A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages

A novel class of linearly implicit energy-preserving schemes for conservative systems

We consider a kind of differential equations d/dt y(t) = R(y(t))y(t) + f(y(t)) with energy conservation. Such conservative models appear for instance in quantum physics, engineering and molecular dynamics. A new class of energy-preserving schemes is constructed by the ideas of scalar auxiliary variable (SAV) and splitting, from which the nonlinearly implicit schemes have been improved to be linearly implicit. The energy conservation and error estimates are rigorously derived. Based on these results, it is shown that the new proposed schemes have unconditionally energy stability and can be implemented with a cost of solving a linearly implicit system. Numerical experiments are done to confirm these good features of the new schemes

An Optimized Two-Step Hybrid Block Method Formulated in Variable Step-Size Mode for Integrating y′′=f(x,y,y′)y''=f(x,y,y') Numerically.

[EN]An optimized two-step hybrid block method is presented for integrating general second order initial value problems numerically. The method considers two intra-step points which are selected adequately in order to optimize the local truncation errors of the main formulas for the solution and the first derivative at the final point of the block. The new proposed method is consistent, zero-stable and has seventh algebraic order of convergence. To illustrate the performance of the method, some numerical experiments are presented for solving this kind of problems, in comparison with methods of similar characteristics in the literature

Symmetric integrators with improved uniform error bounds and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

In this paper, we are concerned with symmetric integrators for the nonlinear relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation has strong nonlinearity when \eps is small. There two aspects bring significantly numerical burdens in designing numerical methods. We propose and analyze a novel class of symmetric integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the proposed symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy in time is improved to be \mathcal{O}(\varepsilon^{3} \hh^2) and \mathcal{O}(\varepsilon^{4} \hh^4) for the time stepsize \hh. The near energy conservation over long times is established for the multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on a NRKG equation show that the proposed integrators have improved uniform error bounds, excellent long time energy conservation and competitive efficiency

A Zero-dissipative Runge-Kutta-Nystrom Method with Minimal Phase-lag

An explicit Runge-Kutta-Nyström method is developed for solving second-order differential equations of the form q f t, q where the solutions are oscillatory. The method has zero-dissipation with minimal phase-lag at a cost of three-function evaluations per step of integration. Numerical comparisons with RKN3HS, RKN3V, RKN4G, and RKN4C methods show the preciseness and effectiveness of the method developed