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MPFC_2D_CN_SAV_adaptive_time.m
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MPFC_2D_CN_SAV_adaptive_time.m
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function phi = MPFC_2D_CN_SAV_adaptive_time(pde,domain,Nx,Ny,time,option)
% Solve 2D phase field crystal equaiton
% (1) phi_tt + alpha\phi_t = \Delta \mu
% (2) \mu = (1+\Delta)^2\phi + (\phi^3 - epsilon\phi)
% (2) \mu = (1+\Delta)^2\phi + \beta*phi + \phi*(\phi^2 - epsilon - \beta)
% Qi Li
% 01/16/2019
global dt epsilon M k2 k4 beta_bar beta alpha C0 hx hy Lx Ly sigma...
dtmin dtmax dtalpha tstart delta_t dttol
if ~exist('option','var'), option = []; end
if ~isfield(option,'tol')
option.tol = 10^-14; % default tol
end
if ~isfield(option,'tolit')
option.tolit = 10^-8; % default tolit
end
if ~isfield(option,'maxit')
option.maxit = 2000; % default maxit
end
if ~isfield(option,'plotflag')
option.plotflag = 0;
end
if ~isfield(option,'saveflag')
option.saveflag = 0;
end
if ~isfield(option,'savefinal')
option.savefinal = 0;
end
if ~isfield(option,'printflag')
option.printflag = 0;
end
if ~isfield(option,'vtkflag')
option.printflag = 0;
end
if ~isfield(option,'energyflag')
option.energyflag = 0;
end
if 1 == option.energyflag
figname_mass = [pde.name,'e',num2str(pde.epsilon),'M',num2str(pde.M),'b_bar',num2str(pde.beta_bar),'b',num2str(pde.beta),'dt',num2str(time.dt),'_mass.txt'];
figname_energy = [pde.name,'e',num2str(pde.epsilon),'M',num2str(pde.M),'b_bar',num2str(pde.beta_bar),'b',num2str(pde.beta),'dt',num2str(time.dt),'_energy.txt'];
out1 = fopen(figname_mass,'w');
out2 = fopen(figname_energy,'w');
end
tol = option.tol;
tolit = option.tolit;
maxit = option.maxit;
%%
T = time.T;
t = time.t0;
dt = time.dt;
tsave = time.tsave;
dir_fig = [pde.name '/fig'];
dir_data = [pde.name '/data'];
epsilon = pde.epsilon;
M = pde.M;
alpha = pde.alpha;
beta_bar = pde.beta_bar;
beta = pde.beta;
sigma = pde.sigma;
Lx = domain.right - domain.left;
Ly = domain.top - domain.bottom;
hx = Lx/Nx;
hy = Ly/Ny;
% x = domain.left + hx*(1:Nx);
% y = domain.bottom + hy*(1:Ny);
x = domain.left + hx*(0:Nx-1);
y = domain.bottom + hy*(0:Ny-1);
[xx,yy] = meshgrid(x,y);
phi0 = pde.init(xx,yy);
psi0 = zeros(size(phi0));
nfigure =1;
[k_x,k_y,kx,ky,kxx,kyy,k2,k4] = prepare_fft2(Lx,Ly,Nx,Ny);
nplot = round((T-t)/dt);
nsave = round(tsave/dt);
tstart = tic;
C0 = (epsilon+alpha).^2*Lx*Ly;
%% initialization phi 1
time1 = time; time1.T = dt; option.energyflag = 0;
[phi1, r1] = MPFC_2D_First_order_SAV(pde,domain,Nx,Ny,time1,option);
[phi1, r1] = MPFC_2D_CN_SAV_PreCor(pde,domain,Nx,Ny,time1,phi1,option);
option.energyflag = 1;
