Finite Difference Calculus Invariant Structure of a Class of Algorithms for the Nonlinear Klein–Gordon Equation

Abstract

Not all integrals of the motion, however, are of equal importance in mechanics. There are some whose constancy is of profound significance, deriving from the fundamental homogeneity and isotropy of space and time." L. D. Landau and E. M. Lifshitz, MechanicsDedicated to Professor Karl. S. Pister on the occasion of his 70th birthday.Abstract. In a previous work, the authors have presented a formalism for deriving systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities. This formalism is herein cast in the context of exact finite difference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus, in the same sense as that of the corresponding differential equation being derivable from its associated energy function (a conserved quantity). A clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities, which are the consistent counterpart of the invariant quantities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the context of finite difference calculus. Some numerical experiments are presented to illustrate the conservation property of the proposed algorithms.

Backward error analysis for multi-symplectic integration methods

“…Further questions to consider are those related to a theoretical and numerical juxtaposition of multi-symplectic schemes with non-symplectic energy/momentum conserving schemes (cf. [10,11,13,14,20,22,23]), or with other non-conservative schemes.…”

A high-accuracy conservative difference approximation for Rosenau-KdV equation

“…Moreover, the conservative scheme is more suitable for long time calculations for large time step. In [15], Li and Vu-Quoc said: "· · · in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation". Therefore, constructing a high-order conservative difference scheme is a significant task.…”

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

“…For a more complete treatment of this subject, see the classic articles by Li and Vu-Quoc [12] and Vu-Quoc and Li [13]. A grid function f n i,j , for a two-dimensional problem expressed in Cartesian coordinates, is defined for integer i, j , and n, and represents an approximation to a continuous variable at spatial location x = ih x and y = jh y , and at time t = nh t ; where h t is the time step, and h x and h y are the spacings between adjacent grid points in the x and y directions.…”

“…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].…”

“…It also appears that the amplification of roundoff error in the energy is not necessarily related to the stability behaviour in a straightforward way-scheme k (a) generates the least such fluctuation, but is most prone to instability, and schemes k (d) and k (e) exhibit the opposite behavior. It would be interesting to make sense of the distinctions among these schemes, perhaps involving other conserved quantities [12].…”