Related papers: Symmetric-adjoint and symplectic-adjoint methods a…
We are studying Runge-Kutta methods along complex paths of integration from a geometric point of view. Thereby we derive special complex time grids, which applied to the problem of integrating a linear autonomous system of ordinary…
Cell collective migration plays a crucial role in a variety of physiological processes. In this work, we propose the Runge-Kutta random feature method to solve the nonlinear and strongly coupled multiphase flow problems of cells, in which…
Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this paper, we explore the geometric properties and develop methods…
We give a detailed study of the symplectic geometry of a family of integrable systems obtained by coupling two angular momenta in a non trivial way. These systems depend on a parameter t $\in$ [0, 1] and exhibit different behaviors…
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…
We present a new approach for constructing covariant symplectic structures for geometrical theories, based on the concept of adjoint operators. Such geometric structures emerge by direct exterior derivation of underlying symplectic…
The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge--Kutta methods, are applicable to general time--reversible differential equations, not just those derived from…
The recently-introduced relaxation approach for Runge-Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge-Kutta methods in…
We propose a family of integrators, Flow-Composed Implicit Runge-Kutta (FCIRK) methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step…
We provide a note on continuous-stage Runge-Kutta methods (csRK) for solving initial value problems of first-order ordinary differential equations. Such methods, as an interesting and creative extension of traditional Runge-Kutta (RK)…
In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on the…
Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for…
In this paper, we study symplectic integration of canonical Hamiltonian systems with Jacobi polynomials. The relevant theoretical results of continuous-stage Runge-Kutta methods are revisited firstly and then symplectic methods with Jacobi…
Based on a variational principle with a stochastic forcing, we indicate that the stochastic Schr\"odinger equation in Stratonovich sense is an infinite-dimensional stochastic Hamiltonian system, whose phase flow preserves symplecticity. We…
Complex dynamical networks appear in a wide range of physical, biological, and engineering systems. The coupling of subsystems with varying time scales often results in multirate behavior. During the simulation of highly integrated…
This paper illuminates the derivation, the applicability, and the efficiency of the Multiplicative Runge-Kutta Method, derived in the frame- work of geometric multiplicative calculus. The removal of the restrictions of geometric…
In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more…
Using simplifying assumptions that are related to the time reversal symmetry, a 1-dimensional family of 8-stage pseudo-symplectic Runge-Kutta methods of order (4, 8), i.e., methods of order 4 that preserve symplectic structure up to order…
Implicit-explicit (IMEX) Runge-Kutta methods play a major rule in the numerical treatment of differential systems governed by stiff and non-stiff terms. This paper discusses order conditions and symplecticity properties of a class of IMEX…
We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential…