5 research outputs found
On the multi-symplectic structure of the Serre-Green-Naghdi equations
In this short note, we present a multi-symplectic structure of the
Serre-Green-Naghdi (SGN) equations modelling nonlinear long surface waves in
shallow water. This multi-symplectic structure allow the use of efficient
finite difference or pseudo-spectral numerical schemes preserving exactly the
multi-symplectic form at the discrete level.Comment: 10 pages, 1 figure, 30 references. Other author's papers can be
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Structure-preserving reduced-order modelling of Korteweg de Vries equation
Computationally efficient, structure-preserving reduced-order methods are
developed for the Korteweg de Vries (KdV) equations in Hamiltonian form. The
KdV equation is discretized in space by finite differences. The resulting
skew-gradient system of ordinary differential equations (ODEs) is integrated
with the linearly implicit Kahan's method, which preserves the Hamiltonian
approximately. We have shown, using proper orthogonal decomposition (POD), the
Hamiltonian structure of the full-order model (FOM) is preserved by the
reduced-order model (ROM). The quadratic nonlinear terms of the KdV equation
are evaluated efficiently by the use of tensorial methods, clearly separating
the offline-online cost of the FOMs and ROMs. The accuracy of the reduced
solutions, preservation of the Hamiltonian, momentum and mass, and
computational speed-up gained by ROMs are demonstrated for the one-dimensional
KdV equation, coupled KdV equations and two-dimensional Zakharov-Kuznetzov
equation with soliton solutionsComment: 20 pages, 10 figures, 1 tabl
The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs
AbstractIn this paper, the multi-symplectic Fourier pseudospectral (MSFP) method is generalized to solve two-dimensional Hamiltonian PDEs with periodic boundary conditions. Using the Fourier pseudospectral method in the space of the two-dimensional Hamiltonian PDE (2D-HPDE), the semi-discrete system obtained is proved to have semi-discrete multi-symplectic conservation laws and a global symplecticity conservation law. Then, the implicit midpoint rule is employed for time integration to obtain the MSFP method for the 2D-HPDE. The fully discrete multi-symplectic conservation laws are also obtained. In addition, the proposed method is applied to solve the Zakharov–Kuznetsov (ZK) equation and the Kadomtsev–Petviashvili (KP) equation. Numerical experiments on soliton solutions of the ZK equation and the KP equation show the high accuracy and effectiveness of the proposed method