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Smol_RK3CN2_pBC.m
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Smol_RK3CN2_pBC.m
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%% Time Stepping solver for the Full Smoluchowski Equation (RK3-CN2)
% Loosely based on Spherefun, this solve uses Double Fourier Sphere (DFS)
% method to tranform the orientational space and time-marched in the DFS
% space. Time stepping is semi-implicit, with Advection terms in RK3 and
% Laplacian term in CN2. Implicit matrix inversion is done using the
% Spherefun Helmholtz Solver (only slightly modified to remove
% chebfun-based activiities).
%% Setting up
% RK3 coeff and constants
alpha=[4/15 1/15 1/6];
gamma=[8/15 5/12 3/4];
rho=[0 -17/60 -5/12];
% Preparing Constants
K2 = (1/(dt*diff_const)); % Helmholtz frequency for BDF1
%% Initialising Matrices
[settings,Mvor,Mgyro,Minert,Mlap,Rdx,Rd2x,Mp1,Mp3,Mp1p3,~]=all_mat_gen(settings);
Mint=settings.Mint;
MintSq=settings.MintSq;
Kp=settings.Kp;
% mats=struct('Mint',Mint,'S_profile',S_profile,'Mvor',Mvor,'Mgyro',Mgyro,'Mlap',Mlap,...
% 'Mp1',Mp1,'Mp3',Mp3,'Rdx',Rdx,'Rd2x',Rd2x);
helm=helmholtz_gen( n, m);
%Swimming and sedimentation
MSwim=Vc*Mp1-(Vsmax-Vsmin)*Mp1p3;
%% Initialise Recorded values
cell_den=NaN(floor(nsteps/saving_rate3),N_mesh);
% PS=PS_RunTime('x','inv',mats,settings,saving_rate1,saving_rate2);
%% Time-Stepping (RK3-CN2)
ucoeff=ucoeff0;
adv_p_coeff =zeros(n*m,N_mesh);
adv_comb_coeff =zeros(n*m,N_mesh);
ucoeff_previous2=NaN(n*m,N_mesh,3);
cell_den_loc=real(Mint*ucoeff*2*pi);
Nint_loc=sum(cell_den_loc,2)*dx;
for i = 1:nsteps
%% RK step 1
k=1;
% Par-For Version
dxu_coeff=ucoeff*Rdx;
dx2u_coeff=ucoeff*Rd2x;
parfor j=1:N_mesh
adv_coeff=S_profile(j)*(Mvor*ucoeff(:,j))+Mgyro*ucoeff(:,j);
adv_coeff=adv_coeff+Minert*ucoeff(:,j);
adv_coeff=adv_coeff-Mint'*(Mint*adv_coeff)/MintSq;
lap_coeff=Mlap*ucoeff(:,j);
lap_coeff=lap_coeff-Mint'*(Mint*lap_coeff)/MintSq;
swim_coeff=MSwim*dxu_coeff(:,j);
DT_coeff=DT*dx2u_coeff(:,j);
adv_p_coeff(:,j)=adv_coeff+swim_coeff-DT_coeff;
rhs_coeff = -K2/alpha(k)*ucoeff(:,j)-lap_coeff+1/diff_const/alpha(k)*(gamma(k)*(adv_p_coeff(:,j)))...
-Kp/alpha(k)*(int_const-Nint_loc)*Mint'.*ucoeff(:,j);
ucoeff(:,j) = helmholtz_cal(rhs_coeff, -K2/alpha(k),helm);
end
cell_den_loc=real(Mint*ucoeff*2*pi);
Nint_loc=sum(cell_den_loc,2)*dx;
%% RK step 2
k=2;
dxu_coeff=ucoeff*Rdx;
dx2u_coeff=ucoeff*Rd2x;
parfor j=1:N_mesh
% adv_p_coeff=adv_coeff+swim_coeff;
adv_coeff=S_profile(j)*(Mvor*ucoeff(:,j))+Mgyro*ucoeff(:,j);
adv_coeff=adv_coeff+Minert*ucoeff(:,j);
adv_coeff=adv_coeff-Mint'*(Mint*adv_coeff)/MintSq;
lap_coeff=Mlap*ucoeff(:,j);
lap_coeff=lap_coeff-Mint'*(Mint*lap_coeff)/MintSq;
swim_coeff=MSwim*dxu_coeff(:,j);
DT_coeff=DT*dx2u_coeff(:,j);
adv_comb_coeff(:,j)=adv_coeff+swim_coeff-DT_coeff;
rhs_coeff = -K2/alpha(k)*ucoeff(:,j)-lap_coeff+1/diff_const/alpha(k)*(gamma(k)*(adv_comb_coeff(:,j))+rho(k)*adv_p_coeff(:,j))...
