--- a/+scheme/Hypsyst2d.m	Wed Jan 25 15:37:12 2017 +0100
+++ b/+scheme/Hypsyst2d.m	Thu Jan 26 09:57:24 2017 +0100
@@ -6,10 +6,10 @@
         x,y % Grid
         X,Y % Values of x and y for each grid point
         order % Order accuracy for the approximation
-
+        
         D % non-stabalized scheme operator
-        A, B, E
-    
+        A, B, E %Coefficient matrices
+        
         H % Discrete norm
         % Norms in the x and y directions
         Hxi,Hyi % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
@@ -17,14 +17,14 @@
         e_w, e_e, e_s, e_n
         params %parameters for the coeficient matrice
     end
-
-
+    
     methods
+        %Solving Hyperbolic systems on the form u_t=-Au_x-Bu_y-Eu
         function obj = Hypsyst2d(m, lim, order, A, B, E, params)
             default_arg('E', [])
             xlim = lim{1};
             ylim = lim{2};
-
+            
             if length(m) == 1
                 m = [m m];
             end
@@ -32,50 +32,50 @@
             obj.A=A;
             obj.B=B;
             obj.E=E;
-
+            
             m_x = m(1);
             m_y = m(2);
             obj.params = params;
-
+            
             ops_x = sbp.D2Standard(m_x,xlim,order);
             ops_y = sbp.D2Standard(m_y,ylim,order);
-
+            
             obj.x = ops_x.x;
             obj.y = ops_y.x;
-
+            
             obj.X = kr(obj.x,ones(m_y,1));
-            obj.Y = kr(ones(m_x,1),obj.y);         
-
+            obj.Y = kr(ones(m_x,1),obj.y);
+            
             Aevaluated = obj.evaluateCoefficientMatrix(A, obj.X, obj.Y);
             Bevaluated = obj.evaluateCoefficientMatrix(B, obj.X, obj.Y);
             Eevaluated = obj.evaluateCoefficientMatrix(E, obj.X, obj.Y);
-
+            
             obj.n = length(A(obj.params,0,0));
-
+            
             I_n = eye(obj.n);I_x = speye(m_x);
             obj.I_x = I_x;
             I_y = speye(m_y);
             obj.I_y = I_y;
-
-
+            
+            
             D1_x = kr(I_n, ops_x.D1, I_y);
             obj.Hxi = kr(I_n, ops_x.HI, I_y);
             D1_y = kr(I_n, I_x, ops_y.D1);
             obj.Hyi = kr(I_n, I_x, ops_y.HI);
-
+            
             obj.e_w = kr(I_n, ops_x.e_l, I_y);
             obj.e_e = kr(I_n, ops_x.e_r, I_y);
             obj.e_s = kr(I_n, I_x, ops_y.e_l);
             obj.e_n = kr(I_n, I_x, ops_y.e_r);
-
-            obj.m=m;
-            obj.h=[ops_x.h ops_y.h];
-            obj.order=order;
-
-            obj.D=-Aevaluated*D1_x-Bevaluated*D1_y-Eevaluated;
-
+            
+            obj.m = m;
+            obj.h = [ops_x.h ops_y.h];
+            obj.order = order;
+            
+            obj.D = -Aevaluated*D1_x-Bevaluated*D1_y-Eevaluated;
+            
         end
-
+        
         % Closure functions return the opertors applied to the own doamin to close the boundary
         % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin.
         %       boundary            is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'.
