Related papers: Factorized Runge-Kutta-Chebyshev Methods

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production-destruction system (PDS) irrespective of the chosen time step size. Although they are now of…

When simulating trajectories by integrating time-continuous car-following models, standard integration schemes such as the forth-order Runge-Kutta method (RK4) are rarely used while the simple Euler's method is popular among researchers. We…

Gradient flows of unit vector fields arise in a wide range of physical models such as harmonic map heat flows, nematic liquid crystals, and magnetization dynamics. Designing numerical schemes that simultaneously preserve the unit length…

Irksome is a library based on the Unified Form Language (UFL) that enables automated generation of Runge--Kutta methods for time-stepping finite element spatial discretizations of partial differential equations (PDE). Allowing users to…

This work introduces a new class of Runge-Kutta methods for solving nonlinearly partitioned initial value problems. These new methods, named nonlinearly partitioned Runge-Kutta (NPRK), generalize existing additive and component-partitioned…

A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\mathbf{u}' = \mathbf{f}(t,\mathbf{u}) + G(t,\mathbf{u}) \mathbf{u}$, where…

In this paper, we obtain results on exponential stability of second order delay differential equations, which are based on a version of the Floquet theory for delay differential equations of the second order we proposed. Our version allows…

We apply the concept of effective order to strong stability preserving (SSP) explicit Runge-Kutta methods. Relative to classical Runge-Kutta methods, methods with an effective order of accuracy are designed to satisfy a relaxed set of order…

We propose entropy-preserving and entropy-stable partitioned Runge--Kutta (RK) methods. In particular, we extend the explicit relaxation Runge--Kutta methods to IMEX--RK methods and a class of explicit second-order multirate methods for…

We propose a physics-informed consistency modeling framework for solving partial differential equations (PDEs) via fast, few-step generative inference. We identify a key stability challenge in physics-constrained consistency training, where…

Segregated Runge-Kutta (SRK) schemes are time integration methods for the incompressible Navier-Stokes equations. In this approach, convection and diffusion can be independently treated either explicitly or implicitly, which in particular…

Explicit integrating factor Runge-Kutta methods are attractive and popular in developing high-order maximum bound principle preserving time-stepping schemes for Allen-Cahn type gradient flows. However, they always suffer from the…

A mixed accuracy framework for Runge--Kutta methods presented in [Grant, JSC 2022] has been shown to speed up the computation in diagonally implicit Runge--Kutta (DIRK) methods by using less expensive low accuracy approaches for the…

The parametric instability arising when ordinary differential equations (ODEs) are numerically integrated with Runge-Kutta-Nystr\"om (RKN) methods with varying step sizes is investigated. It is shown that when linear constant coefficient…

An 11-dimensional family of embedded (4, 5) pairs of explicit 9-stage Runge-Kutta methods with an interpolant of order 5 is derived. Two optimized for efficiency pairs are presented.

Traditional time discretization methods use a single timestep for the entire system of interest and can perform poorly when the dynamics of the system exhibits a wide range of time scales. Multirate infinitesimal step (MIS) methods (Knoth…

Mixed precision Runge--Kutta methods have been recently developed and used for the time-evolution of partial differential equations. Two-derivative Runge--Kutta schemes may offer enhanced stability and accuracy properties compared to…

In [11] and [5], an error estimate of optimal convergence rates and optimal error propagation (optimal^2) was given for the Runge-Kutta discontinuous Galerkin (RKDG) method solving the scalar nonlinear conservation laws in the case of…

Considering the use of dynamical systems in practical applications, often only limited regions in the time or frequency domain are of interest. Therefor, it usually pays off to compute local approximations of the used dynamical systems in…

It is shown that for a parabolic problem with maximal $L^p$-regularity (for $1<p<\infty$), the time discretization by a linear multistep method or Runge--Kutta method has maximal $\ell^p$-regularity uniformly in the stepsize if the method…