Efficient Algorithm For Highly Oscillatory Differential Equations

Highly oscillatory differential equations frequently occur in many fields such as molecular dynamics,celestial mechanics,quantum chemistry,orbital mechanics,radio frequency communication systems,etc.One of the most difficult problems in the numerical solution of ordinary differential equations(ODEs)and in differential-algebraic equations is to propose efficient numerical methods for solving highly oscillatory differential equations.The highly oscillatory nature of the solutions impose a very small step size on standard numerical methods for ODEs.However,since as the step size decreases,the amount of computation will increase rapidly and the round-off error may accumulate enormously to a disaster,which will affect the accuracy and efficiency.So the integration of such equations has been a numerical challenge for a long time.In recent decades,many scholars have paid much attention on the numerical methods for solving highly oscillatory differential equations and made great achievements in this field.In the thesis,we are concerned with numerical treatment of highly oscillatory differential equations.Chapter 1 is devoted to a brief introduction of basic theory of highly oscillatory differential equations and main research contents of this thesis.In Chapter 2,we present adiabatic Filon-type methods for liner and nonlinear highly oscillatory second-order differential equations.We reformulate the second-order differential equations into a first-order system by a variable transformation instead of considering the original one.The solution of the transformed system is a smoother function which is more accessible to numerical approximation.We develop adiabatic Filon-type methods for linear systems by approximating the integral as a linear combination of function values and derivatives.We then present a special combination of Filon-type methods and waveform relaxation methods for nonlinear systems.Both types of methods can be used with far larger step sizes than those required by traditional schemes and their performance drastically improves as frequency grows,as are illustrated by numerical experiments in this chapter.In Chapter 3,we design asymptotic-numerical solvers for second-order highly oscillatory problems with a signal frequency.We transform the second-order highly oscillatory differential equations into a first-order system,present the construction of the asymptotic expansion,and verify the asymptotic expansion with necessary uniform bounds of its coefficients and of the remainder term.Then,we propose an asymptotic-numerical solver to obtain the approximation by truncating the first few terms of the asymptotic expansion.Numerical experiments are carried out to demonstrate the efficiency and accuracy of our proposed method.In Chapter 4,We are concerned with asymptotic-numerical solvers for highly oscillatory ordinary differential equations and Hamiltonian systems.First,we derive an asymptotic expansion of the exact solution in inverse powers of the oscillatory parameter,featuring only a finite number of non-zero expansion coefficients in each term.We then use the truncation with the first few terms of the asymptotic expansion as an effective means to approximate the highly oscillatory problems.The error estimation of the asymptotic-numerical solver is analyzed and near conservation of the energy in the Hamiltonian case is proved.The numerical experiments on the Fermi-Pasta-Ulam problem are implemented to show the efficiency of our proposed methods.