4.2. Field Solver - Poisson Equation

To numerically solve electrostatic problems, PICLas offers a solver for Poisson’s equation

\[\nabla \cdot \textbf{E} = \frac{\rho}{\varepsilon_{0}}\]

where the charge density \(\rho\) is given by the charged particles within the system. To enable a simulation using the Poisson field solver, the code must be compiled using the following setting

PICLAS_EQNSYSNAME     = poisson
PICLAS_TIMEDISCMETHOD = Euler-Explicit, Leapfrog, Boris-Leapfrog, Higuera-Cary, RK3, RK4

where one of the available time discretization methods for the particle evolution must be chosen for PICLAS_TIMEDISCMETHOD, which are also referred to as particle pushers. There are different options available that yield different functionalities.

PICLAS_TIMEDISCMETHOD	Electric field	Magnetic field	Order of Convergence
Euler-Explicit	yes	no	1
Leapfrog	yes	no	2
Boris-Leapfrog	yes	yes	2
Higuera-Cary	yes	yes	2
RK3	yes	yes	3
RK4	yes	yes	4

Note that high-order time discretization methods in general allow for a larger time step and are usually more costly per time step. To see the available parameter input file options, simply run

./bin/piclas --help HDG

4.2.1. CG Solver

The default numerical method for solving the resulting system of linear equations, is the Conjugate Gradient Method. The following input parameter can be set to control the simulation

epsCG     = 1e-6 ! default value for the abort residual
MaxIterCG = 500  ! default value for the number of CG solver iterations

where epsCG is the residual of the CG solver and MaxIterCG are the maximum number of iteration performed in one time step to solve the system. Furthermore, the residual can be either set absolute (default) or relative via

useRelativeAbortCrit = F ! default

4.2.2. PETSc Solver

A multitude of different numerical methods to solve the resulting system of linear equations is given by the implemented PETSc library [7], [8], [9]. For detailed installation steps of PETSc within PICLas, see Section Installing PETSc. To use PETSc, another flag must be set during the compilation of PICLas

PICLAS_PETSC = ON

and the parameter input file for the simulation requires setting

PrecondType = 2 ! default

where the following options are possible

PrecondType	Iterative or Direct	Method
1	iterative	Krylov subspace
2	iterative	Krylov subspace
3	iterative	Krylov subspace
10	direct

Note that the same parameter setting for epsCG will result in a smaller residual with PETSc as compared with the default CG solver without using the PETSc library.