@@ -85,206 +85,217 @@
             default_arg('type','char');
             switch type
                 case{'c','char'}
-                    [closure,penalty]=boundary_condition_char(obj,boundary);
+                    [closure,penalty] = boundary_condition_char(obj,boundary);
                 case{'general'}
-                    [closure,penalty]=boundary_condition_general(obj,boundary,L);
+                    [closure,penalty] = boundary_condition_general(obj,boundary,L);
                 otherwise
                     error('No such boundary condition')
             end
         end
-
+        
         function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
             error('An interface function does not exist yet');
         end
-
+        
         function N = size(obj)
             N = obj.m;
         end
-
+        
         function [ret] = evaluateCoefficientMatrix(obj, mat, X, Y)
-            params=obj.params;
-
+            params = obj.params;
+            
             if isa(mat,'function_handle')
-                [rows,cols]=size(mat(params,0,0));
-                matVec=mat(params,X',Y');
-                matVec=sparse(matVec);
-                side=max(length(X),length(Y));
+                [rows,cols] = size(mat(params,0,0));
+                matVec = mat(params,X',Y');
+                matVec = sparse(matVec);
+                side = max(length(X),length(Y));
             else
-                matVec=mat;
-                [rows,cols]=size(matVec);
-                side=max(length(X),length(Y));
-                cols=cols/side;
+                matVec = mat;
+                [rows,cols] = size(matVec);
+                side = max(length(X),length(Y));
+                cols = cols/side;
             end
-            ret=cell(rows,cols);
-
-            for ii=1:rows
+            ret = cell(rows,cols);
+            
+            for ii = 1:rows
                 for jj=1:cols
-                    ret{ii,jj}=diag(matVec(ii,(jj-1)*side+1:jj*side));
+                    ret{ii,jj} = diag(matVec(ii,(jj-1)*side+1:jj*side));
                 end
             end
-            ret=cell2mat(ret);
+            ret = cell2mat(ret);
         end
-
-
-        function [closure, penalty]=boundary_condition_char(obj,boundary)
-            params=obj.params;
-            x=obj.x; y=obj.y;
-          
+        
+        %Characteristic boundary conditions
+        function [closure, penalty] = boundary_condition_char(obj,boundary)
+            params = obj.params;
+            x = obj.x;
+            y = obj.y;
+            
             switch boundary
                 case {'w','W','west'}
-                    e_=obj.e_w;
-                    mat=obj.A;
-                    boundPos='l';
-                    Hi=obj.Hxi;
-                    [V,Vi,D,signVec]=obj.matrixDiag(mat,x(1),y);
-                    side=max(length(y));
+                    e_ = obj.e_w;
+                    mat = obj.A;
+                    boundPos = 'l';
+                    Hi = obj.Hxi;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,x(1),y);
+                    side = max(length(y));
                 case {'e','E','east'}
-                    e_=obj.e_e;
-                    mat=obj.A;
-                    boundPos='r';
-                    Hi=obj.Hxi;
-                    [V,Vi,D,signVec]=obj.matrixDiag(mat,x(end),y);
-                    side=max(length(y));
+                    e_ = obj.e_e;
+                    mat = obj.A;
+                    boundPos = 'r';
+                    Hi = obj.Hxi;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,x(end),y);
+                    side = max(length(y));
                 case {'s','S','south'}
-                    e_=obj.e_s;
-                    mat=obj.B;
