Conservative numerical methods for

Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string

“…Then, (16) is clearly equivalent to the following nonlinear differential system in R N h (where U h (t) is the vector of the degrees of freedom of u h (t))…”

“…and the positivity property (49) is unconditionally satistied if and only if θ ≥ Before investigating more elaborate discretizations, let us consider the case of the most naïve scheme to discretize (16), which is the most natural extension to the non linear case of the explicit leap frog scheme (θ = 0) for the linear case :…”

“…It appears that results can easily be found in the case of scalar equations (the unknown function takes scalar values), of course in the context of ordinary differential equations ODEs (see for instance [16,26]) but also of some partial differential equations, particularly semi-linear wave equations (see for instance [6,11,12,23,25,31]). The case of systems has been much less investigated and it seems that most results are restricted to very particular systems : see for instance in [27], [15] or [8] in the context of systems of ODE's and [14] (for nonlinear elasticity) or [5] (for nonlinear strings) in the context of systems of PDE's.…”

Conservation properties of a time FE method—part III: Mechanical systems with holonomic constraints

“…Compact description of the augmented system. A more compact description of the augmented system of DAEs treated above can be attained by introducing the vector of coordinates in phase space (26) along with the vector of multipliers (27) The algebraic constraint functions in (16) 3; 4 are collected in (28) Using this notation the augmented Hamiltonian in (25) takes the form…”

A family of conservative finite difference schemes for the dynamical von Karman plate equations

“…In this article, the focus is rather on the issue of numerical stability, and techniques which can be applied to construct algorithms for which numerical stability may be guaranteed. Energy conservation techniques, which have been under study for some time [9][10][11][12][13][14][15] and which relate to early work on the so-called energy-method [16,17], as well as symplectic methods [18,19] are a good match to this particular problem, among others, for various reasons. In general, conservation techniques applied to systems involving low order polynomial approximations to a nonlinearity (such as that which leads to the von Karman system) can yield schemes for which the solution is not only provably numerically stable, but also unique, and efficiently implemented without the need for nonlinear iterative solution techniques; such is not the case for more general types of nonlinearities [20].…”