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view +sbp/+implementations/d4_variable_4.m @ 314:88584b0cfba1 feature/beams

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Corrections and clean up order 4
author Jonatan Werpers <jonatan@werpers.com>
date Fri, 23 Sep 2016 22:55:30 +0200
parents 9230c056a574
children 297d2cbfbe15
line wrap: on
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function [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_4(m,h)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %%% 4:de ordn. SBP Finita differens         %%%
    %%%                                         %%%
    %%% H           (Normen)                    %%%
    %%% D1=H^(-1)Q  (approx f?rsta derivatan)   %%%
    %%% D2          (approx andra derivatan)    %%%
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


    BP = 6;
    if(m<2*BP)
        error(['Operator requires at least ' num2str(2*BP) ' grid points']);
    end

    % Norm
    Hv = ones(m,1);
    Hv(1:4) = [17/48 59/48 43/48 49/48];
    Hv(m-3:m) = rot90(Hv(1:4),2);
    Hv = h*Hv;
    H = spdiag(Hv, 0);
    HI = spdiag(1./Hv, 0);


    % Boundary operators
    e_l = sparse(m,1);
    e_l(1) = 1;
    e_r = rot90(e_l, 2);

    d1_l = sparse(m,1);
    d1_l(1:4) = 1/h*[-11/6 3 -3/2 1/3];
    d1_r = -rot90(d1_l);

    d2_l = sparse(m,1);
    d2_l(1:4) = 1/h^2*[2 -5 4 -1];
    d2_r = rot90(d2_l, 2);

    d3_l = sparse(m,1);
    d3_l(1:4) = 1/h^3*[-1 3 -3 1];
    d3_r = -rot90(d3_l, 2);


    % First derivative SBP operator,
    stencil = [1/12 -2/3 0 2/3 -1/12];
    diags = [-1 0 1];

    Q_U = [
        0 0.59e2/0.96e2 -0.1e1/0.12e2 -0.1e1/0.32e2;
        -0.59e2/0.96e2 0 0.59e2/0.96e2 0;
        0.1e1/0.12e2 -0.59e2/0.96e2 0 0.59e2/0.96e2;
        0.1e1/0.32e2 0 -0.59e2/0.96e2 0;
    ];

    Q = stripeMatrix(stencil, diags, m);
    Q(1:4,1:4)=Q_U;
    Q(m-3:m,m-3:m) = -rot90(Q_U, 2);

    D1 = HI*(Q - 1/2*e_l*e_l' + 1/2*e_r*e_r');


    % Second derivative
    M=sparse(m,m);

    scheme_width = 5;
    scheme_radius = (scheme_width-1)/2;
    r = (1+scheme_radius):(m-scheme_radius);

