Numerical Analysis of Particle in Cell Methods in 2D
Zoe Barbeau, Stanford University
Particle-in-cell (PIC) methods describe advective processes via particle discretizations for partial differential equations. There is not a widely used, formal numerical analysis framework for this method even though PIC methods have been in use since the 1950s. Colella and his collaborators at LBNL, following the ideas of Cottet, have developed such a framework. They have demonstrated its potential for the design of PIC methods with improved accuracy and efficiency for 1+1D kinetic problems. For 1+1D kinetics problems, the deformation matrix is carried as an auxiliary variable and used to remap adaptively when the particles have deformed significantly from the original grid which causes large interpolation errors. We apply this approach to 2D vortex methods and evaluate potential error indicators. The deformation matrix is decomposed via QR factorization where the angle of rotation and degree of stretching are extracted from Q and R respectively. The test cases evaluated are single and double vortex simulations both with smoothly decaying and constant vorticity distributions. Error accrued by interpolation to the grid and back to the particles (PGP) is used to evaluate the size of interpolation error. The approximate error in velocity is determined by taking the difference between grid refinements. The potential error indicators of the angle of Q and eigenvalue of R are compared to the vorticity PGP error and velocity error. The eigenvalue of R shows good correlation with growth in vorticity PGP error and velocity error compared to angle of Q which shows that the dominant source of error is stretching away from the grid. We examine the performance of the eigenvalue of R as an indicator function for remapping.