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This paper analyzes the stability of the class of Time-Accurate and Highly-Stable Explicit Runge-Kutta (TASE-RK) methods, introduced in 2021 by Bassenne et al. (J. Comput. Phys.) for the numerical solution of stiff Initial Value Problems…
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…
A general purpose, modular program package for the integration of large number of independent ordinary differential equation systems capable of using professional graphics cards is presented. The available numerical schemes are the explicit…
An explicit stabilized additive Runge-Kutta scheme is proposed. The method is based on a splitting of the problem in severely stiff and mildly stiff subproblems, which are then independently solved using a Runge-Kutta-Chebyshev scheme. The…
Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were proposed and analyzed in [8]. These specially designed methods use reduced precision or the implicit computations and full…
Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be…
A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schr\"odinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation…
The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of…
Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The…
Finite differences and Runge-Kutta time stepping schemes used in Computational AeroAcoustics simulations are often optimized for low dispersion and dissipation (e.g. DRP or LDDRK schemes) when applied to linear problems in order to…
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…
A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a…
In this paper is described a general 2-nd order accurate (weak sense) procedure for stablizing Monte-Carlo simulations of Ito stochastic differential equations. The splitting procedure includes explicit Runge-Kutta methods, semi-implicit…
A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the…
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…
The task of integrating a large number of independent ODE systems arises in various scientific and engineering areas. For nonstiff systems, common explicit integration algorithms can be used on GPUs, where individual GPU threads…
We apply Runge-Kutta methods to linear partial differential-algebraic equations of the form $Au_t(t,x) + B(u_{xx}(t,x)+ru_x(t,x))+Cu(t,x) = f(t,x)$, where $A,B,C\in\R^{n,n}$ and the matrix $A$ is singular. We prove that under certain…
In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. First of all, we analyze the symplectic conditions of two kinds of exponential integrators, and present a…
In this paper, we study the uniform accuracy of implicit-explicit (IMEX) Runge-Kutta (RK) schemes for general linear hyperbolic relaxation systems satisfying the structural stability condition proposed in \cite{yong_singular_1999}. We…
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,…