[m11_2]Combine three basic equations in geodynamics with self gravitation item.

 Self gravitation:LINE451-486.

%This program complished the process of 2D Eulerian advection with method
%of MIC(allocating memory temporarily).
clear all;clf;
% Set parameters
Nt = 2000;
output_interval = 1;
% Length of model
Lx = 10000000; 
Ly = 10000000;
xmid = Lx/2;
ymid = Ly/2;
% Sun of point number of model
Nx = 101; 
Ny = 101; 
Nx1 = Nx+1;
Ny1 = Ny+1;
% Interval of E-staggered-nodes
dx = Lx / (Nx-1);
dy = Ly / (Ny-1);
% Basic nodes.
x = 0:dx:Lx;
y = 0:dy:Ly;
% Vx staggered nodes.(Nx_node * Ny_node+1)
xvx = 0:dx:Lx;
yvx = -dy/2 : dy : Ly+dy/2;
% Vy staggered nodes.(Nx_node+1 * Ny_node)
xvy = -dx/2 : dx : Lx+dx/2;
yvy = 0:dy:Ly;
% P staggered nodes.(Nx_node+1 * Ny_node+1)
xp = -dx/2 : dx : Lx+dx/2;
yp = -dy/2 : dy : Ly+dy/2;
% Velocity and Pressure matrix
vx_matrix = zeros(Ny1,Nx1);
vy_matrix = zeros(Ny1,Nx1);
p_matrix = zeros(Ny1,Nx1);
% Vx at Pnodes
pvx = zeros(Ny1,Nx1);
pvy = zeros(Ny1,Nx1);
% Gravity acceleration
g_y = zeros(Ny1,Nx1);
g_x = zeros(Ny1,Nx1);
ETA = zeros(Ny,Nx);%density
RHO = zeros(Ny,Nx);%viscosity
% Stokes-equ and continuity-equ
% Temperature-equ
% Amount of markers.
Nx_marker = 4*(Nx-1);
Ny_marker = 4*(Ny-1);
marknum=Nx_marker*Ny_marker;
% Step-length among L-marker points
x_m_step = Lx/Nx_marker;
y_m_step = Ly/Ny_marker;
% Marker points.
% Stokes-equ and continuity-equ
xm=zeros(1,marknum); % Horizontal coordinates, m
ym=zeros(1,marknum);
rhom=zeros(1,marknum); % Density, kg/m^3
etam=zeros(1,marknum);
% Temperature-equ
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
Aefm = zeros(1,marknum); % K/den/Cp  td
Hrm = zeros(1,marknum); % temperature
% Define matrix
hct = zeros(Ny1,Nx1); %Den*Cp
kht = zeros(Ny1,Nx1); %K
T0 = zeros(Ny1,Nx1); %T
td = zeros(Ny1,Nx1); %K/Den*Cp
bt = zeros(Nx1*Ny1, 1);
coet = zeros(Nx1*Ny1, Nx1*Ny1);
bg = zeros(Nx1*Ny1, 1);
coeg = zeros(Nx1*Ny1, Nx1*Ny1);
% Setup of model
% "Squared planet"
den_mantle = 3300;
vis_mantle = 1e+21;
k_mantle = 3;%W/(m*K)
Cp_mantle = 1000;%J/(Kg*K)
t_mantle = 1500;%K 
aef_mantle = 3e-5;%1/K
Hr_mantle = 2e-8;

% "Low density core"
den_plume = 3000;
vis_plume = 1e+17;
k_plume = 2;%W/(m*K)
Cp_plume = 1000;%J/(Kg*K)
t_plume = 1800;%K
aef_plume = 2e-5;%1/K
Hr_plume = 2e-7;

% "Striky space"
den_striky = 0;
vis_striky = 1e+17;
k_striky = 3000;%W/(m*K)
Cp_striky = 1100;%J/(Kg*K)
t_striky = 273;%K
aef_striky = 0;%1/K
Hr_striky = 0;
% radius = 100000;

RHOVY=zeros(Ny1,Nx1); %kg/m^3
RHOVX=zeros(Ny1,Nx1); %kg/m^3
KVY=zeros(Ny1,Nx1); %kg/m^3
KVX=zeros(Ny1,Nx1); %kg/m^3
    
ETAP = zeros(Ny1,Nx1);
RHOP = zeros(Ny1,Nx1);

