#include "FieldSolver/ImplicitSolvers/WarpXSolverVec.H"
#include "ImplicitSolver.H"
#include <AMReX_Array.H>
#include <AMReX_MultiFab.H>
#include <AMReX_REAL.H>

Classes
class	ThetaImplicitEM

Detailed Description

Theta-implicit electromagnetic time solver class. This is a fully implicit algorithm where both the fields and particles are treated implicitly.

The time stencil is Eg^{n+1} = Eg^n + c^2*dt*( curlBg^{n+theta} - mu0*Jg^{n+1/2} ) Bg^{n+1} = Bg^n - dt*curlEg^{n+theta} xp^{n+1} = xp^n + dt*up^{n+1/2}/(0.5*(gammap^n + gammap^{n+1})) up^{n+1} = up^n + dt*qp/mp*(Ep^{n+theta} + up^{n+1/2}/gammap^{n+1/2} x Bp^{n+theta}) where f^{n+theta} = (1.0-theta)*f^{n} + theta*f^{n+1} with 0.5 <= theta <= 1.0

The user-specified time-biasing parameter theta used for the fields on the RHS is bound between 0.5 and 1.0. The algorithm is exactly energy conserving for theta = 0.5. Signifcant damping of high-k modes will occur as theta approaches 1.0. The algorithm is numerially stable for any time step. I.e., the CFL condition for light waves does not have to be satisifed and the time step is not limited by the plasma period. However, how efficiently the algorithm can use large time steps depends strongly on the nonlinear solver. Furthermore, the time step should always be such that particles do not travel outside the ghost region of the box they live in, which is an MPI-related limitation. The time step is always limited by the need to resolve the appropriate physics.

See S. Markidis, G. Lapenta, "The energy conserving particle-in-cell method." JCP 230 (2011).

See G. Chen, L. Chacon, D.C. Barnes, "An energy- and charge-conserving, implicit, elctrostatic particle-in-cell algorithm." JCP 230 (2011).

See J.R. Angus, A. Link, A. Friedman, D. Ghosh, J. D. Johnson, "On numerical energy conservation for an implicit particle-in-cell method coupled with a binary Monte-Carlo algorithm for Coulomb collisions.", JCP 456 (2022).

See J.R. Angus, W. Farmer, A. Friedman, D. Ghosh, D. Grote, D. Larson, A. Link, "An implicit particle code with exact energy and charge conservation for electromagnetic studies of dense plasmas.", JCP 491 (2023).

Classes

Detailed Description