--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Wave2dCurve.m	Thu Sep 17 10:12:50 2015 +0200
@@ -0,0 +1,333 @@
+classdef Wave2dCurve < scheme.Scheme
+    properties
+        m % Number of points in each direction, possibly a vector
+        h % Grid spacing
+        u,v % Grid
+        x,y % Values of x and y for each grid point
+        X,Y % Grid point locations as matrices
+        order % Order accuracy for the approximation
+
+        D % non-stabalized scheme operator
+        M % Derivative norm
+        c
+        J, Ji
+        a11, a12, a22
+
+        H % Discrete norm
+        Hi
+        H_u, H_v % Norms in the x and y directions
+        Hu,Hv % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
+        Hi_u, Hi_v
+        Hiu, Hiv
+        e_w, e_e, e_s, e_n
+        du_w, dv_w
+        du_e, dv_e
+        du_s, dv_s
+        du_n, dv_n
+        gamm_u, gamm_v
+    end
+
+    methods
+        function obj = Wave2dCurve(m,ti,order,c,opSet)
+            default_arg('opSet',@sbp.Variable);
+
+            if length(m) == 1
+                m = [m m];
+            end
+
+            m_u = m(1);
+            m_v = m(2);
+            m_tot = m_u*m_v;
+
+            [u, h_u] = util.get_grid(0, 1, m_u);
+            [v, h_v] = util.get_grid(0, 1, m_v);
+
+
+            % Operators
+            ops_u = opSet(m_u,h_u,order);
+            ops_v = opSet(m_v,h_v,order);
+
+            I_u = speye(m_u);
+            I_v = speye(m_v);
+
+            D1_u = sparse(ops_u.derivatives.D1);
+            D2_u = ops_u.derivatives.D2;
+            H_u =  sparse(ops_u.norms.H);
+            Hi_u = sparse(ops_u.norms.HI);
+            % M_u =  sparse(ops_u.norms.M);
+            e_l_u = sparse(ops_u.boundary.e_1);
+            e_r_u = sparse(ops_u.boundary.e_m);
+            d1_l_u = sparse(ops_u.boundary.S_1);
+            d1_r_u = sparse(ops_u.boundary.S_m);
+
+            D1_v = sparse(ops_v.derivatives.D1);
+            D2_v = ops_v.derivatives.D2;
+            H_v =  sparse(ops_v.norms.H);
+            Hi_v = sparse(ops_v.norms.HI);
+            % M_v =  sparse(ops_v.norms.M);
+            e_l_v = sparse(ops_v.boundary.e_1);
+            e_r_v = sparse(ops_v.boundary.e_m);
+            d1_l_v = sparse(ops_v.boundary.S_1);
+            d1_r_v = sparse(ops_v.boundary.S_m);
+
+
+            % Metric derivatives
+            [X,Y] = ti.map(u,v);
+
+            [x_u,x_v] = gridDerivatives(X,D1_u,D1_v);
+            [y_u,y_v] = gridDerivatives(Y,D1_u,D1_v);
+
+
+
+            J = x_u.*y_v - x_v.*y_u;
+            a11 =  1./J .* (x_v.^2  + y_v.^2);  %% GÖR SOM MATRISER
+            a12 = -1./J .* (x_u.*x_v + y_u.*y_v);
+            a22 =  1./J .* (x_u.^2  + y_u.^2);
+            lambda = 1/2 * (a11 + a22 - sqrt((a11-a22).^2 + 4*a12.^2));
+
+            dof_order = reshape(1:m_u*m_v,m_v,m_u);
+
+            Duu = sparse(m_tot);
+            Dvv = sparse(m_tot);
+
+            for i = 1:m_v
+                D = D2_u(a11(i,:));
+                p = dof_order(i,:);
+                Duu(p,p) = D;
+            end
+
+            for i = 1:m_u
+                D = D2_v(a22(:,i));
+                p = dof_order(:,i);
+                Dvv(p,p) = D;
+            end
+
+            L_12 = spdiags(a12(:),0,m_tot,m_tot);
+            Du = kr(D1_u,I_v);
+            Dv = kr(I_u,D1_v);
+
+            Duv = Du*L_12*Dv;
+            Dvu = Dv*L_12*Du;
+
+
+
+            obj.H = kr(H_u,H_v);
+            obj.Hi = kr(Hi_u,Hi_v);
+            obj.Hu  = kr(H_u,I_v);
