相关论文: Fifth Order Runge-Kutta-Nystr\"om Methods with Com…

Nonlinear parabolic equations are central to numerous applications in science and engineering, posing significant challenges for analytical solutions and necessitating efficient numerical methods. Exponential integrators have recently…

We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step…

We prove the existence of explicit linear multistep methods of any order with positive coefficients. Our approach is based on formulating a linear programming problem and establishing infeasibility of the dual problem. This yields a number…

We consider the Vlasov-Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions,…

This work focuses on the construction of a new class of fourth-order accurate methods for multirate time evolution of systems of ordinary differential equations. We base our work on the Recursive Flux Splitting Multirate (RFSMR) version of…

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 model is solved numerically. We propose to ensure positivity or other…

We propose an extended framework for continuous-stage Runge-Kutta methods which enables us to treat more complicated cases especially for the case weighting on infinite intervals. By doing this, various types of weighted orthogonal…

We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear…

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step…

We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved…

A numerical search approach is used to design high-order diagonally implicit Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of which have identical diagonal elements (SDIRK) and explicit first stage (ESDIRK). In…

We propose an experimental study of adaptive time-stepping methods for efficient modeling of the aggregation-fragmentation kinetics. Precise modeling of this phenomena usually requires utilization of the large systems of nonlinear ordinary…

We consider the construction of semi-implicit linear multistep methods which can be applied to time dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As…

We present a C++ implementation of a fifth order semi-implicit Runge-Kutta algorithm for solving Ordinary Differential Equations. This algorithm can be used for studying many different problems and in particular it can be applied for…

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 this paper, nonstandard multistep methods are considered. It is shown that under some (sufficient and necessary) conditions, these methods attain the same order as their standard counterparts - to prove this statement, a nonstandard…

This work constructs the first-ever sixth-order exponential Runge--Kutta (ExpRK) methods for the time integration of stiff parabolic PDEs. First, we leverage the exponential B-series theory to restate the stiff order conditions for ExpRK…

Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of…

This manuscript introduces a fourth-order Runge-Kutta based implicit-explicit scheme in time along with compact fourth-order finite difference scheme in space for the solution of one-dimensional Kuramoto-Sivashinsky equation with periodic…

This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge-Kutta (GARK) methods (Sandu and Guenther, 2013). Multirate schemes use different step sizes for different components and for…