相关论文: A comparison of high order explicit Runge-Kutta, e…

We study the relationship between local and global error in Runge-Kutta methods for initial-value problems in ordinary differential equations. We show that local error control by means of local extrapolation does not equate to global error…

We develop continuous-stage Runge-Kutta methods based on weighted orthogonal polynomials in this paper. There are two main highlighted merits for developing such methods: Firstly, we do not need to study the tedious solution of…

This survey provides an overview of state-of-the art multirate schemes, which exploit the different time scales in the dynamics of a differential equation model by adapting the computational costs to different activity levels of the system.…

In order to solve continuous-time optimal control problems, direct methods transcribe the infinite-dimensional problem to a nonlinear program (NLP) using numerical integration methods. In cases where the integration error can be manipulated…

This work focuses on the numerical study of a recently published class of Runge-Kutta methods designed for mixed-precision arithmetic. We employ the methods in solving partial differential equations on modern hardware. In particular we…

In current research, we analyse dissipation and dispersion characteristics of most accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K) methods. These methods, known for their $A$-stability and immense accuracy, are…

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…

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…

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,…

High order energy-preserving methods for Hamiltonian systems are presented. For this aim, an energy-preserving condition of continuous stage Runge--Kutta methods is proved. Order conditions are simplified and parallelizable conditions are…

Low-storage explicit Runge-Kutta schemes are particularly popular for the numerical integration of time-dependent partial differential equations based on the method-of-lines due to their efficiency and their reduced memory requirements. We…

Diffusion probabilistic models generate samples by learning to reverse a noise-injection process that transforms data into noise. A key development is the reformulation of the reverse sampling process as a deterministic probability flow…

A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order…

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…

Exponential Runge-Kutta methods constitute efficient integrators for semilinear stiff problems. So far, however, explicit exponential Runge-Kutta methods are available in the literature up to order 4 only. The aim of this paper is to…

In this paper a technique is given to recover the classical order of the method when explicit exponential Runge-Kutta methods integrate reaction-diffusion problems. Although methods of high stiff order for problems with vanishing boundary…

This paper investigates the competitiveness of semi-implicit Runge-Kutta (RK) and spectral deferred correction (SDC) time-integration methods up to order six for incompressible Navier-Stokes problems in conjunction with a high-order…

First-order primal-dual methods are appealing for their low memory overhead, fast iterations, and effective parallelization. However, they are often slow at finding high accuracy solutions, which creates a barrier to their use in…

Explicit Runge-Kutta (RK) integration of hyperbolic initial-boundary value problems with time-dependent Dirichlet data often displays order reduction: the observed convergence order falls below the nominal order because the stage structure…

We introduce a new class of arbitrary-order exponential time differencing methods based on spectral deferred correction (ETDSDC) and describe a simple procedure for initializing the requisite matrix functions. We compare the stability and…