Some Applications Of Runge-Kutta Methods

Hamiltonian systems arise widely in the ?eld of physics,　mechanics, engineering,pure and applied mathematics, etc. It is　generally accepted that all real physical pro-cesses with　negligible dissipation could be expressed, in some way, by　Hamiltonianformalism, so that the research work for corresponding　numerical methods is of impor-tant interest. The important property for Hamiltonian systems is the Poincare and Li-uville's　conservation law of phase areas. In numerically solving　these equations, wehope that the numerical schemes can hold this　property, and the corresponding numeri-cal methods are called as symplectic methods.Since the founding of the symplectic algorithm by famous Chinese scientist Feng Kangand American scientist R.D.Ruth, the resulting theory of structure-preserving algorithm for?nite dimensional Hamiltonian system has been well established. However, much remainsto be done for in?nite dimensional Hamiltonian System (IDHS) is of greater importance dueto its wider application.Recently, Bridges and Marsden have proposed the concepts of the multisymplecticHamilton system and the corresponding multisymplectic schemes for some physical or math-ematical modelings from the different view points. In this thesis, we make some research onthe multisymplectic scheme and prove the conservation law in implicit symplectic Runge-Kutta methods and partition Runge-Kutta methods for Regularized Long-Wave(RLW) equa-tion and Kadomtsev-Petiashvili(KP) equation.Another part in this dissertation is to study the numerical stability of delay differen-tial equations. Actually, many researchers discussed the approximate stability behavior ofRunge-Kutta methods or Linear Multi-step Methods for ?rst order delay differential equa-tion. We will study the stability behavior of numerical solution of DDEs in this thesis, andgive the suf?cient and necessary condition of GP-stability, GPL-stability of Runge-Kuttamethods for second order delay differential equation.