Construction And Analysis Of Structure-Preserving Algorithms For Partial Differential Equations

With the rapid development in the science, more and more physical, chemical and biological processes can be described by using the nonlinear evolution equa-tions or Maxwell’s equations. In many cases, they are conservative systems, there-fore how to establish efficient and conserved algorithms attracts many attention. We devote the present thesis to the local structure-preserving algorithms, including multi-symplectic schemes and local energy-and momentum-preserving schemes, for some nonlinear evolution equations, and efficient structure-preserving algorithms for three-dimensional Maxwell’s equations. The main results are as follows.I. Whether the traditional symplectic algorithm, energy-or momentum-preserving algorithms could be used for the partial differential equations (PDEs) depends on not only the conservative system itself but also the appropriate boundary conditions. That is to say, we cannot apply an structure-preserving algorithms to a conserva-tive system without appropriate boundary conditions. In order to enlarge the appli-cable scope of the structure-preserving algorithms, we propose a series of the local structure-preserving algorithms, including multi-symplectic schemes, local energy-and momentum-preserving schemes, for the coupled nonlinear Schrodinger system, Boussinesq equation and Klein-Gordon-Schrodinger system. The proposed algorithms preserve respectively the corresponding discrete local conservation laws in any local time-space region, which are independent of the boundary conditions. With appro-priate boundaries, such as periodic or homogeneous conditions, the local structure-preserving algorithms will be the global structure-preserving algorithms. We also conduct the nonlinear stability and convergence analysis for some of these schemes. Numerical results verify the theoretical analysis. Numerical comparison with some existing methods show the good performance of the proposed schemes.II. Three-dimensional Maxwell’s equations are bi-Hamiltonian system. We pro-pose two schemes (AVF(2) and AVF(4)) for Maxwell’s equations, by discretizing the Hamiltonian formulations with Fourier pseudospectral method for spatial discretiza-tion and average vector field method for time integration. Both AVF(2) and AVF(4) hold the two Hamiltonian energies automatically, while being energy-, momentum-and divergence-preserving, unconditionally stable, non-dissipative and spectral accu-rate. The error estimates show that the AVF(2)/AVF(4) scheme converges with spectral accuracy in space and second-order/fourth-order accuracy in time, respectively. Nu-merical results are in agreement with the theoretical analysis results.Ⅲ. The AVF(2) and AVF(4) schemes are derived by discretizing the three Maxwell’s equations in space directly, and thus they are inconvenient for coding. For devising more efficient energy-conserved schemes for the three Maxwell’s equa-tions, we utilize the the exponential operator splitting technique, and then obtain tem-poral second-and fourth-order splitting methods. Each subproblem of the splitting methods is a Hamiltonian system with the same Hamiltonian as the original prob-lem. We propose the energy-conserved S-AVF(2) and S-AVF(4) schemes for the three-dimensional Maxwell’s equations, based on the splitting methods, the Fourier pseudo-spectral method and the averaged vector field method. The proposed schemes are energy-conserved, high-order accurate and unconditionally stable, while being imple-mented explicitly. Rigorous error estimates of the schemes are established. Numerical results support the theoretical analysis.