diff -r e8fc3aa1faf6 -r eb2f9233acc3 +scheme/Heat2dVariable.m
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/+scheme/Heat2dVariable.m	Mon Feb 12 13:39:39 2018 -0800
@@ -0,0 +1,262 @@
+classdef Heat2dVariable < scheme.Scheme
+
+% Discretizes the Laplacian with variable coefficent,
+% In the Heat equation way (i.e., the discretization matrix is not necessarily 
+% symmetric)
+% u_t = div * (kappa * grad u ) 
+% opSet should be cell array of opSets, one per dimension. This
+% is useful if we have periodic BC in one direction.
+
+    properties
+        m % Number of points in each direction, possibly a vector
+        h % Grid spacing
+
+        grid
+        dim
+
+        order % Order of accuracy for the approximation
+
+        % Diagonal matrix for variable coefficients
+        KAPPA % Variable coefficient
+
+        D % Total operator
+        D1 % First derivatives
+
+        % Second derivatives
+        D2_kappa
+
+        H, Hi % Inner products
+        e_l, e_r
+        d1_l, d1_r % Normal derivatives at the boundary
+        
+        H_boundary % Boundary inner products
+
+    end
+
+    methods
+
+        function obj = Heat2dVariable(g ,order, kappa_fun, opSet)
+            default_arg('opSet',{@sbp.D2Variable, @sbp.D2Variable});
+            default_arg('kappa_fun', @(x,y) 0*x+1);
+            dim = 2;
+
+            assert(isa(g, 'grid.Cartesian'))
+
+            kappa = grid.evalOn(g, kappa_fun);
+            m = g.size();
+            m_tot = g.N();
+
+            h = g.scaling();
+            lim = g.lim;
+
+            % 1D operators
+            ops = cell(dim,1);
+            for i = 1:dim
+                ops{i} = opSet{i}(m(i), lim{i}, order);
+            end
+
+            I = cell(dim,1);
+            D1 = cell(dim,1);
+            D2 = cell(dim,1);
+            H = cell(dim,1);
+            Hi = cell(dim,1);
+            e_l = cell(dim,1);
+            e_r = cell(dim,1);
+            d1_l = cell(dim,1);
+            d1_r = cell(dim,1);
+
+            for i = 1:dim
+                I{i} = speye(m(i));
+                D1{i} = ops{i}.D1;
+                D2{i} = ops{i}.D2;
+                H{i} =  ops{i}.H;
+                Hi{i} = ops{i}.HI;
+                e_l{i} = ops{i}.e_l;
+                e_r{i} = ops{i}.e_r;
+                d1_l{i} = ops{i}.d1_l;
+                d1_r{i} = ops{i}.d1_r;
+            end
+
+            %====== Assemble full operators ========
+            KAPPA = spdiag(kappa);
+            obj.KAPPA = KAPPA;
+
+            obj.D1 = cell(dim,1);
+            obj.D2_kappa = cell(dim,1);
+            obj.e_l = cell(dim,1);
+            obj.e_r = cell(dim,1);
+            obj.d1_l = cell(dim,1);
+            obj.d1_r = cell(dim,1);
+
+            % D1
+            obj.D1{1} = kron(D1{1},I{2});
+            obj.D1{2} = kron(I{1},D1{2});
+
+            % Boundary operators
+            obj.e_l{1} = kron(e_l{1},I{2});
+            obj.e_l{2} = kron(I{1},e_l{2});
+            obj.e_r{1} = kron(e_r{1},I{2});
+            obj.e_r{2} = kron(I{1},e_r{2});
+
+            obj.d1_l{1} = kron(d1_l{1},I{2});
+            obj.d1_l{2} = kron(I{1},d1_l{2});
+            obj.d1_r{1} = kron(d1_r{1},I{2});
+            obj.d1_r{2} = kron(I{1},d1_r{2});
+
+            % D2
+            for i = 1:dim
+                obj.D2_kappa{i} = sparse(m_tot);
+            end
+            ind = grid.funcToMatrix(g, 1:m_tot);
+
+            for i = 1:m(2)
+                D_kappa = D2{1}(kappa(ind(:,i)));
+                p = ind(:,i);
+                obj.D2_kappa{1}(p,p) = D_kappa;
+            end
+
+            for i = 1:m(1)
+                D_kappa = D2{2}(kappa(ind(i,:)));
+                p = ind(i,:);
+                obj.D2_kappa{2}(p,p) = D_kappa;
+            end
+
+            % Quadratures
+            obj.H = kron(H{1},H{2});
+            obj.Hi = inv(obj.H);
+            obj.H_boundary = cell(dim,1);
+            obj.H_boundary{1} = H{2};
+            obj.H_boundary{2} = H{1};
+
+            % Differentiation matrix D (without SAT)
+            D2_kappa = obj.D2_kappa;
+            D1 = obj.D1;
+            D = sparse(m_tot,m_tot);
+            for i = 1:dim
+                D = D + D2_kappa{i};
+            end
+            obj.D = D;
+            %=========================================%
+
+            % Misc.
