[m10_4](Ibc) Inplicit form solution of T(no advection) with MIC.After correcting subgrid diffsion.

%This program complished the process of 2D Eulerian advection with method
%of allocating memory temporarily.
clear all;clf;

Nt = 30;
output_interval = 1;
% BASIC NODES
Lx = 1000000; 
Ly = 1500000;
x_midd = Lx/2;
y_midd = Ly/2;
Nx = 51; 
Ny = 31; 
Nx1 = Nx+1;
Ny1 = Ny+1;
dx = Lx / (Nx-1);
dy = Ly / (Ny-1);
% Basic E-nodes.
x = 0:dx:Lx;
y = 0:dy:Ly;
g_y = 10;
% PARAMETERS
kw = 10;%W/(m*K)
denw = 3300;%Kg/m^3
Cpw = 1100;%J/(Kg*K)
ks = 3;%W/(m*K)
dens = 3200;%Kg/m^3
Cps = 1000;%J/(Kg*K)
bt = 1000;%K,background temperature
wt = 1300;%K,wave temperature
% L-MARKERS
num_per_cell = 4;
% Nx_marker = (Nx-1)*num_per_cell;
% Ny_marker = (Ny-1)*num_per_cell;
Nx_marker = 400;
Ny_marker = 500;
marknum=Nx_marker*Ny_marker;
x_m_step = Lx/Nx_marker;
y_m_step = Ly/Ny_marker;
% Marker points.
xm=zeros(1,marknum); % Horizontal coordinates, m
ym=zeros(1,marknum);
Km=zeros(1,marknum); % K, kg/m^3  kt
DenCpm=zeros(1,marknum); % den*Cp     hct
tdm = zeros(1,marknum); % K/den/Cp  td
Tm0 = zeros(1,marknum); % temperature
% INITIALIZE K,Den*Cp in the L-markers.
m = 1;
for jm=1:1:Nx_marker
    for im=1:1:Ny_marker
        % Define marker coordinates
        xm(m)=x_m_step/2+(jm-1)*x_m_step+(rand-0.5)*x_m_step;
        ym(m)=y_m_step/2+(im-1)*y_m_step+(rand-0.5)*y_m_step;
        % Marker properties
%         if ( xm(m)<=x_midd+200000 && xm(m)>=x_midd-200000 && ym(m)<=y_midd+300000 && ym(m)>=y_midd-300000)
        if ( xm(m)<=0.7*Lx && xm(m)>=0.3*Lx && ym(m)<=0.7*Ly && ym(m)>=0.3*Lx)
            Km(m)=kw; % K wave
            DenCpm(m)=denw*Cpw; % den*Cp wave
            Tm0(m) = wt;
        else
            Km(m)=ks; % K surround
            DenCpm(m)=dens*Cps; % den*Cp surround
            Tm0(m) = bt;
        end
        % Update marker counter
        m=m+1;
    end
end
% Define matrix
hct = zeros(Ny,Nx); %Den*Cp
kht = zeros(Ny,Nx); %K
T0 = zeros(Ny,Nx); %T
td = zeros(Ny,Nx); %K/Den*Cp
bt = zeros(Nx*Ny, 1);
coet = zeros(Nx*Ny, Nx*Ny);
vx_matrix = zeros(Ny,Nx);
vy_matrix = zeros(Ny,Nx);

for j = 1:Nx
    for i = 1:Ny
        % Constant velocity
        vx_matrix(i,j) = 1e-9;
        vy_matrix(i,j) = 1e-9;
    end
end

% DEFINE dt
kappa=max(max(Km))/min(min(DenCpm)); 
dtexp = min(dx,dy)^2/(3*kappa);
if(vx_matrix(1,1)~=0)
    dtexp=min(dtexp,abs(dx/max(max(vx_matrix)))); % Limitation for horizontal advection timestep
end
if(vy_matrix(1,1)~=0)
    dtexp=min(dtexp,abs(dy/max(max(vy_matrix)))); % Limitation for vertical advection timestep
end
dt_tem = 0.5*dtexp;
subgrdifcoe = 1;
for t = 1:Nt
    % Indispensable
    % Eq.8.18
    w_m_x_node_t = zeros(Ny,Nx);
    K_w_m_t = zeros(Ny,Nx);
    DenCp_w_m_t = zeros(Ny,Nx);
    % Eq.10.12
    T_w_m_t = zeros(Ny,Nx);
    w_m_t_t = zeros(Ny,Nx);
    
