Related papers: A Randomized Runge-Kutta Method for time-irregular…
Relaxation Runge-Kutta methods reproduce a fully discrete dissipation (or conservation) of entropy for entropy stable semi-discretizations of nonlinear conservation laws. In this paper, we derive the discrete adjoint of relaxation…
An error analysis of Runge-Kutta convolution quadrature based on Gauss methods applied to hyperbolic operators is given. The order of convergence relies heavily on the parity of the number of stages, a more favourable situation arising for…
The problem of solving stochastic differential-algebraic equations (SDAEs) of index one with a scalar driving Brownian motion is considered. Recently, the authors proposed a class of stiffly accurate stochastic Runge-Kutta (SRK) methods…
We introduce a family of stochastic optimization methods based on the Runge-Kutta-Chebyshev (RKC) schemes. The RKC methods are explicit methods originally designed for solving stiff ordinary differential equations by ensuring that their…
We analyze the behavior of the Euler method for delay differential equations under nonstandard assumptions on the right-hand-side function f, when evaluations of f are corrupted by informational noise. We provide theoretical upper bounds on…
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…
The use of symplectic numerical schemes on Hamiltonian systems is widely known to lead to favorable long-time behaviour. While this phenomenon is thoroughly understood in the context of finite-dimensional Hamiltonian systems, much less is…
The recently-introduced relaxation approach for Runge-Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge-Kutta methods in…
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…
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…
In this paper we derive and analyze the properties of explicit singly diagonal implicit Runge-Kutta (ESDIRK) integration methods. We discuss the principles for construction of Runge-Kutta methods with embedded methods of different order for…
We consider a one-dimensional stochastic differential equations (SDE) with irregular coefficients. The purpose of this paper is to estimate the $L^p(\Omega)$-difference of SDEs using the norm of the difference of coefficients, where the…
We consider the application of Runge-Kutta (RK) methods to gradient systems $(d/dt)x = -\nabla V(x)$, where, as in many optimization problems, $V$ is convex and $\nabla V$ (globally) Lipschitz-continuous with Lipschitz constant $L$.…
In this note, we connect two different topics from linear algebra and numerical analysis: hypocoercivity of semi-dissipative matrices and strong stability for explicit Runge--Kutta schemes. Linear autonomous ODE systems with a non-coercive…
The phase field crystal (PFC) method is an efficient technique for simulating the evolution of crystalline microstructures at atomistic length scales and diffusive time scales. Due to the high-order derivatives (sixth-order) and the…
This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low-rank…
In this paper, a family of arbitrarily high-order structure-preserving exponential Runge-Kutta methods are developed for the nonlinear Schr\"odinger equation by combining the scalar auxiliary variable approach with the exponential…
A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under exact solution of their governing PDEs. However, standard temporal schemes, such…
Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method…
We further develop a simple modification of Runge--Kutta methods that guarantees conservation or stability with respect to any inner-product norm. The modified methods can be explicit and retain the accuracy and stability properties of the…