--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Hypsyst2d.m	Tue Feb 07 15:47:51 2017 +0100
@@ -0,0 +1,301 @@
+classdef Hypsyst2d < scheme.Scheme
+    properties
+        m % Number of points in each direction, possibly a vector
+        n %size of system
+        h % Grid spacing
+        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 %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.
+        I_x,I_y, I_N
+        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
+            
+            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);
+            
+            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;
+            
+        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'.
+        %       type                is a string specifying the type of boundary condition if there are several.
+        %       data                is a function returning the data that should be applied at the boundary.
+        function [closure, penalty] = boundary_condition(obj,boundary,type,L)
+            default_arg('type','char');
+            switch type
+                case{'c','char'}
+                    [closure,penalty] = boundary_condition_char(obj,boundary);
+                case{'general'}
+                    [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;
+            
+            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));
+            else
+                matVec = mat;
+                [rows,cols] = size(matVec);
+                side = max(length(X),length(Y));
+                cols = cols/side;
+            end
+            ret = cell(rows,cols);
+            
+            for ii = 1:rows
+                for jj=1:cols
+                    ret{ii,jj} = diag(matVec(ii,(jj-1)*side+1:jj*side));
+                end
+            end
+            ret = cell2mat(ret);
+        end
+        
+        %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));
+                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));
+                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));
+                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));
+            end
+            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;
+                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;
+            end
+        end
+        
+        % 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));
+                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));
+                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));
+                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));
+            end
+            
+            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;
+                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;
+            end
+        end
+        
+        % 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));
+                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);
+            
+            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)];
+        end
+        
+    end
+end
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