--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Utux2D.m	Sat Jun 16 14:30:45 2018 -0700
@@ -0,0 +1,286 @@
+classdef Utux2D < scheme.Scheme
+   properties
+        m % Number of points in each direction, possibly a vector
+        h % Grid spacing
+        grid % Grid
+        order % Order accuracy for the approximation
+        v0 % Initial data
+        
+        a % Wave speed a = [a1, a2];
+          % Can either be a constant vector or a cell array of function handles.
+
+        H % Discrete norm
+        H_x, H_y % Norms in the x and y directions
+        Hi, Hx, Hy, Hxi, Hyi % Kroneckered norms
+
+        % Derivatives
+        Dx, Dy
+        
+        % Boundary operators
+        e_w, e_e, e_s, e_n
+        
+        D % Total discrete operator
+
+        % String, type of interface coupling
+        % Default: 'upwind'
+        % Other:   'centered'
+        coupling_type 
+
+        % String, type of interpolation operators
+        % Default: 'AWW' (Almquist Wang Werpers)
+        % Other:   'MC' (Mattsson Carpenter)
+        interpolation_type
+
+        
+        % Cell array, damping on upwstream and downstream sides.
+        interpolation_damping
+
+    end
+
+
+    methods 
+         function obj = Utux2D(g ,order, opSet, a, coupling_type, interpolation_type, interpolation_damping)
+            
+            default_arg('interpolation_damping',{0,0});
+            default_arg('interpolation_type','AWW'); 
+            default_arg('coupling_type','upwind'); 
+            default_arg('a',1/sqrt(2)*[1, 1]); 
+            default_arg('opSet',@sbp.D2Standard);
+
+            assert(isa(g, 'grid.Cartesian'))
+            if iscell(a)
+                a1 = grid.evalOn(g, a{1});
+                a2 = grid.evalOn(g, a{2});
+                a = {spdiag(a1), spdiag(a2)};
+            else
+                a = {a(1), a(2)};
+            end
+             
+            m = g.size();
+            m_x = m(1);
+            m_y = m(2);
+            m_tot = g.N();
+
+            xlim = {g.x{1}(1), g.x{1}(end)};
+            ylim = {g.x{2}(1), g.x{2}(end)};
+            obj.grid = g;
+
+            % Operator sets
+            ops_x = opSet(m_x, xlim, order);
+            ops_y = opSet(m_y, ylim, order);
+            Ix = speye(m_x);
+            Iy = speye(m_y);
+            
+            % Norms
+            Hx = ops_x.H;
+            Hy = ops_y.H;
+            Hxi = ops_x.HI;
+            Hyi = ops_y.HI;
+            
+            obj.H_x = Hx;
+            obj.H_y = Hy;
+            obj.H = kron(Hx,Hy);
+            obj.Hi = kron(Hxi,Hyi);
+            obj.Hx = kron(Hx,Iy);
+            obj.Hy = kron(Ix,Hy);
+            obj.Hxi = kron(Hxi,Iy);
+            obj.Hyi = kron(Ix,Hyi);
+            
+            % Derivatives
+            Dx = ops_x.D1;
+            Dy = ops_y.D1;
+            obj.Dx = kron(Dx,Iy);
+            obj.Dy = kron(Ix,Dy);
+           
+            % Boundary operators
+            obj.e_w = kr(ops_x.e_l, Iy);
+            obj.e_e = kr(ops_x.e_r, Iy);
+            obj.e_s = kr(Ix, ops_y.e_l);
+            obj.e_n = kr(Ix, ops_y.e_r);
+
+            obj.m = m;
+            obj.h = [ops_x.h ops_y.h];
+            obj.order = order;
+            obj.a = a;
+            obj.coupling_type = coupling_type;
+            obj.interpolation_type = interpolation_type;
+            obj.interpolation_damping = interpolation_damping;
+            obj.D = -(a{1}*obj.Dx + a{2}*obj.Dy);
+
+        end
+        % Closure functions return the opertors applied to the own domain 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)
+            default_arg('type','dirichlet');
+            
+            sigma = -1; % Scalar penalty parameter
+            switch boundary
+                case {'w','W','west','West'}
+                    tau = sigma*obj.a{1}*obj.e_w*obj.H_y;
+                    closure = obj.Hi*tau*obj.e_w';
+                    
+                case {'s','S','south','South'}
+                    tau = sigma*obj.a{2}*obj.e_s*obj.H_x;
+                    closure = obj.Hi*tau*obj.e_s';
+            end  
+            penalty = -obj.Hi*tau;
+                
+         end
+          
+         function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
+             
+             % Get neighbour boundary operator
+             switch neighbour_boundary
+                 case {'e','E','east','East'}
+                     e_neighbour = neighbour_scheme.e_e;
+                     m_neighbour = neighbour_scheme.m(2);
+                 case {'w','W','west','West'}
+                     e_neighbour = neighbour_scheme.e_w;
+                     m_neighbour = neighbour_scheme.m(2);
+                 case {'n','N','north','North'}
+                     e_neighbour = neighbour_scheme.e_n;
+                     m_neighbour = neighbour_scheme.m(1);
+                 case {'s','S','south','South'}
+                     e_neighbour = neighbour_scheme.e_s;
+                     m_neighbour = neighbour_scheme.m(1);
+             end
+             
+             switch obj.coupling_type
+             
+             % Upwind coupling (energy dissipation)
+             case 'upwind'
+                 sigma_ds = -1; %"Downstream" penalty
+                 sigma_us = 0; %"Upstream" penalty
+
+             % Energy-preserving coupling (no energy dissipation)
+             case 'centered'
+                 sigma_ds = -1/2; %"Downstream" penalty
+                 sigma_us = 1/2; %"Upstream" penalty
+
+             otherwise
+                error(['Interface coupling type ' coupling_type ' is not available.'])
