Identifier: 2009-34
Author(s): A. Stock
Title: Development and application of a multirate multistep AB method to a discontinuous Galerkin method based particle-in-cell scheme
Page count: 155 pp.
Date: 2009-10-15
Abstract:
A discrete numerical model of a physical problem usual uses a system of equations that has largely varying eigenvalues. These systems are often called stiff. In a physical model the eigenvalues are related to the timescales of the components of the model. A stiff system of equations is used in the particle-in-cell (PIC) method, which is a coupled system of Maxwell’s equation and particles’ equation of motions, modeling the dynamic interaction between electromagnetic fields and charged particles. If we assume that the particles do not move at relativistic speeds, in PIC the fast component is the electrodynamic fields and the slow component is the particles. For the time integration of certain partial differential equations (PDE’s) the time step is restricted by the largest eigenvalue due to the Courant–Friedrichs–Lewy condition (CFL condition), leading to a very small time step. Thus the slow component is inte- grated with an unnecessarily small step. If the calculation of the slow component is very expensive, such as in PIC, the numerical scheme suffers great inefficiencies. These inef- ficiencies could be solved by considering the use of a multirate time integration scheme. In a multirate scheme each component is integrated according to its own timescale. The different components are coupled by using interpolation. Our goal is to develop a multirate multistep Adams-Bashforth (MRAB) method and apply it to PIC for nonrelativistic problems. Multirate linear multistep methods were first mentioned by Gear and Wells in [1]. We use their work to develop a multirate multistep scheme that can be used as a regular time stepper for any application. As most applications of PIC deal with particles at relativistic speeds one might wonder if multirate integrations schemes should be applied to PIC. Thus we want to mention other applications for multirate multistep time integration. One application is local timestepping (LTS) on nonuniform grids focusing on the region of interest (vortices, shocks, walls, etc.). In discontinuous Galerkin (DG) methods with nonuniform grids s strong scale separation occur. This leads to large differences in the DG operator eigenvalues, yielding globally a stiff system of equations. LTS based on a multirate method can make these methods more efficient. Focusing on the numerics of electrodynamics and plasma simulations, such as Mag- netohydrodynamics (MHD) or PIC, the hyperbolic divergence cleaning for the charge conservation error is a reasonable application. The eigenvalue of the cleaning component can have a magnitude that is ten times the speed of light [4], yielding a stiff system. In this thesis we derive a two-rate multistep AB method (TRAB). We shall present 14 different schemes for the TRAB method, differing in the sequence of the components interpolation. We make time step considerations for a two-rate method by performing numerical experiments with linear ODE systems that mimic the behavior of a stiff system of equations similar to what is used for PIC. Finally we apply the two-rate method to the PIC scheme. With the plasma wave test case we compare the accuracy and performance of the TRAB method to a standard Runge-Kutta timestepping method.
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