Related papers: Jacobian-free implicit MDRK methods for stiff syst…
This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions…
Based on the recent development of Jacobian-free Lax-Wendroff (LW) approaches for solving hyperbolic conservation laws [Zorio, Baeza and Mulet, Journal of Scientific Computing 71:246-273, 2017], [Carrillo and Par\'es, Journal of Scientific…
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…
As a method of universal approximation deep neural networks (DNNs) are capable of finding approximate solutions to problems posed with little more constraints than a suitably-posed mathematical system and an objective function.…
Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to…
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…
When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by the classical order condition theory. Commonly, this order reduction phenomenon is…
Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the…
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…
Linearly implicit Runge-Kutta methods with approximate matrix factorization can solve efficiently large systems of differential equations that have a stiff linear part, e.g. reaction-diffusion systems. However, the use of approximate…
Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic…
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a…
Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new…
Dynamical low-rank approximation allows for solving large-scale matrix differential equations (MDEs) with significantly fewer degrees of freedom and has been applied to a growing number of applications. However, most existing techniques…
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a…
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step…
This paper proposes a new method for differentiating through optimal trajectories arising from non-convex, constrained discrete-time optimal control (COC) problems using the implicit function theorem (IFT). Previous works solve a…
This paper develops a new class of linearly implicit time integration schemes called Linearly-Implicit Runge-Kutta-W (LIRK-W) methods. These schemes are based on an implicit-explicit approach which does not require a splitting of the right…
A fast and robust Jacobian-free time-integration method - called Minimum-error Adaptation of a Chemical-Kinetic ODE Solver (MACKS) - for solving stiff ODEs pertaining to chemical-kinetics is proposed herein. The MACKS formulation is based…
The work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are…