+            obj.m = m;
+            obj.h = h;
+            obj.order = order;
+            obj.grid = g;
+            obj.dim = dim;
+
+        end
+
+
+        % Closure functions return the operators applied to the own domain to close the boundary
+        % Penalty functions return the operators 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.
+        %       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, parameter)
+            default_arg('type','Neumann');
+            default_arg('parameter', []);
+
+            % j is the coordinate direction of the boundary
+            % nj: outward unit normal component. 
+            % nj = -1 for west, south, bottom boundaries
+            % nj = 1  for east, north, top boundaries
+            [j, nj] = obj.get_boundary_number(boundary);
+            switch nj
+            case 1
+                e = obj.e_r;
+                d = obj.d1_r;
+            case -1
+                e = obj.e_l;
+                d = obj.d1_l;
+            end
+
+            Hi = obj.Hi;
+            H_gamma = obj.H_boundary{j};
+            KAPPA = obj.KAPPA;
+            kappa_gamma = e{j}'*KAPPA*e{j}; 
+
+            switch type
+
+            % Dirichlet boundary condition
+            case {'D','d','dirichlet','Dirichlet'}
+                    closure = -nj*Hi*d{j}*kappa_gamma*H_gamma*(e{j}' ); 
+                    penalty =  nj*Hi*d{j}*kappa_gamma*H_gamma;
+
+            % Free boundary condition
+            case {'N','n','neumann','Neumann'}
+                    closure = -nj*Hi*e{j}*kappa_gamma*H_gamma*(d{j}' ); 
+                    penalty =  nj*Hi*e{j}*kappa_gamma*H_gamma; 
+
+            % 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
+            error('Interface not implemented');
+        end
+
+        % Returns the coordinate number and outward normal component for the boundary specified by the string boundary.
+        function [j, nj] = get_boundary_number(obj, boundary)
+
+            switch boundary
+                case {'w','W','west','West', 'e', 'E', 'east', 'East'}
+                    j = 1;
+                case {'s','S','south','South', 'n', 'N', 'north', 'North'}
+                    j = 2;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+
+            switch boundary
+                case {'w','W','west','West','s','S','south','South'}
+                    nj = -1;
+                case {'e', 'E', 'east', 'East','n', 'N', 'north', 'North'}
+                    nj = 1;
+            end
+        end
+
+        % Returns the coordinate number and outward normal component for the boundary specified by the string boundary.
+        function [return_op] = get_boundary_operator(obj, op, boundary)
+
+            switch boundary
+                case {'w','W','west','West', 'e', 'E', 'east', 'East'}
+                    j = 1;
+                case {'s','S','south','South', 'n', 'N', 'north', 'North'}
+                    j = 2;
+                otherwise
+                    error('No such boundary: boundary = %s',boundary);
+            end
+
+            switch op
+                case 'e'
+                    switch boundary
+                        case {'w','W','west','West','s','S','south','South'}
+                            return_op = obj.e_l{j};
+                        case {'e', 'E', 'east', 'East','n', 'N', 'north', 'North'}
+                            return_op = obj.e_r{j};
+                    end
+                case 'd'
+                    switch boundary
+                        case {'w','W','west','West','s','S','south','South'}
+                            return_op = obj.d_l{j};
+                        case {'e', 'E', 'east', 'East','n', 'N', 'north', 'North'}
+                            return_op = obj.d_r{j};
+                    end
+                otherwise
+                    error(['No such operator: operatr = ' op]);
+            end
+
+        end
+
+        function N = size(obj)
+            N = prod(obj.m);
+        end
+    end
+end