    function D2 = D2_fun(c)
        % ALTERNATIVES %%%%%%%%%%%%%
        % for i=4:m-3
        %     M(i,i-2:i+2)=[-c(i-1)/0.6e1 + c(i-2)/0.8e1 + c(i)/0.8e1 -c(i-2)/0.6e1 - c(i+1)/0.6e1 - c(i-1)/0.2e1 - c(i)/0.2e1 c(i-2)/0.24e2 + 0.5e1/0.6e1 * c(i-1) + 0.5e1/0.6e1 * c(i+1) + c(i+2)/0.24e2 + 0.3e1/0.4e1 * c(i) -c(i-1)/0.6e1 - c(i+2)/0.6e1 - c(i)/0.2e1 - c(i+1)/0.2e1 -c(i+1)/0.6e1 + c(i)/0.8e1 + c(i+2)/0.8e1;];
        % end
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        % for i=4:m-3
        %     M(i,i-2:i+2)= [
        %      c(i-2)/0.8e1 - c(i-1)/0.6e1 + c(i)  /0.8e1                                   ,
        %     -c(i-2)/0.6e1 - c(i-1)/0.2e1 - c(i)  /0.2e1 - c(i+1)/0.6e1                  ,
        %      c(i-2)/2.4e1 + c(i-1)/1.2e0 + c(i) * 0.3/0.4 + c(i+1)/1.2e0 + c(i+2)/2.4e1 ,
        %                      -c(i-1)/0.6e1 - c(i)  /0.2e1 - c(i+1)/0.2e1 - c(i+2)/0.6e1 ,
        %                                        c(i)  /0.8e1 - c(i+1)/0.6e1 + c(i+2)/0.8e1 ,
        %     ];
        % end
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        Mm2 =  c(r-2)/0.8e1 - c(r-1)/0.6e1 + c(r)  /0.8e1                                  ;
        Mm1 = -c(r-2)/0.6e1 - c(r-1)/0.2e1 - c(r)  /0.2e1 - c(r+1)/0.6e1                 ;
        M0  =  c(r-2)/2.4e1 + c(r-1)/1.2e0 + c(r) * 0.3/0.4 + c(r+1)/1.2e0 + c(r+2)/2.4e1;
        Mp1 =                  -c(r-1)/0.6e1 - c(r)  /0.2e1 - c(r+1)/0.2e1 - c(r+2)/0.6e1;
        Mp2 =                                    c(r)  /0.8e1 - c(r+1)/0.6e1 + c(r+2)/0.8e1;
        % printSize(Mm2);
        % scheme_radius
        % m
        M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],0:2*scheme_radius,length(r),m);
        % M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],(-2:2)+scheme_radius,M(r,:)); % This is slower
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        % %% Somthing is wrong here!!
        % Mm2 =  c(r-2)/0.8e1 - c(r-1)/0.6e1 + c(r)  /0.8e1                                  ;
        % Mm1 = -c(r-2)/0.6e1 - c(r-1)/0.2e1 - c(r)  /0.2e1 - c(r+1)/0.6e1                 ;
        % M0  =  c(r-2)/2.4e1 + c(r-1)/1.2e0 + c(r) * 0.3/0.4 + c(r+1)/1.2e0 + c(r+2)/2.4e1;
        % Mp1 =                  -c(r-1)/0.6e1 - c(r)  /0.2e1 - c(r+1)/0.2e1 - c(r+2)/0.6e1;
        % Mp2 =                                    c(r)  /0.8e1 - c(r+1)/0.6e1 + c(r+2)/0.8e1;
        % % printSize(M_diag_ind);
        % % Mdiags = [Mm2 Mm1 M0  Mp1 Mp2];
        % % printSize(Mdiags);
        % M(M_diag_ind) = [Mm2 Mm1 M0  Mp1 Mp2]; % This is slightly faster
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        % Kan man skriva det som en multiplikation av en 3-dim matris?
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