    % !!!Backup properities K,Den*Cp and T
    kht_backup = kht;
    hct_backup = hct;
    T_backup = T0;
    % Searching the coordinate of upper-left node of every L-marker.
    for m=1:1:marknum
        % !!!Check L-marker in the grid
        if (xm(m)>=x(1) && xm(m)<=x(Nx) && ym(m)>=y(1) && ym(m)<=y(Ny)) 
            % Define i,j indexes for the upper left BASIC NODE(Nx*Ny)
            j=fix((xm(m)-x(1))/dx)+1;
            i=fix((ym(m)-y(1))/dy)+1;
            if(j<1)
                j=1;
            elseif(j>Nx-1)
                j=Nx-1;
            end
            if(i<1)
                i=1;
            elseif(i>Ny-1)
                i=Ny-1;
            end
            % Compute distances between marker and upper-left node.
            dis_x = xm(m)-x(j);
            dis_y = ym(m)-y(i);
            wtmij = (1-dis_x/dx)*(1-dis_y/dy);
            wtmi1j = (1-dis_x/dx)*(dis_y/dy);
            wtmij1 = (dis_x/dx)*(1-dis_y/dy);
            wtmi1j1 = (dis_x/dx)*(dis_y/dy);
            
            % Sum of (K/Den*Cp)*(weight) and sum of weight.
            w_m_x_node_t(i,j) = w_m_x_node_t(i,j) + wtmij;   % denominator% numerator
            K_w_m_t(i,j) = K_w_m_t(i,j) + Km(m).*wtmij;% K
            DenCp_w_m_t(i,j) = DenCp_w_m_t(i,j) + DenCpm(m).*wtmij;% Den*Cp
            T_w_m_t(i,j) = T_w_m_t(i,j) + Tm0(m).*DenCpm(m)*wtmij;% T
            w_m_t_t(i,j) = w_m_t_t(i,j) + DenCpm(m)*wtmij;
            
            w_m_x_node_t(i+1,j) = w_m_x_node_t(i+1,j) + wtmi1j;
            K_w_m_t(i+1,j) = K_w_m_t(i+1,j) + Km(m).*wtmi1j;
            DenCp_w_m_t(i+1,j) = DenCp_w_m_t(i+1,j) + DenCpm(m).*wtmi1j;
            T_w_m_t(i+1,j)   = T_w_m_t(i+1,j) + Tm0(m).*wtmi1j*DenCpm(m);% T
            w_m_t_t(i+1,j) = w_m_t_t(i+1,j) + DenCpm(m)*wtmi1j;
            
            w_m_x_node_t(i,j+1) = w_m_x_node_t(i,j+1) + wtmij1;
            K_w_m_t(i,j+1) = K_w_m_t(i,j+1) + Km(m).*wtmij1;
            DenCp_w_m_t(i,j+1) = DenCp_w_m_t(i,j+1) + DenCpm(m).*wtmij1;
            T_w_m_t(i,j+1)   = T_w_m_t(i,j+1) + Tm0(m).*wtmij1*DenCpm(m);% T
            w_m_t_t(i,j+1) = w_m_t_t(i,j+1) + DenCpm(m)*wtmij1;
            
            w_m_x_node_t(i+1,j+1) = w_m_x_node_t(i+1,j+1) + wtmi1j1;
            K_w_m_t(i+1,j+1) = K_w_m_t(i+1,j+1) + Km(m).*wtmi1j1;
            DenCp_w_m_t(i+1,j+1) = DenCp_w_m_t(i+1,j+1) + DenCpm(m).*wtmi1j1;
            T_w_m_t(i+1,j+1)   = T_w_m_t(i+1,j+1) + Tm0(m).*wtmi1j1*DenCpm(m);% T
            w_m_t_t(i+1,j+1) = w_m_t_t(i+1,j+1) + DenCpm(m)*wtmi1j1;
        end
    end