-                    boundPos='l';
-                    Hi=obj.Hyi;
-                    [V,Vi,D,signVec]=obj.matrixDiag(mat,x,y(1));
-                    side=max(length(x));
+                    e_ = obj.e_s;
+                    mat = obj.B;
+                    boundPos = 'l';
+                    Hi = obj.Hyi;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,x,y(1));
+                    side = max(length(x));
                 case {'n','N','north'}
-                    e_=obj.e_n;
-                    mat=obj.B;
-                    boundPos='r';
-                    Hi=obj.Hyi;
-                    [V,Vi,D,signVec]=obj.matrixDiag(mat,x,y(end));
-                    side=max(length(x));
+                    e_ = obj.e_n;
+                    mat = obj.B;
+                    boundPos = 'r';
+                    Hi = obj.Hyi;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,x,y(end));
+                    side = max(length(x));
             end
-
-            pos=signVec(1); zeroval=signVec(2); neg=signVec(3);
-
+            pos = signVec(1);
+            zeroval = signVec(2);
+            neg = signVec(3);
+            
             switch boundPos
                 case {'l'}
-                    tau=sparse(obj.n*side,pos);
-                    Vi_plus=Vi(1:pos,:);
-                    tau(1:pos,:)=-abs(D(1:pos,1:pos));
-                    closure=Hi*e_*V*tau*Vi_plus*e_';
-                    penalty=-Hi*e_*V*tau*Vi_plus;
+                    tau = sparse(obj.n*side,pos);
+                    Vi_plus = Vi(1:pos,:);
+                    tau(1:pos,:) = -abs(D(1:pos,1:pos));
+                    closure = Hi*e_*V*tau*Vi_plus*e_';
+                    penalty = -Hi*e_*V*tau*Vi_plus;
                 case {'r'}
-                    tau=sparse(obj.n*side,neg);
-                    tau((pos+zeroval)+1:obj.n*side,:)=-abs(D((pos+zeroval)+1:obj.n*side,(pos+zeroval)+1:obj.n*side));
-                    Vi_minus=Vi((pos+zeroval)+1:obj.n*side,:);
-                    closure=Hi*e_*V*tau*Vi_minus*e_';
-                    penalty=-Hi*e_*V*tau*Vi_minus;
+                    tau = sparse(obj.n*side,neg);
+                    tau((pos+zeroval)+1:obj.n*side,:) = -abs(D((pos+zeroval)+1:obj.n*side,(pos+zeroval)+1:obj.n*side));
+                    Vi_minus = Vi((pos+zeroval)+1:obj.n*side,:);
+                    closure = Hi*e_*V*tau*Vi_minus*e_';
+                    penalty = -Hi*e_*V*tau*Vi_minus;
             end
         end
-
-
-        function [closure,penalty]=boundary_condition_general(obj,boundary,L)
-            params=obj.params;
-            x=obj.x; y=obj.y;
-
+        
+        % General boundary condition in the form Lu=g(x)
+        function [closure,penalty] = boundary_condition_general(obj,boundary,L)
+            params = obj.params;
+            x = obj.x;
+            y = obj.y;
+            
             switch boundary
                 case {'w','W','west'}
-                    e_=obj.e_w;
-                    mat=obj.A;
-                    boundPos='l';
-                    Hi=obj.Hxi;
-                    [V,Vi,D,signVec]=obj.matrixDiag(mat,x(1),y);
-                    L=obj.evaluateCoefficientMatrix(L,x(1),y);
-                    side=max(length(y));
+                    e_ = obj.e_w;
+                    mat = obj.A;
+                    boundPos = 'l';
+                    Hi = obj.Hxi;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,x(1),y);
+                    L = obj.evaluateCoefficientMatrix(L,x(1),y);
+                    side = max(length(y));
                 case {'e','E','east'}
-                    e_=obj.e_e;
-                    mat=obj.A;
-                    boundPos='r';
-                    Hi=obj.Hxi;
-                    [V,Vi,D,signVec]=obj.matrixDiag(mat,x(end),y);