% Parameters and matrixs of exercise 9.3
srxy = zeros(Ny1,Nx1);
dsxy = zeros(Ny1,Nx1);
srxx = zeros(Ny,Nx1);
dsxx = zeros(Ny,Nx1);
pxyxy = zeros(Ny1,Nx1);
pxxxx = zeros(Ny1,Nx1);
Hsij = zeros(Ny1,Nx1);
Haij = zeros(Ny1,Nx1);
Hrij = zeros(Ny1,Nx1);
Aefij = zeros(Ny1,Nx1);

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
        % "Squared planet"
        if(xm(m)>=0.2*Lx && xm(m)<=0.8*Lx && ym(m)>=0.2*Ly && ym(m)<=0.8*Ly)
            rhom(m) = den_mantle; % Mantle density mantle == planet
            etam(m) = vis_mantle; % Mantle viscosity
            Km(m)   = k_mantle; % Mantle K 
            DenCpm(m) = den_mantle*Cp_mantle; % Mantle den*Cp
            Tm0(m)  = t_mantle; % Mantle T 
            Aefm(m) = aef_mantle;
            Hrm(m) = Hr_mantle;
            % "Low density core"
            if(xm(m)>=0.35*Lx && xm(m)<=0.65*Lx && ym(m)>=0.35*Ly && ym(m)<=0.65*Ly)
                rhom(m) = den_plume; % Plume density
                etam(m) = vis_plume; % Plume viscosity
                Km(m)   = k_plume; % Plume K
                DenCpm(m) = den_plume*Cp_plume; % Plume den*Cp
                Tm0(m)  = t_plume; %Plume T
                Aefm(m) = aef_plume;
                Hrm(m) = Hr_plume;
            end
        % "Striky space"   
        else
            rhom(m) = den_striky; % Sticky density
            etam(m) = vis_striky; % Sticky viscosity
            Km(m)   = k_striky; % Sticky K 
            DenCpm(m) = 3.3e+6; % Sticky den*Cp
            Tm0(m) = t_striky; % Sticky T
            Aefm(m) = aef_striky;
            Hrm(m) = Hr_striky;
        end
        % Update marker counter
        m=m+1;
    end
end

Kcont = 1e+21/dx;


% Unknown_number is (Ny+1)*(Nx+1)*3
unknown_number = (Nx+1)*(Ny+1)*3;
coe = sparse(unknown_number,unknown_number);
R = zeros(unknown_number, 1);
 
% Boundary condition : free slip = -1 ; no slip = 1
bc_left   = -1;
bc_right  = -1;
bc_top    = -1;
bc_bottom = -1;

dxymax = 0.5;    
dt = 1e+10; % initial timestep
dtkoef=1.2; % timestep increment
DTmax=20; % max temperature change per time step, K
subgrdifcoe = 1;
figure(1);
for t = 1:Nt     
    % "drunken sailor" instability appear if dt = 0 at the next line.
    % dt = 0; 
    % BASCI NODE
    w_m_x_node = zeros(Ny,Nx);
    vis_w_m = zeros(Ny,Nx);
    % rho in vy and vx nodes
    RHOYSUM=zeros(Ny1,Nx1);
    WTYSUM=zeros(Ny1,Nx1);
    RHOXSUM=zeros(Ny1,Nx1);
    WTXSUM=zeros(Ny1,Nx1);
    % K in vy and vx nodes
    KYSUM=zeros(Ny1,Nx1);
    WKYSUM=zeros(Ny1,Nx1);
    KXSUM=zeros(Ny1,Nx1);
    WKXSUM=zeros(Ny1,Nx1);    

