An asynchronous leapfrog method II

A second order explicit one-step numerical method for the initial value problem of the general ordinary differential equation is proposed. It is obtained by natural modifications of the well-known leapfrog method, which is a second order, two-step, explicit method. According to the latter method, the input data for an integration step are two system states, which refer to different times. The usage of two states instead of a single one can be seen as the reason for the robustness of the method. Since the time step size thus is part of the step input data, it is complicated to change this size during the computation of a discrete trajectory. This is a serious drawback when one needs to implement automatic time step control. The proposed modification transforms one of the two input states into a velocity and thus gets rid of the time step dependency in the step input data. For these new step input data, the leapfrog method gives a unique prescription how to evolve them stepwise. The stability properties of this modified method are the same as for the original one: the set of absolute stability is the interval [-i,+i] on the imaginary axis. This implies exponential growth of trajectories in situations where the exact trajectory has an asymptote. By considering new evolution steps that are composed of two consecutive old evolution steps we can average over the velocities of the sub-steps and get an integrator with a much larger set of absolute stability, which is immune to the asymptote problem. The method is exemplified with the equation of motion of a one-dimensional non-linear oscillator describing the radial motion in the Kepler problem.Comment: 41 pages, 25 figure

A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations

This paper presents a new resolution strategy for multi-scale streamer discharge simulations based on a second order time adaptive integration and space adaptive multiresolution. A classical fluid model is used to describe plasma discharges, considering drift-diffusion equations and the computation of electric field. The proposed numerical method provides a time-space accuracy control of the solution, and thus, an effective accurate resolution independent of the fastest physical time scale. An important improvement of the computational efficiency is achieved whenever the required time steps go beyond standard stability constraints associated with mesh size or source time scales for the resolution of the drift-diffusion equations, whereas the stability constraint related to the dielectric relaxation time scale is respected but with a second order precision. Numerical illustrations show that the strategy can be efficiently applied to simulate the propagation of highly nonlinear ionizing waves as streamer discharges, as well as highly multi-scale nanosecond repetitively pulsed discharges, describing consistently a broad spectrum of space and time scales as well as different physical scenarios for consecutive discharge/post-discharge phases, out of reach of standard non-adaptive methods.Comment: Support of Ecole Centrale Paris is gratefully acknowledged for several month stay of Z. Bonaventura at Laboratory EM2C as visiting Professor. Authors express special thanks to Christian Tenaud (LIMSI-CNRS) for providing the basis of the multiresolution kernel of MR CHORUS, code developed for compressible Navier-Stokes equations (D\'eclaration d'Invention DI 03760-01). Accepted for publication; Journal of Computational Physics (2011) 1-2