-                    L=obj.evaluateCoefficientMatrix(L,x(end),y);
-                    side=max(length(y));
+                    e_ = obj.e_e;
+                    mat = obj.A;
+                    boundPos = 'r';
+                    Hi = obj.Hxi;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,x(end),y);
+                    L = obj.evaluateCoefficientMatrix(L,x(end),y);
+                    side = max(length(y));
                 case {'s','S','south'}
-                    e_=obj.e_s;
-                    mat=obj.B;
-                    boundPos='l';
-                    Hi=obj.Hyi;
-                    [V,Vi,D,signVec]=obj.matrixDiag(mat,x,y(1));
-                    L=obj.evaluateCoefficientMatrix(L,x,y(1));
-                    side=max(length(x));
+                    e_ = obj.e_s;
+                    mat = obj.B;
+                    boundPos = 'l';
+                    Hi = obj.Hyi;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,x,y(1));
+                    L = obj.evaluateCoefficientMatrix(L,x,y(1));
+                    side = max(length(x));
                 case {'n','N','north'}
-                    e_=obj.e_n;
-                    mat=obj.B;
-                    boundPos='r';
-                    Hi=obj.Hyi;
-                    [V,Vi,D,signVec]=obj.matrixDiag(mat,x,y(end));
-                    L=obj.evaluateCoefficientMatrix(L,x,y(end));
-                    side=max(length(x));
+                    e_ = obj.e_n;
+                    mat = obj.B;
+                    boundPos = 'r';
+                    Hi = obj.Hyi;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,x,y(end));
+                    L = obj.evaluateCoefficientMatrix(L,x,y(end));      
+                    side = max(length(x));
             end
-
-            pos=signVec(1); zeroval=signVec(2); neg=signVec(3);
-
+            
+            pos = signVec(1);
+            zeroval = signVec(2);
+            neg = signVec(3);
+            
             switch boundPos
                 case {'l'}
-                    tau=sparse(obj.n*side,pos);
-                    Vi_plus=Vi(1:pos,:);
-                    Vi_minus=Vi(pos+zeroval+1:obj.n*side,:);
-                    V_plus=V(:,1:pos);
-                    V_minus=V(:,(pos+zeroval)+1:obj.n*side);
-
-                    tau(1:pos,:)=-abs(D(1:pos,1:pos));
-                    R=-inv(L*V_plus)*(L*V_minus);
-                    closure=Hi*e_*V*tau*(Vi_plus-R*Vi_minus)*e_';
-                    penalty=-Hi*e_*V*tau*inv(L*V_plus)*L;
+                    tau = sparse(obj.n*side,pos);
+                    Vi_plus = Vi(1:pos,:);
+                    Vi_minus = Vi(pos+zeroval+1:obj.n*side,:);
+                    V_plus = V(:,1:pos);
+                    V_minus = V(:,(pos+zeroval)+1:obj.n*side);
+                    
+                    tau(1:pos,:) = -abs(D(1:pos,1:pos));
+                    R = -inv(L*V_plus)*(L*V_minus);
+                    closure = Hi*e_*V*tau*(Vi_plus-R*Vi_minus)*e_';
+                    penalty = -Hi*e_*V*tau*inv(L*V_plus)*L;
                 case {'r'}
-                    tau=sparse(obj.n*side,neg);
-                    tau((pos+zeroval)+1:obj.n*side,:)=-abs(D((pos+zeroval)+1:obj.n*side,(pos+zeroval)+1:obj.n*side));
-                    Vi_plus=Vi(1:pos,:);
-                    Vi_minus=Vi((pos+zeroval)+1:obj.n*side,:);
-
-                    V_plus=V(:,1:pos);
-                    V_minus=V(:,(pos+zeroval)+1:obj.n*side);
-                    R=-inv(L*V_minus)*(L*V_plus);
-                    closure=Hi*e_*V*tau*(Vi_minus-R*Vi_plus)*e_';
-                    penalty=-Hi*e_*V*tau*inv(L*V_minus)*L;
+                    tau = sparse(obj.n*side,neg);
