Related papers: Exploring Exponential Runge-Kutta Methods: A Surve…
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…
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…
In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation…
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…
The recently introduced polynomial time integration framework proposes a novel way to construct time integrators for solving systems of first-order ordinary differential equation by using interpolating polynomials in the complex time plane.…
In this paper we generalize the polynomial time integration framework to additively partitioned initial value problems. The framework we present is general and enables the construction of many new families of additive integrators with…
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.…
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…
Multirate integration is an increasingly relevant tool that enables scientists to simulate multiphysics systems. Existing multirate methods are designed for equations whose fast and slow variables can be linearly separated using additive or…
In a previous paper, a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions. In this paper, we…
Deriving analytical solutions of ordinary differential equations is usually restricted to a small subset of problems and numerical techniques are considered. Inevitably, a numerical simulation of a differential equation will then always be…
A wide range of physical phenomena exhibit auxiliary admissibility criteria, such as conservation of entropy or various energies, which arise implicitly under the exact solution of their governing PDEs. However, standard temporal schemes,…
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…
The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong…
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…
We develop continuous-stage Runge-Kutta-Nystr\"Om (csRKN) methods in this paper. By leading weight function into the formalism of csRKN methods and modifying the original pattern of continuous-stage methods, we establish a new and larger…
The application of Runge-Kutta schemes designed to enjoy a large region of absolute stability can significantly increase the efficiency of numerical methods for PDEs based on a method of lines approach. In this work we investigate the…
Matrix Riccati differential equations arise in many different areas and are particular important within the field of control theory. In this paper we consider numerical integration for large-scale systems of stiff matrix Riccati…
The numerical efficiency of different schemes for solving the Liouville-von Neumann equation within multilevel Redfield theory has been studied. Among the tested algorithms are the well-known Runge-Kutta scheme in two different…
We study the construction and convergence of semi-explicit and iterative decoupling schemes for an elliptic-parabolic problem using higher-order Runge-Kutta methods. For the semi-explicit schemes, which are constructed using a nearby delay…