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assem_mat_vect_gio_stab_i.m
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function [A,rhs] = assem_mat_vect_gio_stab_i(vertices,elements,boundaries,g,dt,un,vn,cn,theta,coeff,h0,h_k,f1,f2,f3,wdn,omega,time...
,un_old,vn_old,cn_old,nx_nodes,ny_nodes,wetnodes,frontnodes,frontwettednodes,drynodes,littlewetnodes,save_path)
%
% assembla la matrice globale e il termine noto del sistema
% con operazioni vettoriali
nov = size(vertices,2);
noe=size(elements,2);
% matrice di advezione: \int_T (\alpha U \cdot grad \phi_j) \phi_i
% A_adve_alpha = ass_adve(vertices,elements,un,vn,alpha,1);
% matrice di advezione: \int_T (U \cdot grad \phi_j) \phi_i
A_adve = ass_adve(vertices,elements,un,vn,1,1,2);
% A_adve_delta = ass_adve(vertices,elements,un,vn,delta,2);
% matrici:
% Gx = \int_T g d(\phi_j)/dx \phi_i
% Gy = \int_T g d(\phi_j)/dy \phi_i
[Gx,Gy] = ass_grad(vertices,elements,1,cn*0.5,2);
GxMom=Gx; GyMom=Gy;
% Per spegnere il termine con c/2 grad c nell'eq del momento sul fronte
% GxMom(frontnodes,:) = 0;
% GyMom(frontnodes,:) = 0;
% GxMom(littlewetnodes,:) = 0;
% GyMom(littlewetnodes,:) = 0;
% GxMom(drynodes,:) = 0;
% GyMom(drynodes,:) = 0;
% matrici:
% Hx = \int_T H d(\phi_j)/dx \phi_i
% Hy = \int_T H d(\phi_j)/dy \phi_i
% [Hx,Hy] = ass_grad(vertices,elements,1,cn/2,1);
% matrice di massa
M=ass_massa(vertices,elements,1,1,2);
% matrice associata alle resistenze al fondo
% Fr = ass_massa(vertices,elements,1,coeff,2);
%
% matrice nulla
Zero = sparse(nov,nov);
[rhs_frx,rhs_fry] = ass_rhs_fr(vertices,elements,coeff,un,vn,4);
rhs_fr = [rhs_frx;rhs_fry];
[rhs_x,rhs_y,rhs_3] = ass_rhs_g(vertices,elements,f1,f2,f3,4);
% [Fx,Fy] = ass_grad(vertices,elements,1,c0*0.5,1);
% rhs_x = Fx*c0;
% rhs_y = Fy*c0;
[N11,N12,N21,N22] = ass_normali_penalty(vertices,elements,nx_nodes,ny_nodes,2);
gamma = 1e10;
N = gamma*[N11 , N12 , Zero;
N21 , N22 , Zero;
Zero, Zero, Zero];
% matrice di massa a blocchi
M3 = [M , Zero, Zero ;
Zero , M , Zero ;
Zero , Zero, M];
% matrice finale
% Dx=Hx-F_x;
% Dy=Hy-F_y;
A = [A_adve ,Zero ,GxMom ;
Zero ,A_adve ,GyMom ;
Gx ,Gy ,A_adve];
% A = A + [Fr , Zero , Zero ;
% Zero , Fr , Zero ;
% Zero , Zero , Zero ];
%
%
A = A + N;
%parametro e matrici stabilizazzione
% Porre delta_s=0 per spegnere tutta la stabilizzazione: sia streamline
% diffusion, sia shock capturing (per spegnere solo shock capturing agire
% in assem_mi, per spegnere solo streamline diffusion BOH)
% delta_s = max(h_k);
delta_s = 1;
% delta_s = 0;
theta_s = theta;
[M_Lh,Lh,f_Lh] = assem_mat_GLS_simm1(vertices,elements,boundaries,g,un,vn,cn,coeff,f1,f2,f3,theta,dt,h_k,omega,time,un_old,vn_old,cn_old,un,vn,drynodes,save_path);
% % Per spegnare la gravità sui frontnodes
% rhs_x(frontnodes) = 0*rhs_x(frontnodes);