    % Computing the value of K and Den*Cp and T of E-nodes. 
    for j = 1:Nx
        for i = 1:Ny
            if(w_m_x_node_t(i,j)>0)
                kht(i,j) = K_w_m_t(i,j)/w_m_x_node_t(i,j);%K
                hct(i,j) = DenCp_w_m_t(i,j)/w_m_x_node_t(i,j);%Den*Cp
            else
                kht(i,j) = kht_backup(i,j);
                hct(i,j) = hct_backup(i,j);
            end
            td(i,j) = kht(i,j)/hct(i,j);%K/(Den*Cp)
            if(w_m_t_t(i,j)>0)
                T0(i,j)  = T_w_m_t(i,j)/w_m_t_t(i,j);%T
            else
                T0(i,j) = T_backup(i,j);
            end
        end
    end
    % Apply the boundary conditions to T at every t+dt.
    % Constant temperature boundary
    % T = constant
%     T0(1,:) = 1000;
%     T0(Ny,:) = 1000;
%     T0(:,1) = 1000;
%     T0(:,Nx) = 1000;
    % Insulting boundary condition
    % T1(boudnary) = T2(internal)
    T0(1,2:Nx-1) = T0(2,2:Nx-1);
    T0(Ny,2:Nx-1) = T0(Ny,2:Nx-1);
    T0(:,1) = T0(:,2);
    T0(:,Nx) = T0(:,Nx-1);
    
    khtdt = kht;
    hctdt = hct;
    % Change k and den*Cp at internal nodes with upwind difference
    % Maybe don't need E-advection in the method of MIC
%     for j = 2:Nx-1
%         for i = 2:Ny-1
%             khtdt(i,j) = dt*( -vx(i,j)*( kht(i,j)-kht(i,j-1) )/dx -vy(i,j)*( kht(i,j)-kht(i-1,j) )/dy ) +kht(i,j) ;
%             hctdt(i,j) = dt*( -vx(i,j)*( hct(i,j)-hct(i,j-1) )/dx -vy(i,j)*( hct(i,j)-hct(i-1,j) )/dy ) +hct(i,j) ;
%         end
%     end
%     kht = khtdt;
%     hct = hctdt;
    % Constant temperature boundary condition.
    % Tbc = constant = 1000
    % up and down
%     for j = 1:Nx
%         i = 1;
%         k = (j-1)*Ny + i;
%         coe(k, k) = 1;
%         b(k, 1) = 1000;
%         i = Ny;
%         k = (j-1)*Ny + i;
%         coe(k, k) = 1;
%         b(k, 1) = 1000;
%     end
%     % left and right
%     for i = 1:Ny
%         j = 1;
%         k = (j-1)*Ny + i;
%         coe(k, k) = 1;
%         b(k, 1) = 1000;
%         j = Nx;
%         k = (j-1)*Ny + i;
%         coe(k, k) = 1;
%         b(k, 1) = 1000;
%     end
    % Insulating boundary condition.
    % Up and down
    for j = 2:Nx-1
        i = 1;
        k = (j-1)*Ny + i;
        % T(1,j) - T(2,j) = 0;
        coet(k, k) = 1;
        coet(k, k+1) = -1;
        bt(k, 1) = 0;
        i = Ny;
        k = (j-1)*Ny + i;
        % T(Ny,j) - T(Ny-1,j) = 0;
        coet(k, k) = 1;  
        coet(k, k-1) = -1;
        bt(k, 1) = 0;
    end
    % left and right
    for i = 1:Ny
        j = 1;
        k = (j-1)*Ny + i;
        coet(k, k) = 1;
        coet(k, k+Ny) = -1;
        bt(k, 1) = 0;
        j = Nx;
        k = (j-1)*Ny + i;
        coet(k, k) = 1;
        coet(k, k-Ny) = -1;
        bt(k, 1) = 0;
    end 