        M(1:6,1:6) = [
            0.12e2/0.17e2 * c(1) + 0.59e2/0.192e3 * c(2) + 0.27010400129e11/0.345067064608e12 * c(3) + 0.69462376031e11/0.2070402387648e13 * c(4) -0.59e2/0.68e2 * c(1) - 0.6025413881e10/0.21126554976e11 * c(3) - 0.537416663e9/0.7042184992e10 * c(4) 0.2e1/0.17e2 * c(1) - 0.59e2/0.192e3 * c(2) + 0.213318005e9/0.16049630912e11 * c(4) + 0.2083938599e10/0.8024815456e10 * c(3) 0.3e1/0.68e2 * c(1) - 0.1244724001e10/0.21126554976e11 * c(3) + 0.752806667e9/0.21126554976e11 * c(4) 0.49579087e8/0.10149031312e11 * c(3) - 0.49579087e8/0.10149031312e11 * c(4) -c(4)/0.784e3 + c(3)/0.784e3;
            -0.59e2/0.68e2 * c(1) - 0.6025413881e10/0.21126554976e11 * c(3) - 0.537416663e9/0.7042184992e10 * c(4) 0.3481e4/0.3264e4 * c(1) + 0.9258282831623875e16/0.7669235228057664e16 * c(3) + 0.236024329996203e15/0.1278205871342944e16 * c(4) -0.59e2/0.408e3 * c(1) - 0.29294615794607e14/0.29725717938208e14 * c(3) - 0.2944673881023e13/0.29725717938208e14 * c(4) -0.59e2/0.1088e4 * c(1) + 0.260297319232891e15/0.2556411742685888e16 * c(3) - 0.60834186813841e14/0.1278205871342944e16 * c(4) -0.1328188692663e13/0.37594290333616e14 * c(3) + 0.1328188692663e13/0.37594290333616e14 * c(4) -0.8673e4/0.2904112e7 * c(3) + 0.8673e4/0.2904112e7 * c(4);
            0.2e1/0.17e2 * c(1) - 0.59e2/0.192e3 * c(2) + 0.213318005e9/0.16049630912e11 * c(4) + 0.2083938599e10/0.8024815456e10 * c(3) -0.59e2/0.408e3 * c(1) - 0.29294615794607e14/0.29725717938208e14 * c(3) - 0.2944673881023e13/0.29725717938208e14 * c(4) c(1)/0.51e2 + 0.59e2/0.192e3 * c(2) + 0.13777050223300597e17/0.26218083221499456e17 * c(4) + 0.564461e6/0.13384296e8 * c(5) + 0.378288882302546512209e21/0.270764341349677687456e21 * c(3) c(1)/0.136e3 - 0.125059e6/0.743572e6 * c(5) - 0.4836340090442187227e19/0.5525802884687299744e19 * c(3) - 0.17220493277981e14/0.89177153814624e14 * c(4) -0.10532412077335e14/0.42840005263888e14 * c(4) + 0.1613976761032884305e19/0.7963657098519931984e19 * c(3) + 0.564461e6/0.4461432e7 * c(5) -0.960119e6/0.1280713392e10 * c(4) - 0.3391e4/0.6692148e7 * c(5) + 0.33235054191e11/0.26452850508784e14 * c(3);
            0.3e1/0.68e2 * c(1) - 0.1244724001e10/0.21126554976e11 * c(3) + 0.752806667e9/0.21126554976e11 * c(4) -0.59e2/0.1088e4 * c(1) + 0.260297319232891e15/0.2556411742685888e16 * c(3) - 0.60834186813841e14/0.1278205871342944e16 * c(4) c(1)/0.136e3 - 0.125059e6/0.743572e6 * c(5) - 0.4836340090442187227e19/0.5525802884687299744e19 * c(3) - 0.17220493277981e14/0.89177153814624e14 * c(4) 0.3e1/0.1088e4 * c(1) + 0.507284006600757858213e21/0.475219048083107777984e21 * c(3) + 0.1869103e7/0.2230716e7 * c(5) + c(6)/0.24e2 + 0.1950062198436997e16/0.3834617614028832e16 * c(4) -0.4959271814984644613e19/0.20965546238960637264e20 * c(3) - c(6)/0.6e1 - 0.15998714909649e14/0.37594290333616e14 * c(4) - 0.375177e6/0.743572e6 * c(5) -0.368395e6/0.2230716e7 * c(5) + 0.752806667e9/0.539854092016e12 * c(3) + 0.1063649e7/0.8712336e7 * c(4) + c(6)/0.8e1;
            0.49579087e8/0.10149031312e11 * c(3) - 0.49579087e8/0.10149031312e11 * c(4) -0.1328188692663e13/0.37594290333616e14 * c(3) + 0.1328188692663e13/0.37594290333616e14 * c(4) -0.10532412077335e14/0.42840005263888e14 * c(4) + 0.1613976761032884305e19/0.7963657098519931984e19 * c(3) + 0.564461e6/0.4461432e7 * c(5) -0.4959271814984644613e19/0.20965546238960637264e20 * c(3) - c(6)/0.6e1 - 0.15998714909649e14/0.37594290333616e14 * c(4) - 0.375177e6/0.743572e6 * c(5) 0.8386761355510099813e19/0.128413970713633903242e21 * c(3) + 0.2224717261773437e16/0.2763180339520776e16 * c(4) + 0.5e1/0.6e1 * c(6) + c(7)/0.24e2 + 0.280535e6/0.371786e6 * c(5) -0.35039615e8/0.213452232e9 * c(4) - c(7)/0.6e1 - 0.13091810925e11/0.13226425254392e14 * c(3) - 0.1118749e7/0.2230716e7 * c(5) - c(6)/0.2e1;
            -c(4)/0.784e3 + c(3)/0.784e3 -0.8673e4/0.2904112e7 * c(3) + 0.8673e4/0.2904112e7 * c(4) -0.960119e6/0.1280713392e10 * c(4) - 0.3391e4/0.6692148e7 * c(5) + 0.33235054191e11/0.26452850508784e14 * c(3) -0.368395e6/0.2230716e7 * c(5) + 0.752806667e9/0.539854092016e12 * c(3) + 0.1063649e7/0.8712336e7 * c(4) + c(6)/0.8e1 -0.35039615e8/0.213452232e9 * c(4) - c(7)/0.6e1 - 0.13091810925e11/0.13226425254392e14 * c(3) - 0.1118749e7/0.2230716e7 * c(5) - c(6)/0.2e1 0.3290636e7/0.80044587e8 * c(4) + 0.5580181e7/0.6692148e7 * c(5) + 0.5e1/0.6e1 * c(7) + c(8)/0.24e2 + 0.660204843e9/0.13226425254392e14 * c(3) + 0.3e1/0.4e1 * c(6);
         ];


