--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Laplace1d.m	Tue Feb 12 17:12:42 2019 +0100
@@ -0,0 +1,158 @@
+classdef Laplace1d < scheme.Scheme
+    properties
+        grid
+        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(grid, order, a)
+            default_arg('a', 1);
+
+            assertType(grid, 'grid.Cartesian');
+
+            ops = sbp.D2Standard(grid.size(), grid.lim{1}, order);
+
+            obj.D2 = sparse(ops.D2);
+            obj.H =  sparse(ops.H);
+            obj.Hi = sparse(ops.HI);
+            obj.M =  sparse(ops.M);
+            obj.e_l = sparse(ops.e_l);
+            obj.e_r = sparse(ops.e_r);
+            obj.d_l = -sparse(ops.d1_l);
+            obj.d_r = sparse(ops.d1_r);
+
+
+            obj.grid = grid;
+            obj.order = order;
+
+            obj.a = a;
+            obj.D = a*obj.D2;
+
+            obj.gamm = grid.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 = obj.getBoundaryOperator('e', boundary);
+            d = obj.getBoundaryOperator('d', boundary);
+            s = obj.getBoundarySign(boundary);
+
+            switch type
+                % Dirichlet boundary condition
+                case {'D','dirichlet'}
+                    tuning = 1.1;
+                    tau1 = -tuning/obj.gamm;
+                    tau2 =  1;
+
+                    tau = tau1*e + tau2*d;
+
+                    closure = obj.a*obj.Hi*tau*e';
+                    penalty = obj.a*obj.Hi*tau;
+
+                % Neumann boundary condition
+                case {'N','neumann'}
+                    tau = -e;
+
+                    closure = obj.a*obj.Hi*tau*d';
+                    penalty = -obj.a*obj.Hi*tau;
+
+                % Unknown, boundary condition
+                otherwise
+                    error('No such boundary condition: type = %s',type);
+            end
+        end
+
+        function [closure, penalty] = interface(obj, boundary, neighbour_scheme, neighbour_boundary, type)
+            % u denotes the solution in the own domain
+            % v denotes the solution in the neighbour domain
+            e_u = obj.getBoundaryOperator('e', boundary);
+            d_u = obj.getBoundaryOperator('d', boundary);
+            s_u = obj.getBoundarySign(boundary);
+
+            e_v = neighbour_scheme.getBoundaryOperator('e', neighbour_boundary);
+            d_v = neighbour_scheme.getBoundaryOperator('d', neighbour_boundary);
+            s_v = neighbour_scheme.getBoundarySign(neighbour_boundary);
+
+            a_u = obj.a;
+            a_v = neighbour_scheme.a;
+
+            gamm_u = obj.gamm;
+            gamm_v = neighbour_scheme.gamm;
+
+            tuning = 1.1;
+
+            tau1 = -(a_u/gamm_u + a_v/gamm_v) * tuning;
+            tau2 = 1/2*a_u;
+            sig1 = -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*a_u*d_u');
+            penalty = obj.Hi*(-tau*e_v' + sig*a_v*d_v');
+        end
+
+        % Returns the boundary operator op for the boundary specified by the string boundary.
+        % op        -- string
+        % boundary  -- string
+        function o = getBoundaryOperator(obj, op, boundary)
+            assertIsMember(op, {'e', 'd'})
+            assertIsMember(boundary, {'l', 'r'})
+
+            o = obj.([op, '_', boundary])
+        end
+
+        % Returns square boundary quadrature matrix, of dimension
+        % corresponding to the number of boundary points
+        %
+        % boundary -- string
+        % Note: for 1d diffOps, the boundary quadrature is the scalar 1.
+        function H_b = getBoundaryQuadrature(obj, boundary)
+            assertIsMember(boundary, {'l', 'r'})
+
+            H_b = 1;
+        end
+
+        % Returns the boundary sign. The right boundary is considered the positive boundary
+        % boundary -- string
+        function s = getBoundarySign(obj, boundary)
+            assertIsMember(boundary, {'l', 'r'})
+
+            switch boundary
+                case {'r'}
+                    s = 1;
+                case {'l'}
+                    s = -1;
+            end
+        end
+
+        function N = size(obj)
+            N = obj.grid.size();
+        end
+
+    end
+end