    % Interpolate viscosity from markers to BASIC NODES.
    % Searching the coordinate of upper-left node of every L-marker.
    for m=1:1:marknum
        % BASIC NODE Interpolate den and vis from markes
        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);
        wtmij = (1-dis_x/dx)*(1-dis_y/dy);
        wtmi1j = (dis_x/dx)*(1-dis_y/dy);
        wtmij1 = (1-dis_x/dx)*(dis_y/dy);
        wtmi1j1 = (dis_x/dx)*(dis_y/dy);
        % |       |
        %     * * * * *   Discard the m-point on the right.
        w_m_x_node(i,j) = w_m_x_node(i,j) + wtmij;
        vis_w_m(i,j) = vis_w_m(i,j) + etam(m).*wtmij;
        % i+1,j
        w_m_x_node(i+1,j) = w_m_x_node(i+1,j) + wtmi1j;
        vis_w_m(i+1,j) = vis_w_m(i+1,j) + etam(m).*wtmi1j;
        % i,j+1
        w_m_x_node(i,j+1) = w_m_x_node(i,j+1) + wtmij1;
        vis_w_m(i,j+1) = vis_w_m(i,j+1) + etam(m).*wtmij1;
        % i+1,j+1
        w_m_x_node(i+1,j+1) = w_m_x_node(i+1,j+1) + wtmi1j1;
        vis_w_m(i+1,j+1) = vis_w_m(i+1,j+1) + etam(m).*wtmi1j1;
            
        % Interpolate Density and K to VY-NODE
        % Vy staggered nodes.(Nx1 * Ny)
        j=fix((xm(m)-xvy(1))/dx)+1;
        i=fix((ym(m)-yvy(1))/dy)+1;
        if(j<1)
            j=1;
        elseif(j>Nx)
            j=Nx;
        end
        if(i<1)
            i=1;
        elseif(i>Ny-1)
            i=Ny-1;
        end
        % Compute distances
        dxmj=xm(m)-xvy(j);
        dymi=ym(m)-yvy(i);
        % Compute weights
        wtmij=(1-dxmj/dx)*(1-dymi/dy);
        wtmi1j=(1-dxmj/dx)*(dymi/dy);
        wtmij1=(dxmj/dx)*(1-dymi/dy);
        wtmi1j1=(dxmj/dx)*(dymi/dy);
        % Update properties
        % i,j Node
        % rho and k in vy nodes
        RHOYSUM(i,j)=RHOYSUM(i,j)+rhom(m)*wtmij;
        WTYSUM(i,j)=WTYSUM(i,j)+wtmij;
        KYSUM(i,j)=KYSUM(i,j)+Km(m)*wtmij;% k in vy nodes
        % i+1,j Node
        RHOYSUM(i+1,j)=RHOYSUM(i+1,j)+rhom(m)*wtmi1j;
        WTYSUM(i+1,j)=WTYSUM(i+1,j)+wtmi1j;
        KYSUM(i+1,j)=KYSUM(i+1,j)+Km(m)*wtmi1j;
        % i,j+1 Node
        RHOYSUM(i,j+1)=RHOYSUM(i,j+1)+rhom(m)*wtmij1;
        WTYSUM(i,j+1)=WTYSUM(i,j+1)+wtmij1;
        KYSUM(i,j+1)=KYSUM(i,j+1)+Km(m)*wtmij1;
        % i+1,j+1 Node
        RHOYSUM(i+1,j+1)=RHOYSUM(i+1,j+1)+rhom(m)*wtmi1j1;
        WTYSUM(i+1,j+1)=WTYSUM(i+1,j+1)+wtmi1j1;
        KYSUM(i+1,j+1)=KYSUM(i+1,j+1)+Km(m)*wtmi1j1;
        % Interpolate Density and K to VX-NODE
        % Vx staggered nodes.(Nx * Ny1)
        j=fix((xm(m)-xvx(1))/dx)+1;
        i=fix((ym(m)-yvx(1))/dy)+1;
        if(j<1)
            j=1;
        elseif(j>Nx-1)
            j=Nx-1;
        end
        if(i<1)
            i=1;
        elseif(i>Ny)
            i=Ny;
        end
        % Compute distances
        dxmj=xm(m)-xvx(j);
        dymi=ym(m)-yvx(i);
        % Compute weights
        wtmij=(1-dxmj/dx)*(1-dymi/dy);
        wtmi1j=(1-dxmj/dx)*(dymi/dy);
        wtmij1=(dxmj/dx)*(1-dymi/dy);
        wtmi1j1=(dxmj/dx)*(dymi/dy);
        % Update properties
        % i,j Node
        RHOXSUM(i,j)=RHOXSUM(i,j)+rhom(m)*wtmij;
        WTXSUM(i,j)=WTXSUM(i,j)+wtmij;
        KXSUM(i,j)=KXSUM(i,j)+Km(m)*wtmij;% k in vx nodes
        % i+1,j Node
        RHOXSUM(i+1,j)=RHOXSUM(i+1,j)+rhom(m)*wtmi1j;
        WTXSUM(i+1,j)=WTXSUM(i+1,j)+wtmi1j;
        KXSUM(i+1,j)=KXSUM(i+1,j)+Km(m)*wtmi1j;
        % i,j+1 Node
        RHOXSUM(i,j+1)=RHOXSUM(i,j+1)+rhom(m)*wtmij1;
        WTXSUM(i,j+1)=WTXSUM(i,j+1)+wtmij1;
        KXSUM(i,j+1)=KXSUM(i,j+1)+Km(m)*wtmij1;
        % i+1,j+1 Node
        RHOXSUM(i+1,j+1)=RHOXSUM(i+1,j+1)+rhom(m)*wtmi1j1;
        WTXSUM(i+1,j+1)=WTXSUM(i+1,j+1)+wtmi1j1;
        KXSUM(i+1,j+1)=KXSUM(i+1,j+1)+Km(m)*wtmi1j1;
    end
    % Computing the value of density and viscosity of BASIC-NODE. 
    for j = 1:Nx
        for i = 1:Ny
            if(w_m_x_node(i,j)>0)
                % Not need RHO on the basic nodes.Just Y nodes.
                ETA(i,j) = vis_w_m(i,j)/w_m_x_node(i,j);
            end
        end
    end
    % Computing Density and K value at the VY and VX-nodes.
    for j=1:1:Nx1
        for i=1:1:Ny1
            % VY-nodes:(Nx1 * Ny) absent Ny1=0 !!!
            if(WTYSUM(i,j)>0)
                % rou_v
                RHOVY(i,j)=RHOYSUM(i,j)/WTYSUM(i,j);
                % K_v
                KVY(i,j)=KYSUM(i,j)/WTYSUM(i,j);
            end
            % VX-nodes:(Nx * Ny1)absent Nx1=0 !!!
            if(WTXSUM(i,j)>0)
                % rou_h
                RHOVX(i,j)=RHOXSUM(i,j)/WTXSUM(i,j);
                % K_h
                KVX(i,j) = KXSUM(i,j)/WTXSUM(i,j);
            end
        end
    end
    % Interpolate ETA from L-markers to P-nodes.
    w_m_p_node = zeros(Ny1,Nx1);% weights except T
    ETAp_w_m = zeros(Ny1,Nx1);% Viscosity at P node
    RHOp_w_m = zeros(Ny1,Nx1);% Density at P node
    Hr_w_m = zeros(Ny1,Nx1);  % Hr at P node
    Aef_w_m = zeros(Ny1,Nx1); % Aef at P node
    K_w_m_t = zeros(Ny1,Nx1); % K at P node
    DenCp_w_m_t = zeros(Ny1,Nx1);% Den*Cp at P node
    T_w_m_t = zeros(Ny1,Nx1); % T at P node
    w_m_t_T = zeros(Ny1,Nx1); % weight of T
    