psi1 = 2*(phi1 - phi0)/dt-psi0;
t = t+dt;
if 1 == option.energyflag
energy_old = calculate_energy_old(out1,out2,hx,hy,t,phi1,psi1,r1,C0);
end
dtmin = time.dtmin;
dtmax = time.dtmax;
dtalpha = time.dtalpha;
dttol = time.dttol;
delta_t = dt;
% for nt = 2:nplot
nt = 2;
while t < T
if t+delta_t>T
delta_t = T - t;
end
t = t+delta_t;
phi_star = 1/2*(3*phi1-phi0);
% step 1
H = fun_H(phi_star);
if isfield(pde,'rhs') && isfield(pde,'exact')
rhs = dt*pde.rhs(xx,yy,t-dt/2);
else
rhs = 0;
end
g = get_rhs(phi1,psi1,r1,H)+rhs;
psiA = inv_A(lap_diff(H));
psiB = inv_A(g);
gamma = -fft2(H.*psiA);
gamma = gamma(1,1)*hx*hy;
% Step 2
Hphi = fft2(H.*psiB);
Hphi = Hphi(1,1)*hx*hy/(1+dt*M*gamma/4);
% Step 3
phi = dt*M/4*Hphi.*psiA + psiB;
%% update phi0
r1 = fun_r(phi,phi1,r1,H);
psi1 = fun_psi(phi,phi1,psi1);
phi0 = phi1;
phi1 = phi;
if 1 == option.printflag
fprintf('epsilon=%.3f,t=%.4f/%.f, dt=%.4f, Nx=%d, Ny=%d \n',epsilon,t,T,delta_t,Nx,Ny);
end
if 1 == option.energyflag
[delta_t,energy_new] = calculate_energy(out1,out2,hx,hy,t,phi1,psi1,r1,C0,phi0,delta_t);
end
energy_old = energy_new;
if 1 == option.saveflag
% fprintf('nt = %d\n',nt);
if 0 == mod(nt,nsave)
ss = [dir_data '/phi_t=' num2str(t) '.txt'];
fid = fopen(ss, 'wt');
fprintf(fid, '%f\n', phi(:));
fclose(fid);
end
nt = nt+1;
end
end
if 1 == option.savefinal
name= [pde.name,'e',num2str(pde.epsilon),'M',num2str(pde.M),'b_bar',num2str(pde.beta_bar),'b',num2str(pde.beta),'Nx=',num2str(Nx),'Ny=',num2str(Ny),'dt=',num2str(dt)];
filename=[name '.mat'];
save(filename,'epsilon','M','x','y','hx','hy','Nx','Ny','dt','T','phi','domain');
% showsolution_2D(nfigure,xx,yy,phi,t,'./','.fig');
end
if 1 == option.energyflag
fclose(out1);
fclose(out2);
end
end
function r = fun_psi(phi,phi0,psi1)
global dt
r = 2*(phi-phi0)/dt-psi1;
end
function r = fun_r_init(phi)
global C0 hx hy
E1 = fft2(F(phi));
r = sqrt(E1(1,1)*hx*hy + C0);
end
function r = fun_r(phi,phi0,r0,b)
global hx hy
bphi0 = fft2(b.*phi0);
bphi0 = bphi0(1,1)*hx*hy;
bphi1 = fft2(b.*phi);
bphi1 = bphi1(1,1)*hx*hy;
g1 = r0 - 1/2*bphi0;
r = 1/2*bphi1+g1;
end
function r = fun_H(phi)
global C0 hx hy
E1 = fft2(F(phi));
r = F_derivative(phi)./sqrt(E1(1,1)*hx*hy+C0);
end
function r = get_rhs(phi1,psi1,r1,b)
global beta_bar beta dt M hx hy alpha sigma Lx Ly
bphiphi = fft2(b.*phi1);
bphiphi = bphiphi(1,1)*hx*hy;
h2 = -2/dt*phi1-psi1;
gg = dt/(2*beta_bar+dt*beta)*fft2(-(beta_bar+dt*beta/2)*h2 + (beta_bar-dt*beta/2)*psi1);
gg = (gg + fft2(phi1))/2;
r = -(beta_bar+beta/2*dt)*h2 +(beta_bar-dt*beta/2)*psi1 ...
+dt*M/2*lap_diff(phi1+2*lap_diff(phi1)+lap_diff(lap_diff(phi1))) ...
+alpha/2*dt*M*lap_diff(phi1) ...
+dt*M/2*(2*r1-1/2*bphiphi).*lap_diff(b)...