    % coefficients of interval points
    for j = 2:Nx-1
        for i = 2:Ny-1
            k = (j-1)*Ny + i; % k is the index
            if vx_matrix(i,j)>0
                dtdx = (T0(i,j)-T0(i,j-1))/dx;
            elseif vx_matrix(i,j)<0
                dtdx = (T0(i,j+1)-T0(i,j))/dx;
            end
            if vy_matrix(i,j)>0
                dtdy = (T0(i,j)-T0(i-1,j))/dy;
            elseif vy_matrix(i,j)<0
                dtdy = (T0(i+1,j)-T0(i,j))/dy;
            end
            coet(k, k) = (khtdt(i,j)+khtdt(i,j+1))/(2.*dx.^2)+(khtdt(i,j-1)+khtdt(i,j))/(2.*dx.^2)...
                +(khtdt(i,j)+khtdt(i+1,j))/(2.*dy.^2)+(khtdt(i-1,j)+khtdt(i,j))/(2.*dy.^2)...
                +hctdt(i,j)/dt_tem; % middle T3
            coet(k, k-1) = -(khtdt(i-1,j)+khtdt(i,j))/(2.*dy.^2); % up T2
            coet(k, k+1) = -(khtdt(i,j)+khtdt(i+1,j))/(2.*dy.^2); % down T4
            coet(k, k-Ny) = -(khtdt(i,j-1)+khtdt(i,j))/(2.*dx.^2); % left T1
            coet(k, k+Ny) = -(khtdt(i,j)+khtdt(i,j+1))/(2.*dx.^2); % right T5
            % !!!Note that there are not advection term because MIC have
            % marker transport.!!!
            bt(k, 1) = T0(i,j)*hctdt(i,j)/dt_tem; %- hctdt(i,j)*(vx(i,j)*dtdx+vy(i,j)*dtdy);
        end
    end
    %direct method:right martix is divided by coe martix on the left
    u = coet \ bt;
    %transform vector to grids
    % U has Ny rows and Nx columns
    U = reshape(u, Ny, Nx);
    T1 = U;
    % Computing dTi,j = dTi,j(subgrid) + dTi,j(remaining) 
    % Eq10.13
    dTij = T1-T0;
    % Computing subgrid diffusion for markers
    if (subgrdifcoe>0)
        % Existence of subgrid diffusion.
        % Page162 How to correct the problem ?
        % operating  consistent subgrid diffusion .
        
        % Clear subgrid temperature changes(dTijsub) for nodes
        dTijsub = zeros(Ny,Nx);
        DWTM = zeros(Ny,Nx);
        % From marker to BASIC NODES
        w_nodes = zeros(Ny,Nx);
        % Interpolate Tm0(nodal) from E-nodes(T0) to L-markers.
        for m = 1:marknum
            if (xm(m)>=x(1) && xm(m)<=x(Nx) && ym(m)>=y(1) && ym(m)<=y(Ny)) 
                % Interpolate T to BASIC NODES(Nx*Ny)
                % Define i,j indexes for the upper left node
                j=fix((xm(m)-x(1))/dx)+1;
                i=fix((ym(m)-y(1))/dy)+1;
                if(j<1)
                    j=1;
                elseif(j>Nx-1)
                    j=Nx-1;
                end
                if(i<1)
                    i=1;
                elseif(i>Ny-1)
                    i=Ny-1;
                end
                % Compute distances
                dis_x=xm(m)-x(j);
                dis_y=ym(m)-y(i);
                % Compute weights
                wtmij=(1-dis_x/dx)*(1-dis_y/dy);
                wtmi1j=(1-dis_x/dx)*(dis_y/dy);
                wtmij1=(dis_x/dx)*(1-dis_y/dy);
                wtmi1j1=(dis_x/dx)*(dis_y/dy);
                % Interpolate Basic node(Tm0nodal) from L-markers(T0i,j)
                Tm0nodal= T0(i,j)*wtmij + T0(i+1,j)*wtmi1j  ...
                         +T0(i,j+1)*wtmij1 + T0(i+1,j+1)*wtmi1j1;
                % Eq.10.14
                dTm = Tm0nodal - Tm0(m);
                % Page163
                tdiff = DenCpm(m)/Km(m)/(2/dx^2+2/dy^2);
                % Computing subgrid diffusion
                sdif=-subgrdifcoe*dt_tem/tdiff;
                if(sdif<-30)
                    sdif=-30;
                end
                % Page162 Eq.10.16
                dTmsub=dTm*(1-exp(sdif));
                % Eq.10.19
                % Tcorrected = Tm0 + dTsubgrid (+ dTremain)
                Tm0(m) = Tm0(m) + dTmsub;
                