    for m=1:1:marknum
        % Reserve original coordinate.
        % Interpolate Hr and aef from L-markers to P-staggered nodes.
        % P staggered nodes.(Nx1 * Ny1)
        j=fix((xm(m)-xp(1))/dx)+1;
        i=fix((ym(m)-yp(1))/dy)+1;
        if(j<1)
            j=1;
        elseif(j>Nx)
            j=Nx;
        end
        % P staggered nodes.(Nx1 * Ny1)
        if(i<1)
            i=1;
        elseif(i>Ny)
            i=Ny;
        end
        % Compute distances
        dis_x=xm(m)-xp(j);
        dis_y=ym(m)-yp(i);
        % Compute weights
        wtmij=(1-dis_x/dx)*(1-dis_y/dy);
        wtmi1j=(dis_x/dx)*(1-dis_y/dy);
        wtmij1=(1-dis_x/dx)*(dis_y/dy);
        wtmi1j1=(dis_x/dx)*(dis_y/dy);
        % Compute vx velocity
        w_m_p_node(i,j)=w_m_p_node(i,j)+wtmij;
        ETAp_w_m(i,j)=ETAp_w_m(i,j)+etam(m)*wtmij;
        RHOp_w_m(i,j)=RHOp_w_m(i,j)+rhom(m)*wtmij;
        Hr_w_m(i,j)=Hr_w_m(i,j)+Hrm(m)*wtmij;
        Aef_w_m(i,j) = Aef_w_m(i,j) + Aefm(m)*wtmij;
        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;
        % i+1,j Node
        w_m_p_node(i+1,j)=w_m_p_node(i+1,j)+wtmi1j;
        ETAp_w_m(i+1,j)=ETAp_w_m(i+1,j)+etam(m)*wtmi1j;
        RHOp_w_m(i+1,j)=RHOp_w_m(i+1,j)+rhom(m)*wtmi1j;
        Hr_w_m(i+1,j)=Hr_w_m(i+1,j)+Hrm(m)*wtmi1j;
        Aef_w_m(i+1,j) = Aef_w_m(i+1,j) + Aefm(m)*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;
        % i,j+1 Node
        w_m_p_node(i,j+1)=w_m_p_node(i,j+1)+wtmij1;
        ETAp_w_m(i,j+1)=ETAp_w_m(i,j+1)+etam(m)*wtmij1;
        RHOp_w_m(i,j+1)=RHOp_w_m(i,j+1)+rhom(m)*wtmij1;
        Hr_w_m(i,j+1)=Hr_w_m(i,j+1)+Hrm(m)*wtmij1;
        Aef_w_m(i,j+1) = Aef_w_m(i,j+1) + Aefm(m)*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;
        % i+1,j+1 Node
        w_m_p_node(i+1,j+1)=w_m_p_node(i+1,j+1)+wtmi1j1;
        ETAp_w_m(i+1,j+1)=ETAp_w_m(i+1,j+1)+etam(m)*wtmi1j1;
        RHOp_w_m(i+1,j+1)=RHOp_w_m(i+1,j+1)+rhom(m)*wtmi1j1;
        Hr_w_m(i+1,j+1)=Hr_w_m(i+1,j+1)+Hrm(m)*wtmi1j1;
        Aef_w_m(i+1,j+1) = Aef_w_m(i+1,j+1) + Aefm(m)*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
    for j = 1:Nx1
        for i = 1:Ny1
            if(w_m_p_node(i,j)>0)
                ETAP(i,j) = ETAp_w_m(i,j)/w_m_p_node(i,j);
                RHOP(i,j) = RHOp_w_m(i,j)/w_m_p_node(i,j);
                Hrij(i,j) = Hr_w_m(i,j)/w_m_p_node(i,j);
                Aefij(i,j) = Aef_w_m(i,j)/w_m_p_node(i,j);
                kht(i,j) = K_w_m_t(i,j)/w_m_p_node(i,j);%K
                hct(i,j) = DenCp_w_m_t(i,j)/w_m_p_node(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 boundary condition after interpolate T0 from markers(any t).
    % top and left Ibc
    T0(1,2:Nx) = T0(2,2:Nx);
    T0(Ny1,2:Nx) = T0(Ny,2:Nx);
    % left and right Ibc
    T0(:,1) = T0(:,2);
    T0(:,Nx1) = T0(:,Nx);
    
