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Diffusion FENNECS: Finite Element Non-Neutral Electron Cloud Simulator (geom_testmumps) Browse

fennecs/matlab/twostream_solution.m
bde8e520bdf2geom_testmumps

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twostream_solution.m
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clear all
close all
%%
L=2*pi; % spatial length of the system
DT=.1; % time step
NT=1000; % number of time steps
NG=128; % number of grid points
N=2; % number of particles
WP=1; % plasma frequency (in normalized units)
QM=-1; % charge of the particles (electrons)
V0=0.0; % mean velocity
VT=0.0; % thermal velocity
XP1=1; % pertubation of positions
Q=WP^2/(QM*N/L); % electron charge in normalized units
rho_back=-Q*N/L; % background charge density (ions)
dx=L/NG; % size of the grid cells
%% initial loading for the 2 Stream instability
xp=transpose(linspace(0,L-L/N,N));
vp=VT*randn(N,1);
pm=transpose([1:N]);pm=1-2*mod(pm,2);
vp=vp+pm.*V0;
% Perturbation
xp=xp+XP1*(L/N)*pm.*rand(N,1);
p=1:N;p=[p p];
un=ones(NG-1,1);
Poisson=spdiags([un -2*un un],[-1 0 1],NG-1,NG-1);
%% Prepare Diagnostics
time = 0:DT:(NT-1)*DT;
Phi=0; mat=0; Eg=0;
NFFT = 2^nextpow2(NG);
k = (2*pi/dx)/2*linspace(0,1,NFFT/2+1);
dv=V0/10; vlim=4*V0; vbin=-vlim:dv:vlim; NV=length(vbin);
mom = zeros(1,NT);
E_kin = zeros(1,NT);
E_pot = zeros(1,NT);
E_the = zeros(1,NT);
dens = zeros(NG,NT);
es_pot = zeros(NG,NT);
es_fld = zeros(NG,NT);
phi_fft = zeros(NG,NT);
vdf = zeros(NV,NT);
%% Main computational cycle
for it=1:NT
%-----DIAGNOSTICS-----
%total momentum
mom(it) = sum(vp);
E_kin(it) = 0.5*mean(vp.^2)*WP^2/QM^2;
E_pot(it) = 0.5*sum(Eg.^2)/NG;
E_the(it) = 0.5*mean((vp-mean(vp)).^2)*WP^2/QM^2;
dens(:,it) = sum(mat)/dx;
es_pot(:,it) = Phi;
es_fld(:,it) = Eg;
vdf(:,it) = histc(vp,vbin);
phi_fft(:,it) = fft(Phi,NFFT)/NG;
%---------------------
% update xp and apply boundary conditions (bc)
xp=mod(xp+vp*DT,L);
% project particles to grid
g1=floor(xp/dx);
g=[g1;g1+1];
% compute what fraction of the particle size that lies on the two nearest cells
fraz1=1-abs(xp/dx-g1);
fraz=[fraz1;1-fraz1];
% apply bc on the projection
out=(g<1);g(out)=g(out)+NG;
out=(g>NG);g(out)=g(out)-NG;
% use represention with a sparse matrix to save up memory
mat=sparse(p,g,fraz,N,NG);
rho=transpose(full((Q/dx)*sum(mat))+rho_back); % charge density
% computing fields
Phi=Poisson\(-rho(1:NG-1)*dx^2);Phi=[Phi;0];
Eg=([Phi(NG); Phi(1:NG-1)]-[Phi(2:NG);Phi(1)])/(2*dx);
% projection q->p and update of vp
vp=vp+mat*QM*Eg*DT;
% plot phase space particle dynamics
scatter(xp,vp,'.')
%ylim([-V0*3 V0*3]);
xlabel('X')
ylabel('V')
pause(0.01)
end
%%
figure;
plot(time,mom)
xlabel('time')
ylabel('momentum')
figure;
hold on
plot(time,E_kin,'k')
plot(time,E_pot,'r')
plot(time,E_the,'m')
plot(time,E_kin+E_pot,'b')
xlabel('time')
ylabel('Energy')
X=[0:dx:L-L/NG];
figure
surfc(time,X,dens(:,:)); shading interp; colorbar;
xlabel('time')
ylabel('X')
title('density')
figure
surfc(time,X,es_pot(:,:)); shading interp; colorbar;
xlabel('time')
ylabel('X')
title('ES potential')
figure
surfc(time,X,es_fld(:,:)); shading interp; colorbar;
xlabel('time')
ylabel('X')
title('ES field')
figure
surfc(time,k,abs(phi_fft(1:NFFT/2+1,:))); shading interp; colorbar;
xlabel('time')
ylabel('k')
title('ES potential (Fourier harmonics)')
figure
surfc(time,vbin,vdf); shading interp; colorbar;
xlabel('time')
ylabel('V')
title('f(v)')
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