% rhs_y(frontnodes) = 0*rhs_y(frontnodes);
% rhs_3(frontnodes) = 0*rhs_3(frontnodes);
%
% % Per spegnare la gravità sui littlewetnodes
% rhs_x(littlewetnodes) = 0*rhs_x(littlewetnodes);
% rhs_y(littlewetnodes) = 0*rhs_y(littlewetnodes);
% rhs_3(littlewetnodes) = 0*rhs_3(littlewetnodes);
% Per la correzione di continuità di cui in [Horritt, eq (12)]
% sui nodi completamente bagnati degli elementi parzialmente bagnati
% aggiungere eta*A_elem/2*(dc1/dt+dc2/dt)
eta = 0;
area = nanmean(pdetrg(vertices,elements));
temp = setdiff(frontnodes,frontwettednodes);
[temp1,temp2] = meshgrid(frontnodes,frontnodes);
CC_A = eta*area/2*M.*sparse(temp1,temp2,1,nov,nov);
CC_A = diag(CC_A*ones(nov,1)); % lumping
CC_A = [Zero Zero Zero; Zero Zero Zero; Zero Zero CC_A];
CC_rhs = CC_A*[0.*un;0.*vn;cn];
% % Per modulare la stabilizzazione sul fronte
% delta_front = 2;
% Lh(frontnodes,:) = delta_front*Lh(frontnodes,:);
% Lh(:,frontnodes) = delta_front*Lh(:,frontnodes);
% Lh(frontnodes,frontnodes) = Lh(frontnodes,frontnodes) / delta_front;
% M_Lh(frontnodes,:) = delta_front*M_Lh(frontnodes,:);
% M_Lh(:,frontnodes) = delta_front*M_Lh(:,frontnodes);
% M_Lh(frontnodes,frontnodes) = M_Lh(frontnodes,frontnodes) / delta_front;
% f_Lh(frontnodes) = delta_front*f_Lh(frontnodes);
% Per imporre u=umax sui dry (ora non fa niente!)
gamma=0;
% [dx_cn_vecchia,dy_cn_vecchia]=pdegrad(vertices,elements,cn_vecchia);
% val = zeros(nov,1);
% val(frontnodes) = (cn(frontnodes)-cn_vecchia(frontnodes))./(dt*dx_cn_vecchia(frontnodes));
% ys = unique(vertices(2,:));
% for i=1:length(ys)
% % [~,idx] = min(abs(vertices(2,:)-(ymax+ymin)/2));
% % ymed = vertices(2,idx);
% % idxs_ymed = find(vertices(2,:) == ymed);
% idxs = find(vertices(2,:) = ys(i);
% idx_front = max(intersect(idxs,frontnodes));
% val(idxs(idxs >= idx_front)) = val(idx_front);
% end
val = max(un(setdiff(wetnodes,frontnodes)));
% [V11,V12,V21,V22] = u_fixval_penalty(vertices,elements,val,2);
V11 = M;
V12 = 0.*V11;
V21 = 0.*V11;
V22 = 0.*V11;
% V(un(setdiff([1:nov],drynodes)),un(setdiff([1:nov],drynodes)))=0*V(un(setdiff([1:nov],drynodes)),un(setdiff([1:nov],drynodes)));
% V(un(setdiff([1:nov],drynodes)),:)=0*V(un(setdiff([1:nov],drynodes)),:);
% V(:,un(setdiff([1:nov],drynodes)))=0*V(:,un(setdiff([1:nov],drynodes)));
V11(setdiff(wetnodes,frontnodes),:)=0;
% V(:,wetnodes)=0*V(:,wetnodes);
V = gamma*[V11 , V12 , Zero;
V21 , V22 , Zero;
Zero, Zero, Zero];
rhs_V = V*[val*ones(nov,1); zeros(nov,1); zeros(nov,1)];
rhs_V(setdiff(wetnodes,frontnodes)) = 0;
% termine noto e matrice IN QUESTO ORDINE
rhs = (M3 - (1-theta)*dt*A)*[un;vn;cn] + delta_s*(M_Lh - dt*(1-theta_s)*Lh)*[un;vn;cn] +dt*([rhs_x;rhs_y;rhs_3] - [rhs_fr;zeros(nov,1)] + delta_s*f_Lh);% + CC_rhs + rhs_V;
A = M3 + theta*dt*A + delta_s*(M_Lh + dt*theta_s*Lh);% + CC_A + V;