Yuezheng Gong

Qi Hong, Jialing Wang, Yuezheng Gong

In this paper, based on the weak form of the Hamiltonian formulation of the regularized long-wave equation and a novel approach of transforming the original Hamiltonian energy into a quadratic functional, a fully implicit and three linear-implicit energy conservation numerical schemes are respectively proposed. The resulting numerical schemes are proved theoretically to satisfy the energy conservation law in the discrete level. Moreover, these linear-implicit schemes are efficient in practical computation because only a linear system need to be solved at each time step. The proposed schemes are both second order accurate in time and space. Numerical experiments are presented to show all the proposed schemes have satisfactory performance in providing accurate solution and the remarkable energy-preserving property.

Qi Hong, Yushun Wang, Yuezheng Gong

In this paper, two novel linear-implicit and momentum-preserving Fourier pseudo-spectral schemes are proposed and analyzed for the regularized long-wave equation. The numerical methods are based on the blend of the Fourier pseudo-spectral method in space and the linear-implicit Crank-Nicolson method or the leap-frog scheme in time. The two fully discrete linear schemes are shown to possess the discrete momentum conservation law, and the linear systems resulting from the schemes are proved uniquely solvable. Due to the momentum conservative property of the proposed schemes, the Fourier pseudo-spectral solution is proved to be bounded in the discrete $L^{\infty}$ norm. Then by using the standard energy method, both the linear-implicit Crank-Nicolson momentum-preserving scheme and the linear-implicit leap-frog momentum-preserving scheme are shown to have the accuracy of $\mathcal{O}(τ^2+N^{-r})$ in the discrete $L^{\infty}$ norm without any restrictions on the grid ratio, where $N$ is the number of nodes and $τ$ is the time step size. Numerical examples are carried out to verify the correction of the theory analysis and the efficiency of the proposed schemes.

Yuezheng Gong, Qi Wang, Zhu Wang

The proper orthogonal decomposition reduced-order models (POD-ROMs) have been widely used as a computationally efficient surrogate models in large-scale numerical simulations of complex systems. However, when it is applied to a Hamiltonian system, a naive application of the POD method can destroy its Hamiltonian structure in the reduced-order model. In this paper, we develop a new reduce-order modeling approach for the Hamiltonian system, which uses the traditional framework of Galerkin projection-based model reduction but modifies the ROM so that the appropriate Hamiltonian structure is preserved. Since the POD truncation can degrade the approximation of the Hamiltonian function, we propose to use the POD basis from shifted snapshots to improve the Hamiltonian function approximation. We further derive a rigorous a priori error estimate of the structure-preserving ROM and demonstrate its effectiveness in several numerical examples. This approach can be readily extended to dissipative Hamiltonian systems, port-Hamiltonian systems etc.