A Step Towards Temporal Multi-Scale Problems

In this talk, we will discuss two classic kinetic plasma problems to motivate the development of a high order Spectral Deferred Correction Lagrangian framework. The two problems are the two stream instability and the dynamics of Bernstine-Greene-Kruskal (BGK) modes (the only known class of fully non-linear solutions to the Vlasov-Poisson (VP) system). These two problems are linked via the notion of particle trapping. Particle trapping is the idea of forming a seperatrix in phase space where electrons on the inside of the seperatrix are trapped at a location in space, while electrons on the out side are able to free stream in space. In particular, we are interested in developing numerical methods that can bridge the electron ion time scale gap which arises in two fluid models of a plasma. The electrons being light, respond quickly to changes in the electric field, while the more massive ions will respond on a much slower time scale. The current objective is to use high order time stepping to begin to bride these vastly different time scales and thereby eliminate an outstanding computational bottle neck for a wide range of interesting plasma problems.

This work will consider two key issues previously neglected in our Lagrangian particle model; the incorporation of arbitrary high order time stepping methods, based on Spectral Deferred Correction (SDC), and the development of sufficiently smooth regularization methods which are consistent with high order time stepping. Spectral Deferred Correction is based on a low order predictor corrector method where a corrector, which is successive applied, is formulated via a Picard integral equation for the residual. We will demonstrate the utility of using high order methods in the correction step by considering both the region of absolute stability and our own metric, which we dub the accuracy plot. In addition, we have formally established that if Runga-Kutta 2, 3 or 4 is used in the correction step, the accuracy of the scheme will increase by 2,3 or 4 respectively upon each iteration of the correction. We present the associated theorems and give some discussion for the spatial case of forward Euler. We next turn our attention to the simulation of $N$ particles interacting via long range forces and demonstrate the need to high continuity of the regularization chosen, so as to maintain high order in SDC time stepping. It will be demonstrated that apparently handling the $N$ particle system is the first step in the development of an arbitrary high order accuracy Lagrangian particle method for the VP system.

This is joint work with R. Krasny, J. Qiu and B. Ong.