    % Solve the possion equation of gravitational potential(Φ)
    % coefficients of boundary
    % up and down
    K = 2/3;% circular
    G = 6.672e-11;% (Nm^2)/(kg^2),gravitational constant.
    for j = 1:Nx1
        for i = 1:Ny1
            k = (j-1)*Ny1 + i;
            distance = ((xp(j)-xmid)^2 + (yp(i)-ymid)^2)^0.5;
            if(distance > Lx/2 || i==1 || i==Ny1 || j==1 || j==Nx1)
                % Graviational = 0
                coeg(k, k) = 1;
                bg(k,1) = 0;
            else
                % Distribution of nodes.
                %       j-1            j           j+1
                %  i-1              2Φ(i-1,j)
                %  i   1Φ(i,j-1)   3Φ(i,j)      5Φ(i,j+1)
                %  i+1              4Φ(i+1,j)
                coeg(k, k) = -2/(dx.^2)-2/(dy.^2); % middleΦ(i,j) 3
                coeg(k, k-1) = 1/(dy.^2);  % up Φ(i-1,j)  2
                coeg(k, k+1) = 1/(dy.^2);  % down Φ(i+1,j) 4
                coeg(k, k-Ny1) = 1/(dx.^2); % left Φ(i,j-1) 1
                coeg(k, k+Ny1) = 1/(dx.^2); % right Φ(i,j+1) 5
                bg(k,1) = 4*K*pi*G*RHOP(i,j);
            end
        end
    end
    ug = coeg\bg;
    Ug = reshape(ug,Ny1,Nx1);
    for j = 1:Nx
        for i = 1:Ny
            g_x(i,j) = -(Ug(i,j+1)-Ug(i,j))/dx;
            g_y(i,j) = -(Ug(i+1,j)-Ug(i,j))/dy;
        end
    end
    % Mechanical Solution
    % Solve Stokes-equ and Continuity-equ.
    for j=1:1:Nx1
        for i=1:1:Ny1
            % Define global indexes in algebraic space
            kvx=((j-1)*Ny1+i-1)*3+1; % Vx
            kvy=kvx+1; % Vy
            kpm=kvx+2; % P
            