+            obj.Hv  = kr(I_u,H_v);
+            obj.Hiu = kr(Hi_u,I_v);
+            obj.Hiv = kr(I_u,Hi_v);
+
+            % obj.M = kr(M_u,H_v)+kr(H_u,M_v);
+            obj.e_w  = kr(e_l_u,I_v);
+            obj.e_e  = kr(e_r_u,I_v);
+            obj.e_s  = kr(I_u,e_l_v);
+            obj.e_n  = kr(I_u,e_r_v);
+            obj.du_w = kr(d1_l_u,I_v);
+            obj.dv_w = (obj.e_w'*Dv)';
+            obj.du_e = kr(d1_r_u,I_v);
+            obj.dv_e = (obj.e_e'*Dv)';
+            obj.du_s = (obj.e_s'*Du)';
+            obj.dv_s = kr(I_u,d1_l_v);
+            obj.du_n = (obj.e_n'*Du)';
+            obj.dv_n = kr(I_u,d1_r_v);
+
+            obj.m = m;
+            obj.h = [h_u h_v];
+            obj.order = order;
+
+
+            obj.c = c;
+            obj.J = spdiags(J(:),0,m_tot,m_tot);
+            obj.Ji = spdiags(1./J(:),0,m_tot,m_tot);
+            obj.a11 = a11;
+            obj.a12 = a12;
+            obj.a22 = a22;
+            obj.D = obj.Ji*c^2*(Duu + Duv + Dvu + Dvv);
+            obj.u = u;
+            obj.v = v;
+            obj.X = X;
+            obj.Y = Y;
+            obj.x = X(:);
+            obj.y = Y(:);
+
+            obj.gamm_u = h_u*ops_u.borrowing.M.S;
+            obj.gamm_v = h_v*ops_v.borrowing.M.S;
+        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.
+        %       neighbour_scheme    is an instance of Scheme that should be interfaced to.
+        %       neighbour_boundary  is a string specifying which boundary to interface to.
+        function [closure, penalty] = boundary_condition(obj,boundary,type,data)
+            default_arg('type','neumann');
+            default_arg('data',0);
+
+            [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv] = obj.get_boundary_ops(boundary);
+
+            switch type
+                % Dirichlet boundary condition
+                case {'D','d','dirichlet'}
+                    error('not implemented')
+                    alpha = obj.alpha;
+
+                    % tau1 < -alpha^2/gamma
+                    tuning = 1.1;
+                    tau1 = -tuning*alpha/gamm;
+                    tau2 =  s*alpha;
+
+                    p = tau1*e + tau2*d;
+
+                    closure = halfnorm_inv*p*e';
+
+                    pp = halfnorm_inv*p;
+                    switch class(data)
+                        case 'double'
+                            penalty = pp*data;
+                        case 'function_handle'
+                            penalty = @(t)pp*data(t);
+                        otherwise
+                            error('Weird data argument!')
+                    end
+
+
+                % Neumann boundary condition
+                case {'N','n','neumann'}
+                    c = obj.c;
+
+
+                    a_n = spdiags(coeff_n,0,length(coeff_n),length(coeff_n));
+                    a_t = spdiags(coeff_t,0,length(coeff_t),length(coeff_t));
+                    d = (a_n * d_n' + a_t*d_t')';
+
+                    tau1 = -s;
+                    tau2 = 0;
+                    tau = c.^2 * obj.Ji*(tau1*e + tau2*d);
+
+                    closure = halfnorm_inv*tau*d';
+
+                    pp = halfnorm_inv*tau;
+                    switch class(data)
+                        case 'double'
+                            penalty = pp*data;
+                        case 'function_handle'
+                            penalty = @(t)pp*data(t);
+                        otherwise
+                            error('Weird data argument!')