+                    tau((pos+zeroval)+1:obj.n*side,:) = -abs(D((pos+zeroval)+1:obj.n*side,(pos+zeroval)+1:obj.n*side));
+                    Vi_plus = Vi(1:pos,:);
+                    Vi_minus = Vi((pos+zeroval)+1:obj.n*side,:);
+                    
+                    V_plus = V(:,1:pos);
+                    V_minus = V(:,(pos+zeroval)+1:obj.n*side);
+                    R = -inv(L*V_minus)*(L*V_plus);
+                    closure = Hi*e_*V*tau*(Vi_minus-R*Vi_plus)*e_';
+                    penalty = -Hi*e_*V*tau*inv(L*V_minus)*L;
             end
         end
-
-
-        function [V,Vi, D,signVec]=matrixDiag(obj,mat,x,y)
-            params=obj.params;
+        
+        % Function that diagonalizes a symbolic matrix A as A=V*D*Vi
+        % D         is a diagonal matrix with the eigenvalues on A on the diagonal sorted by their sign
+        %                                    [d+       ]
+        %                               D =  [   d0    ]
+        %                                    [       d-]
+        % signVec   is a vector specifying the number of possitive, zero and negative eigenvalues of D   
+        function [V,Vi, D,signVec] = matrixDiag(obj,mat,x,y)
+            params = obj.params;
             syms xs ys
-            [V, D]=eig(mat(params,xs,ys));
-            Vi=inv(V);
-            xs=x; 
-            ys=y;
-           
-            side=max(length(x),length(y));
-            Dret=zeros(obj.n,side*obj.n);
-            Vret=zeros(obj.n,side*obj.n);
-            Viret=zeros(obj.n,side*obj.n);
-            for ii=1:obj.n
-                for jj=1:obj.n
-                    Dret(jj,(ii-1)*side+1:side*ii)=eval(D(jj,ii));
-                    Vret(jj,(ii-1)*side+1:side*ii)=eval(V(jj,ii));
-                    Viret(jj,(ii-1)*side+1:side*ii)=eval(Vi(jj,ii));
+            [V, D]= eig(mat(params,xs,ys));
+            Vi = inv(V);
+            xs = x;
+            ys = y;
+            
+            side = max(length(x),length(y));
+            Dret = zeros(obj.n,side*obj.n);
+            Vret = zeros(obj.n,side*obj.n);
+            Viret = zeros(obj.n,side*obj.n);
+            
+            for ii = 1:obj.n
+                for jj = 1:obj.n
+                    Dret(jj,(ii-1)*side+1:side*ii) = eval(D(jj,ii));
+                    Vret(jj,(ii-1)*side+1:side*ii) = eval(V(jj,ii));
+                    Viret(jj,(ii-1)*side+1:side*ii) = eval(Vi(jj,ii));
                 end
             end
-
-            D=sparse(Dret);
-            V=sparse(Vret); 
-            Vi=sparse(Viret);
-            V=obj.evaluateCoefficientMatrix(V,x,y);
-            Vi=obj.evaluateCoefficientMatrix(Vi,x,y);
-            D=obj.evaluateCoefficientMatrix(D,x,y);                       
-            DD=diag(D);
+            
+            D = sparse(Dret);
+            V = sparse(Vret);
+            Vi = sparse(Viret);
+            V = obj.evaluateCoefficientMatrix(V,x,y);
+            Vi = obj.evaluateCoefficientMatrix(Vi,x,y);
+            D = obj.evaluateCoefficientMatrix(D,x,y);
+            DD = diag(D);
             
-            poseig=(DD>0);
-            zeroeig=(DD==0);
-            negeig=(DD<0);
+            poseig = (DD>0);
+            zeroeig = (DD==0);
+            negeig = (DD<0);
             
-            D=diag([DD(poseig); DD(zeroeig); DD(negeig)]);
-            V=[V(:,poseig) V(:,zeroeig) V(:,negeig)];            
-            Vi=[Vi(poseig,:); Vi(zeroeig,:); Vi(negeig,:)];
-            signVec=[sum(poseig),sum(zeroeig),sum(negeig)];
+            D = diag([DD(poseig); DD(zeroeig); DD(negeig)]);
+            V = [V(:,poseig) V(:,zeroeig) V(:,negeig)];
+            Vi = [Vi(poseig,:); Vi(zeroeig,:); Vi(negeig,:)];
+            signVec = [sum(poseig),sum(zeroeig),sum(negeig)];
         end
-
+        
     end
 end
\ No newline at end of file