Related papers: Implicit Runge-Kutta schemes for optimal control p…

The non-differentiability of the singular nonlinearity (such as $f=\ln|u|^2$) at $u=0$ presents significant challenges in devising accurate and efficient numerical schemes for the logarithmic Schr\"{o}dinger equation (LogSE). To address…

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…

A space-time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes,…

In current research, we analyse dissipation and dispersion characteristics of most accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K) methods. These methods, known for their $A$-stability and immense accuracy, are…

Runge-Kutta (RK) methods may exhibit order reduction when applied to certain stiff problems. While fully implicit RK schemes exist that avoid order reduction via high-stage order, DIRK (diagonally implicit Runge-Kutta) schemes are…

This paper investigates the performance of a subclass of exponential integrators, specifically explicit exponential Runge--Kutta methods. It is well known that third-order methods can suffer from order reduction when applied to linearized…

This paper proposes an implicit family of sub-step integration algorithms grounded in the explicit singly diagonally implicit Runge-Kutta (ESDIRK) method. The proposed methods achieve third-order consistency per sub-step and thus the…

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…

This paper concerns the numerical procedure for solving hybrid optimal control problems with sliding modes. The proposed procedure has several features which distinguishes it from the other procedures for the problem. First of all a sliding…

Many control, optimization, and learning algorithms rely on discretizations of continuous-time contracting systems, where preservation of contractivity under numerical integration is key for stability, robustness, and reliable fixed-point…

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive…

In this paper, we extend the Paired-Explicit Runge-Kutta schemes by Vermeire et. al. to fourth-order of consistency. Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which…

A unified theoretical framework is suggested to examine the energy dissipation properties at all stages of additive implicit-explicit Runge-Kutta (IERK) methods up to fourth-order accuracy for gradient flow problems. We construct some…

A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of…

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…

In order to solve continuous-time optimal control problems, direct methods transcribe the infinite-dimensional problem to a nonlinear program (NLP) using numerical integration methods. In cases where the integration error can be manipulated…

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…

Motivated by studies on fully discrete numerical schemes for linear hyperbolic conservation laws, we present a framework on analyzing the strong stability of explicit Runge-Kutta (RK) time discretizations for semi-negative autonomous linear…

We propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable…

In the paper explicit functional continuous Runge-Kutta and Runge-Kutta-Nystr\"om methods for retarded functional differential equations are considered. New methods for first order equations as well as for second order equations of the…