+                    end
+
+                % Unknown, boundary condition
+                otherwise
+                    error('No such boundary condition: type = %s',type);
+            end
+        end
+
+        function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
+            % u denotes the solution in the own domain
+            % v denotes the solution in the neighbour domain
+            [e_u, d_n_u, d_t_u, coeff_n_u, coeff_t_u, s_u, gamm_u, halfnorm_inv_u_n, halfnorm_inv_u_t, halfnorm_u_t] = obj.get_boundary_ops(boundary);
+            [e_v, d_n_v, d_t_v, coeff_n_v, coeff_t_v, s_v, gamm_v, halfnorm_inv_v_n, halfnorm_inv_v_t, halfnorm_v_t] = neighbour_scheme.get_boundary_ops(boundary);
+
+            F_u = s_u * a_n_u * d_n_u' + s_u * a_t_u*d_t_u';
+            F_v = s_v * a_n_v * d_n_v' + s_v * a_t_v*d_t_v';
+
+            tau  = 111111111111111111111111111111;
+            sig1 = 1/2;
+            sig2 = -1/2;
+
+            penalty_parameter_1 = s_u*halfnorm_inv_u_n*(tau + sig1*halfnorm_inv_u_t*F_u'*halfnorm_u_t)*e_u;
+            penalty_parameter_2 = halfnorm_inv_u_n * sig2 * e_u;
+
+            tuning = 1.2;
+
+            alpha_u = obj.alpha;
+            alpha_v = neighbour_scheme.alpha;
+
+            % tau1 < -(alpha_u/gamm_u + alpha_v/gamm_v)
+
+            tau1 = -(alpha_u/gamm_u + alpha_v/gamm_v) * tuning;
+            tau2 = s_u*1/2*alpha_u;
+            sig1 = s_u*(-1/2);
+            sig2 = 0;
+
+            tau = tau1*e_u + tau2*d_u;
+            sig = sig1*e_u + sig2*d_u;
+
+            closure = halfnorm_inv*( tau*e_u' + sig*alpha_u*d_u');
+            penalty = halfnorm_inv*(-tau*e_v' - sig*alpha_v*d_v');
+        end
+
+        % Ruturns the boundary ops and sign for the boundary specified by the string boundary.
+        % The right boundary is considered the positive boundary
+        function [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv_n, halfnorm_inv_t, halfnorm_t] = get_boundary_ops(obj,boundary)
+            switch boundary
+                case 'w'
+                    e = obj.e_w;
+                    d_n = obj.du_w;
+                    d_t = obj.dv_w;
+                    s = -1;
+
+                    coeff_n = obj.a11(:,1);
+                    coeff_t = obj.a12(:,1);
+                case 'e'
+                    e = obj.e_e;
+                    d_n = obj.du_e;
+                    d_t = obj.dv_e;
+                    s = 1;
+
+                    coeff_n = obj.a11(:,end);
+                    coeff_t = obj.a12(:,end);
+                case 's'
+                    e = obj.e_s;
+                    d_n = obj.dv_s;
+                    d_t = obj.du_s;
+                    s = -1;
+
+                    coeff_n = obj.a22(1,:)';
+                    coeff_t = obj.a12(1,:)';
+                case 'n'
+                    e = obj.e_n;
+                    d_n = obj.dv_n;
+                    d_t = obj.du_n;
+                    s = 1;
+
+                    coeff_n = obj.a22(end,:)';
+                    coeff_t = obj.a12(end,:)';
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+
+            switch boundary
+                case {'w','e'}
+                    halfnorm_inv_n = obj.Hiu;
+                    halfnorm_inv_t = obj.Hiv;
+                    halfnorm_t = obj.Hv;
+                    gamm = obj.gamm_u;
+                case {'s','n'}
+                    halfnorm_inv_n = obj.Hiv;
+                    halfnorm_inv_t = obj.Hiu;
+                    halfnorm_t = obj.Hu;
+                    gamm = obj.gamm_v;
+            end
+        end
+
+        function N = size(obj)
+            N = prod(obj.m);
+        end
+
+    end
+
+    methods(Static)
+        % Calculates the matrcis need for the inteface coupling between boundary bound_u of scheme schm_u
+        % and bound_v of scheme schm_v.
+        %   [uu, uv, vv, vu] = inteface_couplong(A,'r',B,'l')
+        function [uu, uv, vv, vu] = interface_coupling(schm_u,bound_u,schm_v,bound_v)
+            [uu,uv] = schm_u.interface(bound_u,schm_v,bound_v);
+            [vv,vu] = schm_v.interface(bound_v,schm_u,bound_u);
+        end
+    end
+end
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