We introduce a new class of parametricization structure-preserving partitioned Runge-Kutta ($α$-PRK) methods for Hamiltonian systems with holonomic constraints. When the scalar parameter $α=0$, the methods are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs, which can preserve all the quadratic invariants and the constraints. When $α\neq 0$, the methods are also shown to preserve all the quadratic invariants and the constraints manifold exactly. At the same time, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $α^*=α(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the parametrized PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. The parametric $α$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.
Consider the problem of finding the maximal nonpositive solvent $Φ$ of the quadratic matrix equation (QME) $X^2 + BX + C =0$ with $B$ being a nonsingular $M$-matrix and $C$ an $M$-matrix such that $B^{-1}C\ge 0$, and $B - C - I$ a nonsingular $M$-matrix. Such QME arises from an overdamped vibrating system. Recently, Yu et al. ({\em Appl. Math. Comput.}, 218: 3303--3310, 2011) proved that $ρ(Φ)\le 1$ for this QME. In this paper, we slightly improve their result and prove $ρ(Φ)< 1$, which is important for the quadratic convergence of the structure-preserving doubling algorithm. Then, a new globa…
A novel class of explicit high-order energy-preserving methods are proposed for general Hamiltonian partial differential equations with non-canonical structure matrix. When the energy is not quadratic, it is firstly done that the original system is reformulated into an equivalent form with a modified quadratic energy conservation law by the energy quadratization approach. Then the resulting system that satisfies the quadratic energy conservation law is discretized in time by combining explicit high-order Runge-Kutta methods with orthogonal projection techniques. The proposed schemes are shown …
Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods
Dipartimento di Matematica e Informatica “U. Dini”, Viale Morgagni 67/a, 50134 Firenze, Italy（都尼数学与信息系）；Dipartimento di Matematica, Via Orabona 4, 70125 Bari, Italy（巴里大学数学系）
We perform a numerical analysis of a class of randomly perturbed {H}amiltonian systems and {P}oisson systems. For the considered additive noise perturbation of such systems, we show the long time behavior of the energy and quadratic Casimirs for the exact solution. We then propose and analyze a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence one, weak order of convergence two. These properties are illustrated with numerical experiments.
In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system, which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system. Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem. Under t…
Structure-preserving deflation of critical eigenvalues in quadratic eigenvalue problems associated with damped mass-spring systems
IIT Guwahati（印度理工学院古瓦哈提分校）；TU Berlin（柏林工业大学）；Josip Juraj Strossmayer University of Osijek, School of Applied Mathematics and Informatics（奥西耶克约瑟普尤拉伊斯特罗斯梅亚耶尔大学应用数学与信息学院）
In this paper, we present a novel class of high-order energy-preserving schemes for solving the Zakharov-Rubenchik equations. The main idea of the scheme is first to introduce an quadratic auxiliary variable to transform the Hamiltonian energy into a modified quadratic energy and the original system is then reformulated into an equivalent system which satisfies the mass, modified energy as well as two linear invariants. The symplectic Runge-Kutta method in time, together with the Fourier pseudo-spectral method in space is employed to compute the solution of the reformulated system. The main be…
In this paper, we are concerned with arbitrarily high-order momentum-preserving and energy-preserving schemes for solving the generalized Rosenau-type equation, respectively. The derivation of the momentum-preserving schemes is made within the symplectic Runge-Kutta method, coupled with the standard Fourier pseudo-spectral method in space. Then, combined with the quadratic auxiliary variable approach and the symplectic Runge-Kutta method, together with the standard Fourier pseudo-spectral method, we present a class of high-order mass- and energy-preserving schemes for the Rosenau equation. Fin…
In this paper, we present two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition/discrete empirical interpolation method (POD/DEIM) that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit …
In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for preserving them have been extensively studied. Since linear invariants can be easily kept after discretisation, quadratic invariants are essentially the simplest ones. Quadratic invariants are important objects that appear not only in many physical examples but also in the computationally efficient conservative schemes for general invariants such as scalar auxili…
In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, the key idea is based on the extrapolation/prediction-correction technique and the symplectic Runge-Kutta method in time, together with the standard Fourier pseudo-spectral method in space. We show that the scheme is linear, high-order, unconditionally stable and preserves the discrete momentum of the system. For the energy-preserving scheme, it is mainly based on the energy quad…
In this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an equivalent system, which inherits a quadratic energy. {The new system is then discretized} by the standard Fourier pseudo-spectral method, which can exactly preserve the semi-discrete energy conservation law. Subsequently, { a symplectic Runge-Kutta method such as the Gauss collocation method is applied} for the resulting semi-discrete system to arrive at a…
This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the literature preserve fully discrete local conservation laws whose densities are either quadratic or a Hamiltonian. The approach generalizes to time integrators with more steps and conservation laws of other kinds; higher-dimensional PDEs can be treated by iterating the new strategy. We use the Boussinesq equation as a benchmark and introduce new families of schemes …
Computationally efficient, structure-preserving reduced-order methods are developed for the Korteweg de Vries (KdV) equations in Hamiltonian form. The KdV equation is discretized in space by finite differences. The resulting skew-gradient system of ordinary differential equations (ODEs) is integrated with the linearly implicit Kahan's method, which preserves the Hamiltonian approximately. We have shown, using proper orthogonal decomposition (POD), the Hamiltonian structure of the full-order model (FOM) is preserved by the reduced-order model (ROM). The quadratic nonlinear terms of the KdV equa…
Second-order linear structure-preserving modified finite volume schemes for the regularized long-wave equation
Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）；College of Science, Nanjing University of Aeronautics and Astronautics（南京航空航天大学理学院）
A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schrödinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation is first reformulated into an equivalent form which satisfies a quadratic energy. The spatial derivatives of the system are then approximated with the standard Fourier pseudo-spectral method. Subsequently, we apply the extrapolation technique/prediction-correction strategy to the nonlinear terms of the semi-discretized system and a linearized energy-conservin…
Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
Università di Firenze（佛罗伦萨大学）；Università di Bari（巴里大学）
