9 research outputs found
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
, which is inversely proportional to the speed of light.
The highly oscillatory property in time of this model corresponds to the
parameter 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
Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations
We apply the Fourier-least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods