classdef LaplaceCurvilinearMin < scheme.Scheme
    properties
        m % Number of points in each direction, possibly a vector
        h % Grid spacing
        dim % Number of spatial dimensions

        grid

        order % Order of accuracy for the approximation

        a,b % Parameters of the operator


        % Inner products and operators for physical coordinates
        D % Laplace operator
        H, Hi % Inner product
        e_w, e_e, e_s, e_n
        d_w, d_e, d_s, d_n % Normal derivatives at the boundary
        H_w, H_e, H_s, H_n % Boundary inner products
        Dx, Dy % Physical derivatives
        M % Gradient inner product

        % Metric coefficients
        J, Ji
        a11, a12, a22
        K
        x_u
        x_v
        y_u
        y_v
        s_w, s_e, s_s, s_n % Boundary integral scale factors

        % Inner product and operators for logical coordinates
        H_u, H_v % Norms in the x and y directions
        Hi_u, Hi_v
        Hu,Hv % Kroneckerd norms. 1'*Hx*v corresponds to integration in the x dir.
        Hiu, Hiv
        du_w, dv_w
        du_e, dv_e
        du_s, dv_s
        du_n, dv_n

        % Borrowing constants
        theta_M_u, theta_M_v
        theta_R_u, theta_R_v
        theta_H_u, theta_H_v

        % Temporary, only used for nonconforming interfaces but should be removed.
        lambda

        % Number of boundary points in minumum check
        bp
    end

    methods
        % Implements  a*div(b*grad(u)) as a SBP scheme
        % TODO: Implement proper H, it should be the real physical quadrature, the logic quadrature may be but in a separate variable (H_logic?)

        function obj = LaplaceCurvilinearMin(g, order, a, b, opSet)
            default_arg('opSet',@sbp.D2Variable);
            default_arg('a', 1);
            default_arg('b', @(x,y) 0*x + 1);

            % Number of boundary points in minimum check
            switch order
            case 2
                obj.bp = 2;
            case 4
                obj.bp = 4;
            case 6
                obj.bp = 7;
            end

            % assert(isa(g, 'grid.Curvilinear'))
            if isa(a, 'function_handle')
                a = grid.evalOn(g, a);
            end
            a = spdiag(a);

            if isa(b, 'function_handle')
                b = grid.evalOn(g, b);
            end

            % If b is scalar
            if length(b) == 1
                % b = b*speye(g.N(), g.N());
                b = b*ones(g.N(), 1);
            end
            b = spdiag(b);

            dim = 2;
            m = g.size();
            m_u = m(1);
            m_v = m(2);
            m_tot = g.N();

            % 1D operators
            ops_u = opSet(m_u, {0, 1}, order);
            ops_v = opSet(m_v, {0, 1}, order);

            h_u = ops_u.h;
            h_v = ops_v.h;

            I_u = speye(m_u);
            I_v = speye(m_v);

            D1_u = ops_u.D1;
            D2_u = ops_u.D2;
            H_u =  ops_u.H;
            Hi_u = ops_u.HI;
            e_l_u = ops_u.e_l;
            e_r_u = ops_u.e_r;
            d1_l_u = ops_u.d1_l;
            d1_r_u = ops_u.d1_r;

            D1_v = ops_v.D1;
            D2_v = ops_v.D2;
            H_v =  ops_v.H;
            Hi_v = ops_v.HI;
            e_l_v = ops_v.e_l;
            e_r_v = ops_v.e_r;
            d1_l_v = ops_v.d1_l;
            d1_r_v = ops_v.d1_r;


