相关论文: Variational symplectic diagonally implicit Runge-K…

In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge-Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems,…

We prove that a class of A-stable symplectic Runge--Kutta time semidiscretizations (including the Gauss--Legendre methods) applied to a class of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic functions with analytic…

A variational integrator of arbitrarily high-order on the special orthogonal group $SO(n)$ is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second-order derivative of the…

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and…

One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. Recently, a new class of methods, named Hamiltonian Boundary…

In this paper, we develop a new type of Runge--Kutta (RK) discontinuous Galerkin (DG) method for solving hyperbolic conservation laws. Compared with the original RKDG method, the new method features improved compactness and allows simple…

It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but it has been shown that no B-Series method can be volume…

For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge-Kutta methods such as the Radau IIA methods…

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $\epsilon$, and the schemes under study preserve the…

The recent interest in structure preserving stochastic Lagrangian and Hamiltonian systems raises questions regarding how such models are to be understood and the principles through which they are to be derived. By considering a…

This work deals with two groups of spectral analysis results for matrices arising in fully implicit Runge-Kutta methods used for linear time-dependent partial differential equations. These were applied for different formulations of the same…

Lagrangian systems subject to fractional damping can be incorporated into a variational formalism. The construction can be made by doubling the state variables and introducing fractional derivatives \cite{JiOb2}. The main objective of this…

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…

Classical mathematical techniques such as discrete integration, gradient descent optimization, and state estimation (exemplified by the Runge-Kutta method, Gauss-Newton minimization, and extended Kalman filter or EKF, respectively), rely on…

This paper presents a systematic theoretical framework to derive the energy identities of general implicit and explicit Runge--Kutta (RK) methods for linear seminegative systems. It generalizes the stability analysis of explicit RK methods…

We study symplectic numerical integration of mechanical systems with a Hamiltonian specified in non-canonical coordinates and its application to guiding-center motion of charged plasma particles in magnetic confinement devices. The…

In this paper, we present continuous-stage partitioned Runge-Kutta (csPRK) methods for energy-preserving integration of Hamiltonian systems. A sufficient condition for the energy preservation of the csPRK methods is derived. It is shown…

We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…

In this paper we discuss the use of implicit Runge-Kutta schemes for the time discretization of optimal control problems with evolution equations. The specialty of the considered discretizations is that the discretizations schemes for the…

This article uses Cartan-K\"ahler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the…