            % Vx equation External points
            if(i==1 || i==Ny1 || j==1 || j==Nx || j==Nx1)
                % Boundary Condition
                % 1*Vx=0
                coe(kvx,kvx)=1; % Left part
                R(kvx)=0; % Right part
                % Top boundary
                if(i==1 && j>1 && j1 && j1 && i1 && idxymax*dx)
        dt=dxymax*dx/maxvx;
    end
    if(dt*maxvy>dxymax*dy)
        dt=dxymax*dy/maxvy;
    end
    khtdt = kht;
    hctdt = hct;
    for titer = 1:1:2
        % Ctbc
        % Why???
        % Up and down
        for j = 2:Nx
            i = 1;
            k = (j-1)*Ny1 + i;
            % T(1,j) - T(2,j) = 0;
            coet(k, k) = 1;
            coet(k, k+1) = -1;
            bt(k, 1) = 0;
            i = Ny1;
            k = (j-1)*Ny1 + 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:Ny1
            j = 1;
            k = (j-1)*Ny1 + i;
            coet(k, k) = 1;
            coet(k, k+Ny1) = -1;
            bt(k, 1) = 0;
            j = Nx1;
            k = (j-1)*Ny1 + i;
            coet(k, k) = 1;
            coet(k, k-Ny1) = -1;
            bt(k, 1) = 0;
        end

