相关论文: Lyapunov and Sylvester equations: A quadrature fra…

We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential…

This work proposes and analyzes a new class of numerical integrators for computing low-rank approximations to solutions of matrix differential equation. We combine an explicit Runge-Kutta method with repeated randomized low-rank…

The alternating direction implicit (ADI) methods are computationally efficient and numerically effective tools for computing low-rank solutions of large-scale linear matrix equations. It is known in the literature that the low-rank ADI…

In this paper we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework, initially devised for the approximation of ordinary differential equations, is…

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…

There exist many Runge-Kutta methods (explicit or implicit), more or less adapted to specific problems. Some of them have interesting properties, such as stability for stiff problems or symplectic capability for problems with energy…

Implicit Runge--Kutta (IRK) methods are highly effective for solving stiff ordinary differential equations (ODEs) but can be computationally expensive for large-scale problems due to the need of solving coupled algebraic equations at each…

We derive the alternating-directions implicit (ADI) method based on a commuting operator split and apply the results in detail to the continuous time algebraic Lyapunov equation with low-rank constant term and approximate solution, giving…

We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation…

A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of…

This paper is devoted to examining the stability of Runge-Kutta methods for solving nonlinear Volterra delay-integro-differential-algebraic equations (DIDAEs) with constant delay. Hybrid numerical schemes combining Runge-Kutta methods and…

In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…

This paper illuminates the derivation, the applicability, and the efficiency of the Multiplicative Runge-Kutta Method, derived in the frame- work of geometric multiplicative calculus. The removal of the restrictions of geometric…

Symplectic partitioned Runge--Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the…

This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving…

The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and…

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…

Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…

In the present paper, we present some numerical methods for computing approximate solutions to some large differential linear matrix equations. In the first part of this work, we deal with differential generalized Sylvester matrix equations…

Recently, an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial-value problems, using Runge-Kutta methods. We generalize this approach to multistep…