In this paper, we present an abstract framework which describes algebraically the derivation of order conditions independently of the nature of differential equations considered or the type of integrators used to solve them. Our structure includes a Hopf algebra of functions, whose properties are used to answer several questions of prime interest in numerical analysis. In particular, we show that, under some mild assumptions, there exist integrators of arbitrarily high orders for arbitrary (modified)...
In this paper, we consider
Hamiltonian systems. We prove the existence of a first derivative of the flow with respect to initial values and show that it satisfies the symplecticity condition almost everywhere in the phase-space. In a second step, we present a geometric integrator for such systems (called the method) based on B-splines interpolation and a splitting method introduced by McLachlan and Quispel [. (2003) 411–418], and we prove it is convergent, and that it preserves...
In this paper we study a discrete Raman laser amplification model given as a Lotka-Volterra system. We show that in an ideal situation, the equations can be written as a Poisson system with boundary conditions using a global change of coordinates. We address the questions of existence and uniqueness of a solution. We deduce numerical schemes for the approximation of the solution that have good stability.
In this paper we study a discrete Raman laser amplification model
given as a Lotka-Volterra system.
We show that in an ideal situation,
the equations can be written as a Poisson system with
boundary conditions using a global change of coordinates.
We address the questions of existence and uniqueness of a solution.
We deduce numerical schemes for
the approximation of the solution that have good stability.
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