        % coefficients of interval points
        for j = 2:Nx
            for i = 2:Ny
                k = (j-1)*Ny1 + 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
                % KVX == Kh;KVY == Kv
                khij = KVX(i,j);
                khij1 = KVX(i,j-1);
                kvij = KVY(i,j);
                kvi1j = KVY(i-1,j);
                % Not 2* why?? page177
                coet(k, k) = (khij+khij1)/dx^2+hctdt(i,j)/dt...
                            +(kvij+kvi1j)/dy^2;% middle T3 i,j
                coet(k, k-1) = -(kvi1j)/(2.*dy.^2); % up T2 i-1,j
                coet(k, k+1) = -(kvij)/(2.*dy.^2); % down T4 i+1,j 
                coet(k, k-Ny1) = -(khij1)/(dx.^2); % left T1 i,j-1
                coet(k, k+Ny1) = -(khij)/(dx.^2); % right T5 i,j+1
                % !!!Note that there are not advection term because MIC have
                % marker transport.!!!
                % Adding H(P-nodes)
                bt(k, 1) = T0(i,j)*hctdt(i,j)/dt + Hrij(i,j) + Haij(i,j) + Hsij(i,j); %- 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 Ny1 rows and Nx1 columns
        U = reshape(u, Ny1, Nx1);
        T1 = U;
        % Computing dTi,j = dTi,j(subgrid) + dTi,j(remaining)
        % Eq10.13
        dTij = T1-T0;
        maxDTcurrent = max(max(abs(dTij)));
        if(titer<2 && maxDTcurrent>DTmax)
            dt=dt/maxDTcurrent*DTmax;
        else
            break;
        end
    end
    % 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
        % From marker to P-NODES
        w_Pnodes = zeros(Ny1,Nx1);
        DWTM = zeros(Ny1,Nx1);
        % Interpolate Tm0(nodal) from E-nodes(T0) to L-markers.
        for m = 1:marknum
            % Interpolate T to P-NODES(Nx*Ny)
            % Define i,j indexes for the upper left node
            j=fix((xm(m)-xp(1))/dx)+1;
            i=fix((ym(m)-yp(1))/dy)+1;
            if(j<1)
                j=1;
            elseif(j>Nx)
                j=Nx;
            end
            if(i<1)
                i=1;
            elseif(i>Ny)
                i=Ny;
            end
            % Compute distances
            dis_x=xm(m)-xp(j);
            dis_y=ym(m)-yp(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/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_Pnodes(i,j) = w_Pnodes(i,j) + DenCpm(m)*wtmij;
            DWTM(i+1,j) = DWTM(i+1,j) + dTmsub*DenCpm(m)*wtmi1j;
            w_Pnodes(i+1,j) = w_Pnodes(i+1,j) + DenCpm(m)*wtmi1j;
            DWTM(i,j+1) = DWTM(i,j+1) + dTmsub*DenCpm(m)*wtmij1;
            w_Pnodes(i,j+1) = w_Pnodes(i,j+1) + DenCpm(m)*wtmij1;
            DWTM(i+1,j+1) = DWTM(i+1,j+1) + dTmsub*DenCpm(m)*wtmi1j1;
            w_Pnodes(i+1,j+1) = w_Pnodes(i+1,j+1) + DenCpm(m)*wtmi1j1;
        end
        dTijsub = zeros(Ny1,Nx1);
        for j = 1:Nx1
            for i = 1:Ny1
                if(w_Pnodes(i,j)>0)
                    % Compting dTijsub
                    % Eq.10.17
                    dTijsub(i,j) = DWTM(i,j)/w_Pnodes(i,j);
                end
            end
        end
        % Eq.10.18
        % Update dTij
        dTijremain = dTij - dTijsub;
    end
    % Interpolate L-markers(dTremain) from P-node(dTijremain)
    for m = 1:marknum
        % Define i,j indexes for the upper left node
        j=fix((xm(m)-xp(1))/dx)+1;
        i=fix((ym(m)-yp(1))/dy)+1;
        if(j<1)
            j=1;
        elseif(j>Nx)
            j=Nx;
        end
        if(i<1)
            i=1;
        elseif(i>Ny)
            i=Ny;
        end
        % Compute distances
        dis_x=xm(m)-xp(j);
        dis_y=ym(m)-yp(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
    % Figure
    subplot(1,3,1);
    colormap('Jet');
    pcolor(x,y,log10(ETA));
    hold on;
    %P nodes:(Nx1 * Ny1).
    quiver(xp(1:2:Nx1),yp(1:2:Ny1),pvx(1:2:Ny1,1:2:Nx1),pvy(1:2:Ny1,1:2:Nx1),'w')
    xlabel('Horizontal(m)')
    ylabel('Vertical(m)')
    caxis([17 21]);
    set(gca,'xaxislocation','top');
    set (gca,'YDir','reverse')
    shading interp;
    axis ij image;
    colorbar
    title(['log10(ETA) after ',num2str(t),' deltat'])
    % Ha
    subplot(1,3,2);
    colormap('Jet');
    pcolor(xp,yp,Hrij);
    xlabel('Horizontal(m)')
    ylabel('Vertical(m)')
    set(gca,'xaxislocation','top');
    set (gca,'YDir','reverse')
    shading interp;
    colorbar;
    axis ij image;
    title(['Ha after ',num2str(t),' deltat'])
    subplot(1,3,3);
    pcolor(xp,yp,T1);colorbar;
    xlabel('Horizontal(Km)');
    ylabel('Vertical(Km)');
    zlabel('Gravitational potential');
    title(['(Ibc)(MIC) Temperature after',num2str(t),' deltat'])
    set(gca,'xaxislocation','top');
    set (gca,'YDir','reverse')
    axis ij image;
    shading interp;
    pause(0.01);
    % The classical Runge-Kutta scheme,Fourth-order in sapce First-order in time
    % Define Vxa(Vya)1,Vxb(Vyb)2,Vxc(Vyc)3,Vxd(Vyd)4.
    vxm=zeros(4,1);
    vym=zeros(4,1);
    % Interpolate vx and vy components from E-nodes to L-markers.
    for m=1:1:marknum
        % Reserve original coordinate.
        xa = xm(m);
        ya = ym(m);
        for rk = 1:1:4
            % Interpolate vx-p and vy-p from P-staggered nodes to L-markers.
            % P staggered nodes.(Nx1 * Ny1)
            j=fix((xm(m)-xp(1))/dx)+1;
            i=fix((ym(m)-yp(1))/dy)+1;
            if(j<1)
                j=1;
            elseif(j>Nx)
                j=Nx;
            end
            if(i<1)
                i=1;
            elseif(i>Ny)
                i=Ny;
            end
            % Compute distances
            dis_x=xm(m)-xp(j);
            dis_y=ym(m)-yp(i);
            % Compute weights
            wtmij=(1-dis_x/dx)*(1-dis_y/dy);
            wtmi1j=(dis_x/dx)*(1-dis_y/dy);
            wtmij1=(1-dis_x/dx)*(dis_y/dy);
            wtmi1j1=(dis_x/dx)*(dis_y/dy);
            % Compute vxm and vym velocity
            vxm(rk) = pvx(i,j)*wtmij+pvx(i+1,j)*wtmi1j...
                     +pvx(i,j+1)*wtmij1+pvx(i+1,j+1)*wtmi1j1;
            vym(rk) = pvy(i,j)*wtmij+pvy(i+1,j)*wtmi1j...
                     +pvy(i,j+1)*wtmij1+pvy(i+1,j+1)*wtmi1j1;
            % Interpolate vx from Vx-staggered nodes to L-markers.
            % Define i,j indexes for the upper left node
            j=fix((xm(m)-xvx(1))/dx)+1;
            i=fix((ym(m)-yvx(1))/dy)+1;
            if(j<1)
                j=1;
            elseif(j>Nx-1)
                j=Nx-1;
            end
            % Vx staggered nodes.(Nx * Ny+1)
            if(i<1)
                i=1;
            elseif(i>Ny)
                i=Ny;
            end
            % Compute distances
            dis_x=xm(m)-xvx(j);
            dis_y=ym(m)-yvx(i);
            % Compute weights
            wtmij=(1-dis_x/dx)*(1-dis_y/dy);
            wtmi1j=(dis_x/dx)*(1-dis_y/dy);
            wtmij1=(1-dis_x/dx)*(dis_y/dy);
            wtmi1j1=(dis_x/dx)*(dis_y/dy);
            % Compute vx velocity
            vxcij = (1/3)*(2*vx_matrix(i,j));
            vxci1j = (1/3)*(2*vx_matrix(i+1,j));
            vxcij1 = (1/3)*(2*vx_matrix(i,j+1));
            vxci1j1 = (1/3)*(2*vx_matrix(i+1,j+1));
            vxm(rk) = (1/3)*vxm(rk) + (vxcij*wtmij+vxci1j*wtmi1j...
                     +vxcij1*wtmij1+vxci1j1*wtmi1j1);
            