            % Logical operators
            Du = kr(D1_u,I_v);
            Dv = kr(I_u,D1_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);

            e_w  = kr(e_l_u,I_v);
            e_e  = kr(e_r_u,I_v);
            e_s  = kr(I_u,e_l_v);
            e_n  = kr(I_u,e_r_v);
            obj.du_w = kr(d1_l_u,I_v);
            obj.dv_w = (e_w'*Dv)';
            obj.du_e = kr(d1_r_u,I_v);
            obj.dv_e = (e_e'*Dv)';
            obj.du_s = (e_s'*Du)';
            obj.dv_s = kr(I_u,d1_l_v);
            obj.du_n = (e_n'*Du)';
            obj.dv_n = kr(I_u,d1_r_v);


            % Metric coefficients
            coords = g.points();
            x = coords(:,1);
            y = coords(:,2);

            x_u = Du*x;
            x_v = Dv*x;
            y_u = Du*y;
            y_v = Dv*y;

            J = x_u.*y_v - x_v.*y_u;
            a11 =  1./J .* (x_v.^2  + y_v.^2);
            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));

            K = cell(dim, dim);
            K{1,1} = spdiag(y_v./J);
            K{1,2} = spdiag(-y_u./J);
            K{2,1} = spdiag(-x_v./J);
            K{2,2} = spdiag(x_u./J);
            obj.K = K;

            obj.x_u = x_u;
            obj.x_v = x_v;
            obj.y_u = y_u;
            obj.y_v = y_v;

            % Assemble full operators
            L_12 = spdiag(a12);
            Duv = Du*b*L_12*Dv;
            Dvu = Dv*b*L_12*Du;

            Duu = sparse(m_tot);
            Dvv = sparse(m_tot);
            ind = grid.funcToMatrix(g, 1:m_tot);

            for i = 1:m_v
                b_a11 = b*a11;
                D = D2_u(b_a11(ind(:,i)));
                p = ind(:,i);
                Duu(p,p) = D;
            end

            for i = 1:m_u
                b_a22 = b*a22;
                D = D2_v(b_a22(ind(i,:)));
                p = ind(i,:);
                Dvv(p,p) = D;
            end


            % Physical operators
            obj.J = spdiag(J);
            obj.Ji = spdiag(1./J);

            obj.D = obj.Ji*a*(Duu + Duv + Dvu + Dvv);
            obj.H = obj.J*kr(H_u,H_v);
            obj.Hi = obj.Ji*kr(Hi_u,Hi_v);

            obj.e_w = e_w;
            obj.e_e = e_e;
            obj.e_s = e_s;
            obj.e_n = e_n;

            %% normal derivatives
            I_w = ind(1,:);
            I_e = ind(end,:);
            I_s = ind(:,1);
            I_n = ind(:,end);

            a11_w = spdiag(a11(I_w));
            a12_w = spdiag(a12(I_w));
            a11_e = spdiag(a11(I_e));
            a12_e = spdiag(a12(I_e));
            a22_s = spdiag(a22(I_s));
            a12_s = spdiag(a12(I_s));
            a22_n = spdiag(a22(I_n));
            a12_n = spdiag(a12(I_n));

            s_w = sqrt((e_w'*x_v).^2 + (e_w'*y_v).^2);
            s_e = sqrt((e_e'*x_v).^2 + (e_e'*y_v).^2);
            s_s = sqrt((e_s'*x_u).^2 + (e_s'*y_u).^2);
            s_n = sqrt((e_n'*x_u).^2 + (e_n'*y_u).^2);

            obj.d_w = -1*(spdiag(1./s_w)*(a11_w*obj.du_w' + a12_w*obj.dv_w'))';
            obj.d_e =    (spdiag(1./s_e)*(a11_e*obj.du_e' + a12_e*obj.dv_e'))';
            obj.d_s = -1*(spdiag(1./s_s)*(a22_s*obj.dv_s' + a12_s*obj.du_s'))';
            obj.d_n =    (spdiag(1./s_n)*(a22_n*obj.dv_n' + a12_n*obj.du_n'))';

            obj.Dx = spdiag( y_v./J)*Du + spdiag(-y_u./J)*Dv;
            obj.Dy = spdiag(-x_v./J)*Du + spdiag( x_u./J)*Dv;

            %% Boundary inner products
            obj.H_w = H_v*spdiag(s_w);
            obj.H_e = H_v*spdiag(s_e);
            obj.H_s = H_u*spdiag(s_s);
            obj.H_n = H_u*spdiag(s_n);

            % Misc.
            obj.m = m;
            obj.h = [h_u h_v];
            obj.order = order;
            obj.grid = g;
            obj.dim = dim;

            obj.a = a;
            obj.b = b;
            obj.a11 = a11;
            obj.a12 = a12;
            obj.a22 = a22;
            obj.s_w = spdiag(s_w);
            obj.s_e = spdiag(s_e);
            obj.s_s = spdiag(s_s);
            obj.s_n = spdiag(s_n);

            obj.theta_M_u = h_u*ops_u.borrowing.M.d1;
            obj.theta_M_v = h_v*ops_v.borrowing.M.d1;

            obj.theta_R_u = h_u*ops_u.borrowing.R.delta_D;
            obj.theta_R_v = h_v*ops_v.borrowing.R.delta_D;

            obj.theta_H_u = h_u*ops_u.borrowing.H11;
            obj.theta_H_v = h_v*ops_v.borrowing.H11;

            % Temporary
            obj.lambda = lambda;
        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, parameter)
            default_arg('type','neumann');
            default_arg('parameter', []);

            e               = obj.getBoundaryOperator('e', boundary);
            d               = obj.getBoundaryOperator('d', boundary);
            H_b             = obj.getBoundaryQuadrature(boundary);
            s_b             = obj.getBoundaryScaling(boundary);
            [th_H, ~, th_R] = obj.getBoundaryBorrowing(boundary);
            m               = obj.getBoundaryNumber(boundary);

            K = obj.K;
            J = obj.J;
            Hi = obj.Hi;
            a = obj.a;
            b_b = e'*obj.b*e;
            b = obj.b;

            switch type
                % Dirichlet boundary condition
                case {'D','d','dirichlet'}
                    tuning = 1.0;

                    sigma = 0*b;
                    for i = 1:obj.dim
                        sigma = sigma + b*J*K{i,m}*K{i,m};
                    end

                    % Minimum check on sigma
                    mx = obj.m(1);
                    my = obj.m(2);
                    bp = obj.bp;
                    sigma_mat = reshape(diag(sigma), my, mx);
                    switch boundary
                    case 'w'
                        sigma_min = min(sigma_mat(:,1:bp), [], 2);
                        sigma_mat = repmat(sigma_min, 1, mx);
                    case 'e'
                        sigma_min = min(sigma_mat(:,end-bp+1:end), [], 2);
                        sigma_mat = repmat(sigma_min, 1, mx);
                    case 's'
                        sigma_min = min(sigma_mat(1:bp,:), [], 1);
                        sigma_mat = repmat(sigma_min, my, 1);
                    case 'n'
                        sigma_min = min(sigma_mat(end-bp+1:end,:), [], 1);
                        sigma_mat = repmat(sigma_min, my, 1);
                    end
                    sigma_min = sigma_mat(:);
                    sigma_min = spdiag(sigma_min);

                    % Window
                    sigma = e'*sigma*e;
                    sigma_min = e'*sigma_min*e;

                    sigma = sigma/s_b;
                    sigma_min = sigma_min/s_b;

                    tau = tuning*(1/th_R*sigma/sigma_min*sigma + obj.dim/th_H*sigma);

                    closure = a*Hi*d*b_b*H_b*e' ...
                             -a*Hi*e*tau*H_b*e';

                    penalty = -a*Hi*d*b_b*H_b ...
                              +a*Hi*e*tau*H_b;


                % Neumann boundary condition. Note that the penalty is for du/dn and not b*du/dn.
                case {'N','n','neumann'}
                    tau1 = -1;
                    tau2 = 0;
                    tau = (tau1*e + tau2*d)*H_b;

                    closure =  a*Hi*tau*b_b*d';
                    penalty = -a*Hi*tau*b_b;


                % Unknown, boundary condition
                otherwise
                    error('No such boundary condition: type = %s',type);
            end
        end

        % type     Struct that specifies the interface coupling.
        %          Fields:
        %          -- tuning:           penalty strength, defaults to 1.2
        %          -- interpolation:    type of interpolation, default 'none'
        function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary,type)

            % error('Not implemented')

            defaultType.tuning = 1.0;
            defaultType.interpolation = 'none';
            default_struct('type', defaultType);

            switch type.interpolation
            case {'none', ''}
                [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary,type);
            case {'op','OP'}
                [closure, penalty] = interfaceNonConforming(obj,boundary,neighbour_scheme,neighbour_boundary,type);
            otherwise
                error('Unknown type of interpolation: %s ', type.interpolation);
            end
        end

        function [closure, penalty] = interfaceStandard(obj,boundary,neighbour_scheme,neighbour_boundary,type)
            tuning = type.tuning;

            dim = obj.dim;
            % u denotes the solution in the own domain
            % v denotes the solution in the neighbour domain
            u = obj;
            v = neighbour_scheme;

            % Boundary operators, u
            e_u     = u.getBoundaryOperator('e', boundary);
            d_u     = u.getBoundaryOperator('d', boundary);
            s_b_u   = u.getBoundaryScaling(boundary);
            [th_H_u, ~, th_R_u] = u.getBoundaryBorrowing(boundary);
            m_u = u.getBoundaryNumber(boundary);

            % Coefficients, u
            K_u = u.K;
            J_u = u.J;
            b_u = u.b;
            b_b_u = e_u'*u.b*e_u;

            % Boundary operators, v
            e_v     = v.getBoundaryOperator('e', neighbour_boundary);
            d_v     = v.getBoundaryOperator('d', neighbour_boundary);
            s_b_v   = v.getBoundaryScaling(neighbour_boundary);
            [th_H_v, ~, th_R_v] = v.getBoundaryBorrowing(neighbour_boundary);
            m_v = v.getBoundaryNumber(neighbour_boundary);

            % BUGFIX?!?!?
            if (strcmp(boundary,'s') && strcmp(neighbour_boundary,'e')) || (strcmp(boundary,'e') && strcmp(neighbour_boundary,'s'))
                e_v = fliplr(e_v);
                d_v = fliplr(d_v);
                s_b_v = rot90(s_b_v,2);
            end

            % Coefficients, v
            K_v = v.K;
            J_v = v.J;
            b_v = v.b;
            b_b_v = e_v'*v.b*e_v;

            %--- Penalty strength tau -------------
            sigma_u = 0*b_u;
            sigma_v = 0*b_v;
            for i = 1:obj.dim
                sigma_u = sigma_u + b_u*J_u*K_u{i,m_u}*K_u{i,m_u};
                sigma_v = sigma_v + b_v*J_v*K_v{i,m_v}*K_v{i,m_v};
            end

            %--- Minimum check on sigma_u ----
            mx = u.m(1);
            my = u.m(2);
            bp = u.bp;
            sigma_mat = reshape(diag(sigma_u), my, mx);
            switch boundary
            case 'w'
                sigma_min = min(sigma_mat(:,1:bp), [], 2);
                sigma_mat = repmat(sigma_min, 1, mx);
            case 'e'
                sigma_min = min(sigma_mat(:,end-bp+1:end), [], 2);
                sigma_mat = repmat(sigma_min, 1, mx);
            case 's'
                sigma_min = min(sigma_mat(1:bp,:), [], 1);
                sigma_mat = repmat(sigma_min, my, 1);
            case 'n'
                sigma_min = min(sigma_mat(end-bp+1:end,:), [], 1);
                sigma_mat = repmat(sigma_min, my, 1);
            end
            sigma_min_u = sigma_mat(:);
            sigma_min_u = spdiag(sigma_min_u);

            % Window
            sigma_u = e_u'*sigma_u*e_u;
            sigma_min_u = e_u'*sigma_min_u*e_u;
            % -------------------------------

            %--- Minimum check on sigma_v ----
            mx = v.m(1);
            my = v.m(2);
            bp = v.bp;
            sigma_mat = reshape(diag(sigma_v), my, mx);
            switch neighbour_boundary
            case 'w'
                sigma_min = min(sigma_mat(:,1:bp), [], 2);
                sigma_mat = repmat(sigma_min, 1, mx);
            case 'e'
                sigma_min = min(sigma_mat(:,end-bp+1:end), [], 2);
                sigma_mat = repmat(sigma_min, 1, mx);
            case 's'
                sigma_min = min(sigma_mat(1:bp,:), [], 1);
                sigma_mat = repmat(sigma_min, my, 1);
            case 'n'
                sigma_min = min(sigma_mat(end-bp+1:end,:), [], 1);
                sigma_mat = repmat(sigma_min, my, 1);
            end
            sigma_min_v = sigma_mat(:);
            sigma_min_v = spdiag(sigma_min_v);

            % Window
            sigma_v = e_v'*sigma_v*e_v;
            sigma_min_v = e_v'*sigma_min_v*e_v;
            % -------------------------------

            sigma_u = sigma_u/s_b_u;
            sigma_min_u = sigma_min_u/s_b_u;

            sigma_v = sigma_v/s_b_v;
            sigma_min_v = sigma_min_v/s_b_v;

            tau_R_u = 1/th_R_u*sigma_u/sigma_min_u*sigma_u;
            tau_R_v = 1/th_R_v*sigma_v/sigma_min_v*sigma_v;

            tau_H_u = dim*1/th_H_u*sigma_u;
            tau_H_v = dim*1/th_H_v*sigma_v;

            tau = 1/4*tuning*((tau_R_u + tau_H_u) + (tau_R_v + tau_H_v));
            %--------------------------------------

            % Operators/coefficients that are only required from this side
            Hi = u.Hi;
            H_b = u.getBoundaryQuadrature(boundary);
            a = u.a;

            closure = 1/2*a*Hi*d_u*b_b_u*H_b*e_u' ...
                     -1/2*a*Hi*e_u*H_b*b_b_u*d_u' ...
                         -a*Hi*e_u*tau*H_b*e_u';

            penalty = -1/2*a*Hi*d_u*b_b_u*H_b*e_v' ...
                      -1/2*a*Hi*e_u*H_b*b_b_v*d_v' ...
                          +a*Hi*e_u*tau*H_b*e_v';
        end

        function [closure, penalty] = interfaceNonConforming(obj,boundary,neighbour_scheme,neighbour_boundary,type)

            % TODO: Make this work for curvilinear grids
            warning('LaplaceCurvilinear: Non-conforming grid interpolation only works for Cartesian grids.');
            warning('LaplaceCurvilinear: Non-conforming interface uses Virtas penalty strength');
            warning('LaplaceCurvilinear: Non-conforming interface assumes that b is constant');

            % User can request special interpolation operators by specifying type.interpOpSet
            default_field(type, 'interpOpSet', @sbp.InterpOpsOP);
            interpOpSet = type.interpOpSet;
            tuning = type.tuning;


            % 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);
            H_b_u       = obj.getBoundaryQuadrature(boundary);
            I_u         = obj.getBoundaryIndices(boundary);
            [~, gamm_u] = obj.getBoundaryBorrowing(boundary);

            e_v         = neighbour_scheme.getBoundaryOperator('e', neighbour_boundary);
            d_v         = neighbour_scheme.getBoundaryOperator('d', neighbour_boundary);
            H_b_v       = neighbour_scheme.getBoundaryQuadrature(neighbour_boundary);
            I_v         = neighbour_scheme.getBoundaryIndices(neighbour_boundary);
            [~, gamm_v] = neighbour_scheme.getBoundaryBorrowing(neighbour_boundary);