-Kp/alpha(k)*(int_const-Nint_loc)*Mint'.*ucoeff(:,j); %#ok<*PFBNS>
ucoeff(:,j) = helmholtz_cal(rhs_coeff, -K2/alpha(k),helm);
end
cell_den_loc=real(Mint*ucoeff*2*pi);
Nint_loc=sum(cell_den_loc,2)*dx;
%% RK step 3
k=3;
dxu_coeff=ucoeff*Rdx;
dx2u_coeff=ucoeff*Rd2x;
adv_p_coeff=adv_comb_coeff;
parfor j=1:N_mesh
adv_coeff=S_profile(j)*(Mvor*ucoeff(:,j))+Mgyro*ucoeff(:,j);
adv_coeff=adv_coeff+Minert*ucoeff(:,j);
adv_coeff=adv_coeff-Mint'*(Mint*adv_coeff)/MintSq;
lap_coeff=Mlap*ucoeff(:,j);
lap_coeff=lap_coeff-Mint'*(Mint*lap_coeff)/MintSq;
swim_coeff=MSwim*dxu_coeff(:,j);
DT_coeff=DT*dx2u_coeff(:,j);
adv_comb_coeff(:,j)=adv_coeff+swim_coeff-DT_coeff;
rhs_coeff = -K2/alpha(k)*ucoeff(:,j)-lap_coeff+1/diff_const/alpha(k)*(gamma(k)*(adv_comb_coeff(:,j))+rho(k)*adv_p_coeff(:,j))...
-Kp/alpha(k)*(int_const-Nint_loc)*Mint'.*ucoeff(:,j);
ucoeff(:,j) = helmholtz_cal(rhs_coeff, -K2/alpha(k),helm);
end
cell_den_loc=real(Mint*ucoeff*2*pi);
Nint_loc=sum(cell_den_loc,2)*dx;
%% Saving for Post-Processing
% Saving full Psi and it time derivative
% PS=PS.RunTimeCall(ucoeff,i);
% if ( mod(i, saving_rate2) == 0 )
% ufull_save=ucoeff;
% t=i*dt;
% end
% if ( mod(i, saving_rate2) == 2 )&& i~=2
% fdt_full_save=((-ucoeff./(real(Mint*ucoeff*2*pi))...
% + ucoeff_previous2(:,:,1)./(real(Mint*ucoeff_previous2(:,:,1)*2*pi)))/12 ...
% +(ucoeff_previous2(:,:,3)./(real(Mint*ucoeff_previous2(:,:,3)*2*pi))...
% -ucoeff_previous2(:,:,2)./(real(Mint*ucoeff_previous2(:,:,2)*2*pi)))*(2/3))/dt;
% udt_full_save=((-ucoeff...
% + ucoeff_previous2(:,:,1))/12 ...
% +(ucoeff_previous2(:,:,3)...
% -ucoeff_previous2(:,:,2))*(2/3))/dt;
% save(['t' num2str(t) '.mat'],'t','ufull_save','fdt_full_save','udt_full_save');
% end
% if ( mod(i, saving_rate2) == 1 )&& i~=1
% ucoeff_previous2(:,:,3)=ucoeff;
% end
% if ( mod(i, saving_rate2) == saving_rate2-1 )
% ucoeff_previous2(:,:,2)=ucoeff;
% end
% if ( mod(i, saving_rate2) == saving_rate2-2 )
% ucoeff_previous2(:,:,1)=ucoeff;
% end
% Saving Cell Density
if ( mod(i, saving_rate3) == 0 )
cell_den(i/saving_rate3,:)=cell_den_loc;
disp([num2str(i) '/' num2str(nsteps)]);
if any(isnan(cell_den(i/saving_rate3,:)))
disp('NaN detected in cell_den');
break;
end
end
end