                % Interpolate Basic node(dTijsub) from L-markers(dTmsub)
                % Eq.10.17
                DWTM(i,j) = DWTM(i,j) + dTmsub*DenCpm(m)*wtmij;
                w_nodes(i,j) = w_nodes(i,j) + DenCpm(m)*wtmij;
                DWTM(i+1,j) = DWTM(i+1,j) + dTmsub*DenCpm(m)*wtmi1j;
                w_nodes(i+1,j) = w_nodes(i+1,j) + DenCpm(m)*wtmi1j;
                DWTM(i,j+1) = DWTM(i,j+1) + dTmsub*DenCpm(m)*wtmij1;
                w_nodes(i,j+1) = w_nodes(i,j+1) + DenCpm(m)*wtmij1;
                DWTM(i+1,j+1) = DWTM(i+1,j+1) + dTmsub*DenCpm(m)*wtmi1j1;
                w_nodes(i+1,j+1) = w_nodes(i+1,j+1) + DenCpm(m)*wtmi1j1;
            end
        end
        for j = 1:Nx
            for i = 1:Ny
                if(w_nodes(i,j)>0)
                    % Compting dTijsub
                    % Eq.10.17
                    dTijsub(i,j) = DWTM(i,j)/w_nodes(i,j);
%                 else
%                     dTijsub(i,j) = dTijsub(i,j);
                end
            end
        end
        % Eq.10.18
        % Update dTij
        dTijremain = dTij - dTijsub;
    end
    % Interpolate L-markers(dTremain) from Basicnode(dTijremain)
    for m = 1:marknum
        if (xm(m)>=x(1) && xm(m)<=x(Nx) && ym(m)>=y(1) && ym(m)<=y(Ny)) 
            % Define i,j indexes for the upper left node
            j=fix((xm(m)-x(1))/dx)+1;
            i=fix((ym(m)-y(1))/dy)+1;
            if(j<1)
                j=1;
            elseif(j>Nx-1)
                j=Nx-1;
            end
            if(i<1)
                i=1;
            elseif(i>Ny-1)
                i=Ny-1;
            end
            % Compute distances
            dis_x=xm(m)-x(j);
            dis_y=ym(m)-y(i);
            % Compute weights
            wtmij=(1-dis_x/dx)*(1-dis_y/dy);
            wtmi1j=(1-dis_x/dx)*(dis_y/dy);
            wtmij1=(dis_x/dx)*(1-dis_y/dy);
            wtmi1j1=(dis_x/dx)*(dis_y/dy);
            % Interpolate Basic node(Tm0nodal) from L-markers(T0i,j)
            dTmremain=dTijremain(i,j)*wtmij + dTijremain(i+1,j)*wtmi1j  ...
                +dTijremain(i,j+1)*wtmij1 + dTijremain(i+1,j+1)*wtmi1j1;
            % Tcorrected = Tm0(Tm0+ dTsubgrid) + dTremain
            % Eq.10.19
            if(t>1)
                Tm0(m) = Tm0(m) + dTmremain;
            else
                % Interpolate new temperature for the 1st timestep
                Tm0(m) = T1(i,j)*wtmij + T1(i+1,j)*wtmi1j  ...
                        +T1(i,j+1)*wtmij1 + T1(i+1,j+1)*wtmi1j1;
            end
        end
    end
    % Marker transport
    for m = 1:marknum
        vxm = 1e-9;
        vym = 1e-9;
        xm(m) = xm(m)+vxm*dt_tem;
        ym(m) = ym(m)+vym*dt_tem;
        if (xm(m) > Lx)
            xm(m) = xm(m) - Lx;
        elseif(xm(m) < 0)
            xm(m) = xm(m) + Lx;
        end
        if (ym(m) > Ly)
            ym(m) = ym(m) - Ly;
        elseif(ym(m) < 0)
            ym(m) = ym(m) + Ly;
        end
    end
    figure(1);
    pcolor(x,y,T1);colorbar;
    xlabel('Horizontal(Km)');
    ylabel('Vertical(Km)');
    zlabel('Gravitational potential');
    title(['(CTbc) Inplicit form solution of T(advection and MIC) after',num2str(t),' deltat'])
    set(gca,'xaxislocation','top');
    set (gca,'YDir','reverse')
    colormap('Jet');
    shading interp;
    pause(0.01);

end