        M(m-5:m,m-5:m) = [
            c(m-7)/0.24e2 + 0.5e1/0.6e1 * c(m-6) + 0.5580181e7/0.6692148e7 * c(m-4) + 0.4887707739997e13/0.119037827289528e15 * c(m-3) + 0.3e1/0.4e1 * c(m-5) + 0.660204843e9/0.13226425254392e14 * c(m-2) + 0.660204843e9/0.13226425254392e14 * c(m-1) -c(m-6)/0.6e1 - 0.1618585929605e13/0.9919818940794e13 * c(m-3) - c(m-5)/0.2e1 - 0.1118749e7/0.2230716e7 * c(m-4) - 0.13091810925e11/0.13226425254392e14 * c(m-2) - 0.13091810925e11/0.13226425254392e14 * c(m-1) -0.368395e6/0.2230716e7 * c(m-4) + c(m-5)/0.8e1 + 0.48866620889e11/0.404890569012e12 * c(m-3) + 0.752806667e9/0.539854092016e12 * c(m-2) + 0.752806667e9/0.539854092016e12 * c(m-1) -0.3391e4/0.6692148e7 * c(m-4) - 0.238797444493e12/0.119037827289528e15 * c(m-3) + 0.33235054191e11/0.26452850508784e14 * c(m-2) + 0.33235054191e11/0.26452850508784e14 * c(m-1) -0.8673e4/0.2904112e7 * c(m-2) - 0.8673e4/0.2904112e7 * c(m-1) + 0.8673e4/0.1452056e7 * c(m-3) -c(m-3)/0.392e3 + c(m-2)/0.784e3 + c(m-1)/0.784e3;
             -c(m-6)/0.6e1 - 0.1618585929605e13/0.9919818940794e13 * c(m-3) - c(m-5)/0.2e1 - 0.1118749e7/0.2230716e7 * c(m-4) - 0.13091810925e11/0.13226425254392e14 * c(m-2) - 0.13091810925e11/0.13226425254392e14 * c(m-1) c(m-6)/0.24e2 + 0.5e1/0.6e1 * c(m-5) + 0.3896014498639e13/0.4959909470397e13 * c(m-3) + 0.8386761355510099813e19/0.128413970713633903242e21 * c(m-2) + 0.280535e6/0.371786e6 * c(m-4) + 0.3360696339136261875e19/0.171218627618178537656e21 * c(m-1) -c(m-5)/0.6e1 - 0.4959271814984644613e19/0.20965546238960637264e20 * c(m-2) - 0.375177e6/0.743572e6 * c(m-4) - 0.13425842714e11/0.33740880751e11 * c(m-3) - 0.193247108773400725e18/0.6988515412986879088e19 * c(m-1) -0.365281640980e12/0.1653303156799e13 * c(m-3) + 0.564461e6/0.4461432e7 * c(m-4) + 0.1613976761032884305e19/0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18/0.7963657098519931984e19 * c(m-1) -0.1328188692663e13/0.37594290333616e14 * c(m-2) + 0.2226377963775e13/0.37594290333616e14 * c(m-1) - 0.8673e4/0.363014e6 * c(m-3) c(m-3)/0.49e2 + 0.49579087e8/0.10149031312e11 * c(m-2) - 0.256702175e9/0.10149031312e11 * c(m-1);
             -0.368395e6/0.2230716e7 * c(m-4) + c(m-5)/0.8e1 + 0.48866620889e11/0.404890569012e12 * c(m-3) + 0.752806667e9/0.539854092016e12 * c(m-2) + 0.752806667e9/0.539854092016e12 * c(m-1) -c(m-5)/0.6e1 - 0.4959271814984644613e19/0.20965546238960637264e20 * c(m-2) - 0.375177e6/0.743572e6 * c(m-4) - 0.13425842714e11/0.33740880751e11 * c(m-3) - 0.193247108773400725e18/0.6988515412986879088e19 * c(m-1) c(m-5)/0.24e2 + 0.1869103e7/0.2230716e7 * c(m-4) + 0.507284006600757858213e21/0.475219048083107777984e21 * c(m-2) + 0.3e1/0.1088e4 * c(m) + 0.31688435395e11/0.67481761502e11 * c(m-3) + 0.27769176016102795561e20/0.712828572124661666976e21 * c(m-1) -0.125059e6/0.743572e6 * c(m-4) + c(m)/0.136e3 - 0.23099342648e11/0.101222642253e12 * c(m-3) - 0.4836340090442187227e19/0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18/0.5525802884687299744e19 * c(m-1) 0.260297319232891e15/0.2556411742685888e16 * c(m-2) - 0.59e2/0.1088e4 * c(m) - 0.106641839640553e15/0.1278205871342944e16 * c(m-1) + 0.26019e5/0.726028e6 * c(m-3) -0.1244724001e10/0.21126554976e11 * c(m-2) + 0.3e1/0.68e2 * c(m) + 0.752806667e9/0.21126554976e11 * c(m-1);
