Efficient structure‐preserving schemes for good Boussinesq equation

Abstract

The good Boussinesq equation is endowed with symplectic conservation law and energy conservation law. In this paper, some new highly efficient structure‐preserving methods for the good Boussinesq equation are proposed by improving the standard finite difference method (FDM). The new methods only use and calculate values at the odd (or even) nodes to reduce the computational cost. We call this kind of methods odd‐even method (OEM). Numerical results show that the OEM and the standard FDM have nearly the same numerical errors under the same mesh partition. However, the OEM is much more efficient than the standard FDM, such as the consumed CPU time and occupied memory.

Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

“…Theorem 2 Assume that the solution of (4) is C 3 [a, b] as a periodic function. Then, the Hamiltonian functional (10) is constant along the solution of (4).…”

“…Proof. In fact, using arguments similar to those used in the previous theorem, one has: (7), (10), and (11) represents the relevant geometric properties of the solution we are interested in, which we shall reproduce in the discrete approximation.…”

“…This latter, in turn, is equivalent, up to a constant, to the functional (10), via the transformations (15)- (19).…”

“…Hereafter, we shall assume that u 0 (x) and v 0 (x) are such that the solution of problem (4) and (5) is regular enough, as a periodic function on [a, b], for all t ≥ 0. The numerical solution of (1), (3) or (4) has been developed along different directions, ranging from the pseudo-spectral or splitting approach [13][14][15][16][17][18][19]46], up to finite-difference and finite-element schemes [20][21][22][23][24]47], as well as structure-preserving methods [10,25,26] and energy-preserving methods [27,28]. In particular, [11,12] consider an energy-conserving strategy based on the Hamiltonian boundary value methods (HBVMs) for the "good" Boussinesq and the improved Boussinesq equation, respectively, while a second-order symplectic method preserving the energy and the momentum is considered in [29].…”

Efficient energy‐preserving scheme of the three‐coupled nonlinear Schrödinger equation

“…AVF method upon preserving some structure characters of a dynamic system is a suitable candidate to design energy‐preserving schemes. () It is widely used to construct structure‐preserving schemes for partial differential equations . To improve the computational efficiency, high‐order compact method aims to design high‐order schemes with smaller stencils .…”