--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Laplace1D.m	Thu Nov 22 07:58:11 2018 +0100
@@ -0,0 +1,171 @@
+classdef Laplace1D < scheme.Scheme
+    properties
+        g
+        order % Order accuracy for the approximation
+
+        D % non-stabalized scheme operator
+        H % Discrete norm
+        M % Derivative norm
+        a
+
+        D2
+        Hi
+        e_l
+        e_r
+        d_l
+        d_r
+        gamm
+    end
+
+    methods
+        function obj = Laplace1D(g, order, a)
+            default_arg('a', 1);
+
+            assertType(g, 'grid.Cartesian');
+
+            ops = sbp.Ordinary(g.size(), g.h, order);
+
+            obj.D2 = sparse(ops.derivatives.D2);
+            obj.H =  sparse(ops.norms.H);
+            obj.Hi = sparse(ops.norms.HI);
+            obj.M =  sparse(ops.norms.M);
+            obj.e_l = sparse(ops.boundary.e_1);
+            obj.e_r = sparse(ops.boundary.e_m);
+            obj.d_l = sparse(ops.boundary.S_1);
+            obj.d_r = sparse(ops.boundary.S_m);
+
+
+            obj.g = g;
+            obj.order = order;
+
+            obj.a = a;
+            obj.D = a*obj.D2;
+
+            obj.gamm = h*ops.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,s] = obj.get_boundary_ops(boundary);
+
+            switch type
+                % Dirichlet boundary condition
+                case {'D','dirichlet'}
+                    alpha = obj.alpha;
+
+                    % tau1 < -alpha^2/gamma
+                    tuning = 1.1;
+                    tau1 = -tuning*alpha/obj.gamm;
+                    tau2 =  s*alpha;
+
+                    p = tau1*e + tau2*d;
+
+                    closure = obj.Hi*p*e';
+
+                    pp = obj.Hi*p;
+                    switch class(data)
+                        case 'double'
+                            penalty = pp*data;
+                        case 'function_handle'
+                            penalty = @(t)pp*data(t);
+                        otherwise
+                            error('Wierd data argument!')
+                    end
+
+
+                % Neumann boundary condition
+                case {'N','neumann'}
+                    alpha = obj.alpha;
+                    tau1 = -s*alpha;
+                    tau2 = 0;
+                    tau = tau1*e + tau2*d;
+
+                    closure = obj.Hi*tau*d';
+
+                    pp = obj.Hi*tau;
+                    switch class(data)
+                        case 'double'
+                            penalty = pp*data;
+                        case 'function_handle'
+                            penalty = @(t)pp*data(t);
+                        otherwise
+                            error('Wierd 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_u,s_u] = obj.get_boundary_ops(boundary);
+            [e_v,d_v,s_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary);
+
+            tuning = 1.1;
+
+            alpha_u = obj.alpha;
+            alpha_v = neighbour_scheme.alpha;
+
+            gamm_u = obj.gamm;
+            gamm_v = neighbour_scheme.gamm;
+
+            % 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 = obj.Hi*( tau*e_u' + sig*alpha_u*d_u');
+            penalty = obj.Hi*(-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,s] = get_boundary_ops(obj,boundary)
+            switch boundary
+                case 'l'
+                    e = obj.e_l;
+                    d = obj.d_l;
+                    s = -1;
+                case 'r'
+                    e = obj.e_r;
+                    d = obj.d_r;
+                    s = 1;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+        end
+
+        function N = size(obj)
+            N = 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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