On The Study Of Energy-preserving Methods For Differential Equations

Conservative and dissipative systems naturally arise from a wide range of scientif-ic fields such as celestial mechanics, molecular dynamics, circuit simulation, quantum mechanics, and electromagnetics. Energy is one of the most important quantity charac-terizing conservative and dissipative systems. In the spirit of structure-preserving algo-rithm, a good numerical method should preserve as many discrete geometric/physical structures corresponding to the continuous structures of original systems as possible. In particular, energy-preserving or-decaying methods are designed to preserve the first integral or Lyapunov function of conservative or dissipative systems. Numerous the-oretical and experimental results show that the energy-preserving or energy-decaying property guarantees good numerical properties such as the linear error growth, correct qualitative property, and strong stability property.This thesis focuses on constructing and analyzing energy-preserving or-decaying methods having good geometric properties and high algebraic accuracy for conservative or dissipative ODEs as well as PDEs. Chapter 1 briefly introduces background and mo-tivation about energy-preserving methods and relevant numerical methods. Our original published/accepted papers on energy-preserving/decaying methods are also mentioned. The remainder of this thesis is as follows:Chapter 2 is concerned with the conservative or dissipative system having a major linear term in the form y’=Q(My+▽U(y)), y(t0)=y0∈Rd, where Q is a skew-symmetric or negative semi-definite matrix and M is symmetric matrix. For this system, we propose and analyze a symmetric second-order exponential integrator pre-serving the first integral or Lyapunov function H(y)=182τ My+U(y). Comparing to standard methods, this exponential integrator permits larger stepsize and achieve higher solution accuracy provided ‖QM‖?>‖QHessU(y)‖.In Chapter 3, we construct symmetric functionally fitted energy-preserving meth-ods for general Hamiltonian systems y’=J-1▽H(y), y(to)= yo∈Rd, where J is a canonical symplectic matrix. Enlarging the functionally fitted space, we rigorously prove that these methods can achieve arbitrarily high order. By adding trigonometric functions{sin(ωt),cos(wt)} to the functionally fitted space, we obtain trigonometri-cally fitted methods of arbitrarily high order. Comparing to standard energy-preserving method, our trigonometrically fitted methods of the same order are much more efficient for solving oscillatory Hamiltonian systems with a fixed frequency ω.Chapter 4 concerns general multi-symplectic Hamiltonian PDEs with one temporal variable t and two spatial variables x, y Mzt+Kzx+Lzy= ▽zS(z, x, y), z= z(x, y, t) ∈Rd, where M, K, and L are skew-symmetric matrices. Combining the continuous-stage Runge-Kutta temporal discretization and the pseudospectral/Gauss-Legendre spatial discretization, we present a general approach to constructing high-order methods pre-serving a discrete local energy conservation law, which approximates the local energy conservation law (?)t[S-1/2τKzx-1/2zτLzy)+(?)x[1/2zτKzt)+(?)y(1/2zτLzt)=0.Moreover, a number of numerical examples including the non-separable triatomic molecule Hamiltonian system, wind-induced oscillation, a-Fermi-Pasta-Ulam system, perturbed Kepler problem, Duffing equation, Fermi-Pasta-Ulam problem with very high frequency, and (coupled) nonlinear Schrodinger equations with one and two spa-tial variables are provided, demonstrating the excellent numerical behavior of our new methods.