--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Hypsyst2dCurve.m	Thu Jan 26 13:07:51 2017 +0000
@@ -0,0 +1,378 @@
+classdef Hypsyst2dCurve < scheme.Scheme
+    properties
+        m % Number of points in each direction, possibly a vector
+        n % size of system
+        h % Grid spacing
+        X,Y % Values of x and y for each grid point
+        
+        J, Ji % Jacobaian and inverse Jacobian
+        xi,eta
+        Xi,Eta
+        
+        A,B
+        X_eta, Y_eta
+        X_xi,Y_xi
+        order % Order accuracy for the approximation
+        
+        D % non-stabalized scheme operator
+        Ahat, Bhat, E
+        
+        H % Discrete norm
+        Hxii,Hetai % Kroneckerd norms in xi and eta.
+        I_xi,I_eta, I_N, onesN
+        e_w, e_e, e_s, e_n
+        index_w, index_e,index_s,index_n
+        params % Parameters for the coeficient matrice
+    end
+    
+    
+    methods
+        % Solving Hyperbolic systems on the form u_t=-Au_x-Bu_y-Eu
+        function obj = Hypsyst2dCurve(m, order, A, B, E, params, ti)
+            default_arg('E', [])
+            xilim = {0 1};
+            etalim = {0 1};
+            
+            if length(m) == 1
+                m = [m m];
+            end
+            obj.params = params;
+            obj.A=A;
+            obj.B=B;
+            
+            obj.Ahat=@(params,x,y,x_eta,y_eta)(A(params,x,y).*y_eta-B(params,x,y).*x_eta);
+            obj.Bhat=@(params,x,y,x_xi,y_xi)(B(params,x,y).*x_xi-A(params,x,y).*y_xi);
+            obj.E=@(params,x,y,~,~)E(params,x,y);
+            
+            m_xi = m(1);
+            m_eta = m(2);
+            m_tot=m_xi*m_eta;
+            
+            ops_xi = sbp.D2Standard(m_xi,xilim,order);
+            ops_eta = sbp.D2Standard(m_eta,etalim,order);
+            
+            obj.xi = ops_xi.x;
+            obj.eta = ops_eta.x;
+            
+            obj.Xi = kr(obj.xi,ones(m_eta,1));
+            obj.Eta = kr(ones(m_xi,1),obj.eta);
+            
+            obj.n = length(A(obj.params,0,0));
+            obj.onesN=ones(obj.n);
+            
+            obj.index_w=1:m_eta;
+            obj.index_e=(m_tot-m_e        
+        
+        metric_termsta+1):m_tot;
+            obj.index_s=1:m_eta:(m_tot-m_eta+1);
+            obj.index_n=(m_eta):m_eta:m_tot;
+            
+            I_n = eye(obj.n);
+            I_xi = speye(m_xi);
+            obj.I_xi = I_xi;
+            I_eta = speye(m_eta);
+            obj.I_eta = I_eta;
+            
+            D1_xi = kr(I_n, ops_xi.D1, I_eta);
+            obj.Hxii = kr(I_n, ops_xi.HI, I_eta);
+            D1_eta = kr(I_n, I_xi, ops_eta.D1);
+            obj.Hetai = kr(I_n, I_xi, ops_eta.HI);
+            
+            obj.e_w = kr(I_n, ops_xi.e_l, I_eta);
+            obj.e_e = kr(I_n, ops_xi.e_r, I_eta);
+            obj.e_s = kr(I_n, I_xi, ops_eta.e_l);
+            obj.e_n = kr(I_n, I_xi,         
+        
+        metric_termsops_eta.e_r);
+            
+            [X,Y] = ti.map(obj.xi,obj.eta);
+            
+            [x_xi,x_eta] = gridDerivatives(X,ops_xi.D1,ops_eta.D1);
+            [y_xi,y_eta] = gridDerivatives(Y,ops_xi.D1, ops_eta.D1);
+            
+            obj.X = reshape(X,m_tot,1);
+            obj.Y = reshape(Y,m_tot,1);
+            obj.X_xi = reshape(x_xi,m_tot,1);
+            obj.Y_xi = reshape(y_xi,m_tot,1);
+            obj.X_eta = reshape(x_eta,m_tot,1);
+            obj.Y_eta = reshape(y_eta,m_tot,1);
+            
+            Ahat_evaluated = obj.evaluateCoefficientMatrix(obj.Ahat, obj.X, obj.Y,obj.X_eta,obj.Y_eta);
+            Bhat_evaluated = obj.evaluateCoefficientMatrix(obj.Bhat, obj.X, obj.Y,obj.X_xi,obj.Y_xi);
+            E_evaluated = obj.evaluateCoefficientMatrix(obj.E, obj.X, obj.Y,[],[]);
+            
+            obj.m = m;
+            obj.h = [ops_xi.h ops_eta.h];
+            obj.order = order;
+            obj.J = obj.X_xi.*obj.Y_eta - obj.X_eta.*obj.Y_xi;
+            obj.Ji = kr(I_n,spdiags(1./obj.J,0,m_tot,m_tot));
+            
+            obj.D = obj.Ji*(-Ahat_evaluated*D1_xi-Bhat_evaluated*D1_eta)-E_evaluated;
+            
+        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',General boundary conditions'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_boundaryGeneral boundary conditions)
+            error('An interface function does not exist yet');
+        end
+        
+        function N = size(obj)
+            N = obj.m;
+        end
+        
+        function [ret] = evaluateCoefficientMatrix(obj, mat, X, Y,x_,y_)
+            params = obj.params;
+            
+            if isa(mat,'function_handle')
+                [rows,cols] = size(mat(params,0,0,0,0));
+                x_ = kr(obj.onesN,x_);
+                y_ = kr(obj.onesN,y_);
+                matVec = mat(params,X',Y',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;
+            xi = obj.xi;
+            eta = obj.eta;
+            
+            switch boundary
+                case {'w','W','west'}
+                    e_ = obj.e_w;
+                    mat = obj.Ahat;
+                    boundPos = 'l';
