for ordinary differential equations (odes) can be modified to solve stochastic differential equations (sdes) with strong solutions provided that appropriate changes are made to the way stepsizes are…
This work gives the asymptotic error distribution of the stochastic Runge--Kutta (SRK) method of strong order $1$ applied to Stratonovich-type stochastic differential equations. For dealing with the implicitness introduced in the diffusion…
In this paper, authors successfully construct a new algorithm for the new higher order scheme of weak approximation of SDEs. The algorithm presented here is based on [1][2]. Although this algorithm shares some features with the algorithm…
Hamiltonian systems are one of the most important class of dynamical systems with a geometric structure called symplecticity and the numerical algorithms which can preserve such geometric structure are of interest. In this article we study…
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a…