            % Find the number of grid points along the interface
            m_u = size(e_u, 2);
            m_v = size(e_v, 2);

            Hi = obj.Hi;
            a = obj.a;

            u = obj;
            v = neighbour_scheme;

            b1_u = gamm_u*u.lambda(I_u)./u.a11(I_u).^2;
            b2_u = gamm_u*u.lambda(I_u)./u.a22(I_u).^2;
            b1_v = gamm_v*v.lambda(I_v)./v.a11(I_v).^2;
            b2_v = gamm_v*v.lambda(I_v)./v.a22(I_v).^2;

            tau_u = -1./(4*b1_u) -1./(4*b2_u);
            tau_v = -1./(4*b1_v) -1./(4*b2_v);

            tau_u = tuning * spdiag(tau_u);
            tau_v = tuning * spdiag(tau_v);
            beta_u = tau_v;

            % Build interpolation operators
            intOps = interpOpSet(m_u, m_v, obj.order, neighbour_scheme.order);
            Iu2v = intOps.Iu2v;
            Iv2u = intOps.Iv2u;

            closure = a*Hi*e_u*tau_u*H_b_u*e_u' + ...
                      a*Hi*e_u*H_b_u*Iv2u.bad*beta_u*Iu2v.good*e_u' + ...
                      a*1/2*Hi*d_u*H_b_u*e_u' + ...
                      -a*1/2*Hi*e_u*H_b_u*d_u';

            penalty = -a*Hi*e_u*tau_u*H_b_u*Iv2u.good*e_v' + ...
                      -a*Hi*e_u*H_b_u*Iv2u.bad*beta_u*e_v' + ...
                      -a*1/2*Hi*d_u*H_b_u*Iv2u.good*e_v' + ...
                      -a*1/2*Hi*e_u*H_b_u*Iv2u.bad*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, {'w', 'e', 's', 'n'})

            o = obj.([op, '_', boundary]);
        end

        % Returns square boundary quadrature matrix, of dimension
        % corresponding to the number of boundary points
        %
        % boundary -- string
        function H_b = getBoundaryQuadrature(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            H_b = obj.(['H_', boundary]);
        end

        % Returns square boundary quadrature scaling matrix, of dimension
        % corresponding to the number of boundary points
        %
        % boundary -- string
        function s_b = getBoundaryScaling(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            s_b = obj.(['s_', boundary]);
        end

        % Returns the coordinate number corresponding to the boundary
        %
        % boundary -- string
        function m = getBoundaryNumber(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            switch boundary
                case {'w', 'e'}
                    m = 1;
                case {'s', 'n'}
                    m = 2;
            end
        end

        % Returns the indices of the boundary points in the grid matrix
        % boundary -- string
        function I = getBoundaryIndices(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            ind = grid.funcToMatrix(obj.grid, 1:prod(obj.m));
            switch boundary
                case 'w'
                    I = ind(1,:);
                case 'e'
                    I = ind(end,:);
                case 's'
                    I = ind(:,1)';
                case 'n'
                    I = ind(:,end)';
            end
        end

        % Returns borrowing constant gamma
        % boundary -- string
        function [theta_H, theta_M, theta_R] = getBoundaryBorrowing(obj, boundary)
            assertIsMember(boundary, {'w', 'e', 's', 'n'})

            switch boundary
                case {'w','e'}
                    theta_H = obj.theta_H_u;
                    theta_M = obj.theta_M_u;
                    theta_R = obj.theta_R_u;
                case {'s','n'}
                    theta_H = obj.theta_H_v;
                    theta_M = obj.theta_M_v;
                    theta_R = obj.theta_R_v;
            end
        end

        function N = size(obj)
            N = prod(obj.m);
        end
    end
end