Projection methods and discrete gradient methods for preserving first integrals of ODEs
Richard A. Norton David I. McLaren G. R. W. Quispel Ari Stern Antonella Zanna
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 2079-2098 doi: 10.3934/dcds.2015.35.2079
In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that each (linear) projection method is equivalent to a class of discrete gradient methods, in both single and multiple first integral cases, and known results for discrete gradient methods also apply to projection methods. Thus we prove that in the single first integral case, under certain mild conditions, the numerical solution for a projection method exists and is locally unique, and preserves the order of accuracy of the underlying method. Our results allow considerable freedom for the choice of projection direction and do not have a time step restriction close to critical points.
keywords: discrete gradients Hamiltonian systems. projection energy preserving integrators Geometric integration
Discrete gradient methods have an energy conservation law
Robert I. McLachlan G. R. W. Quispel
Discrete & Continuous Dynamical Systems - A 2014, 34(3): 1099-1104 doi: 10.3934/dcds.2014.34.1099
We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservation law. The discrete conservation law has the same conserved density as the continuous conservation law, while its flux is found by replacing all derivatives of the conserved density appearing in the continuous flux by discrete gradients.
keywords: Discrete conservation laws energy method. discrete gradient methods
Discrete gradient methods for preserving a first integral of an ordinary differential equation
Richard A. Norton G. R. W. Quispel
Discrete & Continuous Dynamical Systems - A 2014, 34(3): 1147-1170 doi: 10.3934/dcds.2014.34.1147
In this paper we consider discrete gradient methods for approximating the solution and preserving a first integral (also called a constant of motion) of autonomous ordinary differential equations. We prove under mild conditions for a large class of discrete gradient methods that the numerical solution exists and is locally unique, and that for arbitrary $p\in \mathbb{N}$ we may construct a method that is of order $p$. In the proofs of these results we also show that the constants in the time step constraint and the error bounds may be chosen independently from the distance to critical points of the first integral.
    In the case when the first integral is quadratic, for arbitrary $p \in \mathbb{N}$, we have devised a new method that is linearly implicit at each time step and of order $p$. A numerical example suggests that this new method has advantages in terms of efficiency.
keywords: discrete gradients linearly implicit methods. Hamiltonian systems order of accuracy Geometric integration energy preserving integrators

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