            % Interpolate vy
            % Define i,j indexes for the upper left node
            j=fix((xm(m)-xvy(1))/dx)+1;
            i=fix((ym(m)-yvy(1))/dy)+1;
            if(j<1)
                j=1;
            elseif(j>Nx)
                j=Nx;
            end
            if(i<1)
                i=1;
            elseif(i>Ny-1)
                i=Ny-1;
            end
            % Compute distances
            dis_x=xm(m)-xvy(j);
            dis_y=ym(m)-yvy(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);
            % Compute vx velocity
            vycij = (1/3)*(2*vy_matrix(i,j));
            vyci1j = (1/3)*(2*vy_matrix(i+1,j));
            vycij1 = (1/3)*(2*vy_matrix(i,j+1));
            vyci1j1 = (1/3)*(2*vy_matrix(i+1,j+1));
            vym(rk) = (1/3)*vym(rk) + (vycij*wtmij+vyci1j*wtmi1j...
                     +vycij1*wtmij1+vyci1j1*wtmi1j1);
                 
            if(rk == 1 || rk == 2)
                xm(m) = vx_matrix(rk)*dt/2 + xa;
                ym(m) = vy_matrix(rk)*dt/2 + ya;
            elseif(rk == 3)
                xm(m) = vx_matrix(rk)*dt + xa;
                ym(m) = vy_matrix(rk)*dt + ya;
            end
        end
        % Return original coordinate.
        xm(m) = xa;
        ym(m) = ya;
        % Move markers
        vxefft = (1/6)*(vxm(1)+2*vxm(2)+2*vxm(3)+vxm(4));
        vyefft = (1/6)*(vym(1)+2*vym(2)+2*vym(3)+vym(4));        
        xm(m)=xm(m)+dt*vxefft;
        ym(m)=ym(m)+dt*vyefft;
    end
end

 

 Initial model：

 

Model after 61 dt：