Structure-preserving Algorithms For Some Partial Differential Equations

Plenty of important equations in mathematical physics can be written as Hamiltonian systems, which preserve energy conservation laws and symplectic geometric structures. The basic principle of modern numerical algorithms is to preserve the structures of the original problems. Therefore, it is meaningful to study the numerical algorithms which preserve the energy conservation laws or the symplectic geometric structures of the Hamiltonian partial differential equations. This dissertation is devoted to construct novel structure-preserving algorithms for some important nonlinear partial differential equations. Meanwhile, we investigate the responding discrete conservative properties,convergence and numerical stability of proposed algorithms theoretically, which would guarantee their effectiveness during long-time computations. Main contributions of this dissertation are as follows:1. The explicit multi-symplectic algorithm is proposed to solve the two-dimensional Zakharov–Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral scheme for spatial discretization and the symplectic Euler scheme for temporal discretization. It is verified that the proposed method has corresponding discrete multisymplectic conservation laws. Numerical results indicate that the proposed algorithm has high accuracy, the property of structure-preservation, and numerical stability during longtime computations. Meanwhile, the CPU time by proposed explicit algorithm is much less than that via the implicit multi-symplectic algorithm.2. All kinds of novel efficient structure-preserving algorithms for one-dimensional and two-dimensional Schr?dinger equations are constructed. Specifically, firstly we propose a multi-symplectic wavelet splitting method to solve the strongly coupled nonlinear Schr?dinger equations. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation. Secondly, based on the wavelet collocation scheme, Fourier pseudospectral scheme and split-step technique, two novel conservative algorithms are constructed for solving Schr?dinger equations with variable coefficients, including cubic nonlinear Schr?dinger equation, Gross-Pitaevskii equation, and two-dimensional time variable Schr?dinger equation. It is proved that they can preserve the charge conservation exactly. The global energy and momentum conservation laws can be preserved under several conditions. Thirdly, we propose two local conservative algorithms for solving two-dimensional nonlinear Schr?dinger equation. Without consideration of the boundary conditions, they can preserve corresponding local energy and momentum conservation laws exactly at arbitrary spatial-temporal regions. Meanwhile,the charge, global energy and global momentum conservation laws can be conserved under periodic boundary conditions. Theoretical analyses, including conservative properties,error estimation and stability analysis, are investigated. Finally, considering the coupled nonlinear Schr?dinger system with multiply components, we provide a novel framework for constructing energy-preserving algorithms. In detail, based on the high order compact method, Fourier pseudospectral method and wavelet collocation method, a series of high accurate conservative algorithms is presented. We prove the proposed framework can preserve the corresponding discrete charge and energy conservation laws exactly.3.We propose energy-preserving and multi-symplectic wavelet algorithms for solving the nonlinear Dirac model in quantum physics. We theoretically analyse that the proposed algorithms can preserve crucial conservative properties and the inner geometric structures,respectively. Meanwhile, we take the splitting technique and explicit strategy into original algorithms for the improvement of efficiency. Numerical results reveal that the proposed algorithms are stable during long time computations, and the modified algorithms are more efficient.4. Based on the hyperbolic-elliptic formulation of the Degasperis–Procesi equation,we propose a hyperbolic-elliptic split-step algorithm. The Degasperis–Procesi equation can be split into one Burgers equation and one Benjamin–Bona–Mahony equation. For the Burgers subsystem, essentially non-oscillatory finite-volume method is employed. For the Benjamin–Bona–Mahony subsystem, multi-symplectic Fourier pseudospectral method is proposed. Numerical experiments show the proposed algorithm can treat discontinuities well and be effective during long-time numerical computations.