+             end
+
+             % Check grid ratio for interpolation
+             switch boundary
+                 case {'w','W','west','West','e','E','east','East'}
+                     m = obj.m(2);       
+                 case {'s','S','south','South','n','N','north','North'}
+                     m = obj.m(1);
+             end
+             grid_ratio = m/m_neighbour;
+             if grid_ratio ~= 1
+
+                [ms, index] = sort([m, m_neighbour]);
+                orders = [obj.order, neighbour_scheme.order];
+                orders = orders(index);
+
+                switch obj.interpolation_type
+                case 'MC'
+                    interpOpSet = sbp.InterpMC(ms(1),ms(2),orders(1),orders(2));
+                    if grid_ratio < 1
+                        I_neighbour2local_us = interpOpSet.IF2C;
+                        I_neighbour2local_ds = interpOpSet.IF2C;
+                        I_local2neighbour_us = interpOpSet.IC2F;
+                        I_local2neighbour_ds = interpOpSet.IC2F;
+                    elseif grid_ratio > 1
+                        I_neighbour2local_us = interpOpSet.IC2F;
+                        I_neighbour2local_ds = interpOpSet.IC2F;
+                        I_local2neighbour_us = interpOpSet.IF2C;
+                        I_local2neighbour_ds = interpOpSet.IF2C;
+                    end
+                case 'AWW'
+                    %String 'C2F' indicates that ICF2 is more accurate.
+                    interpOpSetF2C = sbp.InterpAWW(ms(1),ms(2),orders(1),orders(2),'F2C');
+                    interpOpSetC2F = sbp.InterpAWW(ms(1),ms(2),orders(1),orders(2),'C2F'); 
+                    if grid_ratio < 1 
+                        % Local is coarser than neighbour
+                        I_neighbour2local_us = interpOpSetC2F.IF2C;
+                        I_neighbour2local_ds = interpOpSetF2C.IF2C;
+                        I_local2neighbour_us = interpOpSetC2F.IC2F;
+                        I_local2neighbour_ds = interpOpSetF2C.IC2F;
+                    elseif grid_ratio > 1
+                        % Local is finer than neighbour 
+                        I_neighbour2local_us = interpOpSetF2C.IC2F;
+                        I_neighbour2local_ds = interpOpSetC2F.IC2F;
+                        I_local2neighbour_us = interpOpSetF2C.IF2C;
+                        I_local2neighbour_ds = interpOpSetC2F.IF2C;
+                    end
+                otherwise
+                    error(['Interpolation type ' obj.interpolation_type ...
+                         ' is not available.' ]);
+                end
+
+             else 
+                % No interpolation required
+                I_neighbour2local_us = speye(m,m);
+                I_neighbour2local_ds = speye(m,m);
+            end    
+             
+             int_damp_us = obj.interpolation_damping{1};
+             int_damp_ds = obj.interpolation_damping{2};
+
+             I = speye(m,m);
+             I_back_forth_us = I_neighbour2local_us*I_local2neighbour_us;
+             I_back_forth_ds = I_neighbour2local_ds*I_local2neighbour_ds;
+
+
+             switch boundary
+                 case {'w','W','west','West'}
+                     tau = sigma_ds*obj.a{1}*obj.e_w*obj.H_y;
+                     closure = obj.Hi*tau*obj.e_w';
+                     penalty = -obj.Hi*tau*I_neighbour2local_ds*e_neighbour';
+
+                     beta = int_damp_ds*obj.a{1}...
+                            *obj.e_w*obj.H_y;
+                     closure = closure + obj.Hi*beta*(I_back_forth_ds - I)*obj.e_w';     
+                 case {'e','E','east','East'}
+                     tau = sigma_us*obj.a{1}*obj.e_e*obj.H_y;
+                     closure = obj.Hi*tau*obj.e_e';
+                     penalty = -obj.Hi*tau*I_neighbour2local_us*e_neighbour';
+
+                     beta = int_damp_us*obj.a{1}...
+                            *obj.e_e*obj.H_y;
+                     closure = closure + obj.Hi*beta*(I_back_forth_us - I)*obj.e_e'; 
+                 case {'s','S','south','South'}
+                     tau = sigma_ds*obj.a{2}*obj.e_s*obj.H_x;
+                     closure = obj.Hi*tau*obj.e_s'; 
+                     penalty = -obj.Hi*tau*I_neighbour2local_ds*e_neighbour';
+
+                     beta = int_damp_ds*obj.a{2}...
+                            *obj.e_s*obj.H_x;
+                     closure = closure + obj.Hi*beta*(I_back_forth_ds - I)*obj.e_s';
+                 case {'n','N','north','North'}
+                     tau = sigma_us*obj.a{2}*obj.e_n*obj.H_x;
+                     closure = obj.Hi*tau*obj.e_n';
+                     penalty = -obj.Hi*tau*I_neighbour2local_us*e_neighbour';
+
+                     beta = int_damp_us*obj.a{2}...
+                            *obj.e_n*obj.H_x;
+                     closure = closure + obj.Hi*beta*(I_back_forth_us - I)*obj.e_n'; 
+             end
+             
+                 
+         end
+      
+        function N = size(obj)
+            N = obj.m;
+        end
+
+    end
+
+    methods(Static)
+        % Calculates the matrices needed for the inteface coupling between boundary bound_u of scheme schm_u
+        % and bound_v of scheme schm_v.
+        %   [uu, uv, vv, vu] = inteface_coupling(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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