-sigma*dt*M/2*phi1 + sigma*dt*M*gg(1,1)*hx*hy/(Lx*Ly);
end
function lap=lap_diff(phi)
global k2
lap=real(ifft2((k2.*fft2(phi))));
end
function r = inv_A(phi)
global dt M k2 alpha beta_bar beta sigma
r = real(ifft2(fft2(phi)./((beta_bar+dt*beta/2)*2/dt-dt*M/2*k2.*(1+k2).^2-dt*M/2*alpha.*k2+sigma*dt*M/2)));
end
function energy3 = calculate_energy_old(out1,out2,hx,hy,t,phi,psi,r,C0)
global M epsilon alpha beta_bar k2 Lx Ly sigma tstart dt
energy1 = hx*hy*sum(sum( 1/4*phi.^4 - epsilon/2*phi.^2+ 1/2*phi.^2 + lap_diff(phi).*phi + 1/2*lap_diff(phi).^2 + alpha/2*phi.^2));
Fpsi = fft2(psi)./k2; Fpsi(1,1) = 0; psi_inv_lap = real(ifft2(Fpsi));
ene = -0.5/M*beta_bar*hx*hy*sum(sum(psi_inv_lap.*psi));
phi_hat = fft2(phi);
psi_OK = phi - phi_hat(1,1)*hx*hy/(Lx*Ly);
Fpsi = fft2(-psi_OK)./k2; Fpsi(1,1) = 0; psi_OK = real(ifft2(Fpsi));
psi_hat = fft2(-lap_diff(psi_OK).*psi_OK);
ene2 = 0.5*sigma*psi_hat(1,1)*hx*hy;
energy2 = energy1 + ene;
energy3 = hx*hy*sum(sum(1/2*phi.^2 + lap_diff(phi).*phi + 1/2*lap_diff(phi).^2 + alpha/2*phi.^2)) + r.^2 - C0 + ene;
mass = hx*hy*sum(sum( phi ));
fprintf(out1,'%14.6e %f \n',t,mass);
energy1 = energy1+ene2;
energy2 = energy2+ene2;
energy3 = energy3+ene2;
fprintf(out2,'%14.6e %f %f %f %f %f \n',t,energy1,energy2,energy3,dt,toc(tstart));
end
function [delta_t,energy3] = calculate_energy(out1,out2,hx,hy,t,phi,psi,r,C0,phiold,delta_t_old)
global M epsilon alpha beta_bar k2 Lx Ly sigma dtmin dtmax dtalpha dttol tstart
energy1 = hx*hy*sum(sum( 1/4*phi.^4 - epsilon/2*phi.^2+ 1/2*phi.^2 + lap_diff(phi).*phi + 1/2*lap_diff(phi).^2 + alpha/2*phi.^2));
Fpsi = fft2(psi)./k2; Fpsi(1,1) = 0; psi_inv_lap = real(ifft2(Fpsi));
ene = -0.5/M*beta_bar*hx*hy*sum(sum(psi_inv_lap.*psi));
phi_hat = fft2(phi);
psi_OK = phi - phi_hat(1,1)*hx*hy/(Lx*Ly);
Fpsi = fft2(-psi_OK)./k2; Fpsi(1,1) = 0; psi_OK = real(ifft2(Fpsi));
psi_hat = fft2(-lap_diff(psi_OK).*psi_OK);
ene2 = 0.5*sigma*psi_hat(1,1)*hx*hy;
energy2 = energy1 + ene;
energy3 = hx*hy*sum(sum(1/2*phi.^2 + lap_diff(phi).*phi + 1/2*lap_diff(phi).^2 + alpha/2*phi.^2)) + r.^2 - C0 + ene;
mass = hx*hy*sum(sum( phi ));
fprintf(out1,'%14.6e %f \n',t,mass);
energy1 = energy1+ene2;
energy2 = energy2+ene2;
energy3 = energy3+ene2;
% delta_t = max([dtmin,dtmax/sqrt(1+dtalpha*(abs(energy3-energy_old)/delta_t_old))]);
error = hx*hy*sum(sum( (phi-phiold).^2 ))./(hx*hy*sum(sum( (phiold).^2 )));
delta_t = max([dtmin,min([dtalpha*delta_t_old*(dttol/error).^(1/2),dtmax])]);
fprintf(out2,'%14.6e %f %f %f %f %f \n',t,energy1,energy2,energy3,delta_t,toc(tstart));
end
function r = F_derivative(phi0)
global alpha epsilon
r = phi0.*(phi0.^2 - epsilon - alpha);
end
function r = F(phi)
global alpha epsilon
r = 1/4*phi.^4 - epsilon/2*phi.^2 - alpha/2*phi.^2;
end