+                    Hi = obj.Hxii;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,X(obj.index_w),Y(obj.index_w),obj.X_eta(obj.index_w),obj.Y_eta(obj.index_w));
+                    side = max(length(eta));
+                case {'e','E','east'}
+                    e_ = obj.e_e;
+                    mat = obj.Ahat;
+                    boundPos = 'r';
+                    Hi = obj.Hxii;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,X(obj.index_e),Y(obj.index_e),obj.X_eta(obj.index_e),obj.Y_eta(obj.index_e));
+                    side = max(length(eta));
+                case {'s','S','south'}
+                    e_ = obj.e_s;
+                    mat = obj.Bhat;
+                    boundPos = 'l';
+                    Hi = obj.Hetai;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,X(obj.index_s),Y(obj.index_s),obj.X_xi(obj.index_s),obj.Y_xi(obj.index_s));
+                    side = max(length(xi));
+                case {'n','N','north'}
+                    e_ = obj.e_n;
+                    mat = obj.Bhat;
+                    boundPos = 'r';
+                    Hi = obj.Hetai;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,X(obj.index_n),Y(obj.index_n),obj.X_xi(obj.index_n),obj.Y_xi(obj.index_n));
+                    side = max(length(xi));
+            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;
+            xi = obj.xi;
+            eta = obj.eta;
+            
+            switch boundary
+                case {'w','W','west'}
+                    e_ = obj.e_w;
+                    mat = obj.Ahat;
+                    boundPos = 'l';
+                    Hi = obj.Hxii;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,X(obj.index_w),Y(obj.index_w),obj.X_eta(obj.index_w),obj.Y_eta(obj.index_w));
+                    
+                    Ji_vec = diag(obj.Ji);
+                    Ji = diag(Ji_vec(obj.index_w));
+                    xi_x = Ji*obj.Y_eta(obj.index_w);
+                    xi_y = -Ji*obj.X_eta(obj.index_w);
+                    L = obj.evaluateCoefficientMatrix(L,xi_x,xi_y,[],[]);
+                    side = max(length(eta));
+                case {'e','E','east'}
+                    e_ = obj.e_e;
+                    mat = obj.Ahat;
+                    boundPos = 'r';
+                    Hi = obj.Hxii;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,X(obj.index_e),Y(obj.index_e),obj.X_eta(obj.index_e),obj.Y_eta(obj.index_e));
+                    
+                    Ji_vec = diag(obj.Ji);
+                    Ji = diag(Ji_vec(obj.index_e));
+                    xi_x = Ji*obj.Y_eta(obj.index_e);
+                    xi_y = -Ji*obj.X_eta(obj.index_e);
+                    L = obj.evaluateCoefficientMatrix(L,-xi_x,-xi_y,[],[]);
+                    side = max(length(eta));
+                case {'s','S','south'}
+                    e_ = obj.e_s;
+                    mat = obj.Bhat;
+                    boundPos = 'l';
+                    Hi = obj.Hetai;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,X(obj.index_s),Y(obj.index_s),obj.X_xi(obj.index_s),obj.Y_xi(obj.index_s));
+                    
+                    Ji_vec = diag(obj.Ji);
+                    Ji = diag(Ji_vec(obj.index_s));
+                    eta_x = Ji*obj.Y_xi(obj.index_s);
+                    eta_y = -Ji*obj.X_xi(obj.index_s);
+                    L = obj.evaluateCoefficientMatrix(L,eta_x,eta_y,[],[]);
+                    side = max(length(xi));
+                case {'n','N','north'}
+                    e_ = obj.e_n;
+                    mat = obj.Bhat;
+                    boundPos = 'r';
+                    Hi = obj.Hetai;
+                    [V,Vi,D,signVec] = obj.matrixDiag(mat,X(obj.index_n),Y(obj.index_n),obj.X_xi(obj.index_n),obj.Y_xi(obj.index_n));
+                    
+                    Ji_vec = diag(obj.Ji);
+                    Ji = diag(Ji_vec(obj.index_n));
+                    eta_x = Ji*obj.Y_xi(obj.index_n);
+                    eta_y = -Ji*obj.X_xi(obj.index_n);
+                    L = obj.evaluateCoefficientMatrix(L,-eta_x,-eta_y,[],[]);
+                    side = max(length(xi));
+            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+1:obj.n*side,:);
+                    V_plus = V(:,1:pos);
+                    V_minus = V(:,(pos)+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,x_,y_)
+            params = obj.params;
+            syms xs ys
+            if(sum(abs(x_)) ~= 0)
+                syms xs_
+            else
+                xs_ = 0;
+            end
+            
+            if(sum(abs(y_))~= 0)
+                syms ys_;
+            else
+                ys_ = 0;
+            end
+            
+            [V, D] = eig(mat(params,xs,ys,xs_,ys_));
+            Vi = inv(V);
+            syms xs ys xs_ ys_
+            
+            xs = x;
+            ys = y;
+            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,x_,y_);
+            D = obj.evaluateCoefficientMatrix(D,x,y,x_,y_);
+            Vi = obj.evaluateCoefficientMatrix(Vi,x,y,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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