             -0.3391e4/0.6692148e7 * c(m-4) - 0.238797444493e12/0.119037827289528e15 * c(m-3) + 0.33235054191e11/0.26452850508784e14 * c(m-2) + 0.33235054191e11/0.26452850508784e14 * c(m-1) -0.365281640980e12/0.1653303156799e13 * c(m-3) + 0.564461e6/0.4461432e7 * c(m-4) + 0.1613976761032884305e19/0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18/0.7963657098519931984e19 * c(m-1) -0.125059e6/0.743572e6 * c(m-4) + c(m)/0.136e3 - 0.23099342648e11/0.101222642253e12 * c(m-3) - 0.4836340090442187227e19/0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18/0.5525802884687299744e19 * c(m-1) 0.564461e6/0.13384296e8 * c(m-4) + 0.470299699916357e15/0.952302618316224e15 * c(m-3) + 0.550597048646198778781e21/0.1624586048098066124736e22 * c(m-1) + c(m)/0.51e2 + 0.378288882302546512209e21/0.270764341349677687456e21 * c(m-2) -0.59e2/0.408e3 * c(m) - 0.29294615794607e14/0.29725717938208e14 * c(m-2) - 0.2234477713167e13/0.29725717938208e14 * c(m-1) - 0.8673e4/0.363014e6 * c(m-3) -0.59e2/0.3136e4 * c(m-3) - 0.13249937023e11/0.48148892736e11 * c(m-1) + 0.2e1/0.17e2 * c(m) + 0.2083938599e10/0.8024815456e10 * c(m-2);
             -0.8673e4/0.2904112e7 * c(m-2) - 0.8673e4/0.2904112e7 * c(m-1) + 0.8673e4/0.1452056e7 * c(m-3) -0.1328188692663e13/0.37594290333616e14 * c(m-2) + 0.2226377963775e13/0.37594290333616e14 * c(m-1) - 0.8673e4/0.363014e6 * c(m-3) 0.260297319232891e15/0.2556411742685888e16 * c(m-2) - 0.59e2/0.1088e4 * c(m) - 0.106641839640553e15/0.1278205871342944e16 * c(m-1) + 0.26019e5/0.726028e6 * c(m-3) -0.59e2/0.408e3 * c(m) - 0.29294615794607e14/0.29725717938208e14 * c(m-2) - 0.2234477713167e13/0.29725717938208e14 * c(m-1) - 0.8673e4/0.363014e6 * c(m-3) 0.9258282831623875e16/0.7669235228057664e16 * c(m-2) + 0.3481e4/0.3264e4 * c(m) + 0.228389721191751e15/0.1278205871342944e16 * c(m-1) + 0.8673e4/0.1452056e7 * c(m-3) -0.6025413881e10/0.21126554976e11 * c(m-2) - 0.59e2/0.68e2 * c(m) - 0.537416663e9/0.7042184992e10 * c(m-1);
             -c(m-3)/0.392e3 + c(m-2)/0.784e3 + c(m-1)/0.784e3 c(m-3)/0.49e2 + 0.49579087e8/0.10149031312e11 * c(m-2) - 0.256702175e9/0.10149031312e11 * c(m-1) -0.1244724001e10/0.21126554976e11 * c(m-2) + 0.3e1/0.68e2 * c(m) + 0.752806667e9/0.21126554976e11 * c(m-1) -0.59e2/0.3136e4 * c(m-3) - 0.13249937023e11/0.48148892736e11 * c(m-1) + 0.2e1/0.17e2 * c(m) + 0.2083938599e10/0.8024815456e10 * c(m-2) -0.6025413881e10/0.21126554976e11 * c(m-2) - 0.59e2/0.68e2 * c(m) - 0.537416663e9/0.7042184992e10 * c(m-1) 0.3e1/0.3136e4 * c(m-3) + 0.27010400129e11/0.345067064608e12 * c(m-2) + 0.234566387291e12/0.690134129216e12 * c(m-1) + 0.12e2/0.17e2 * c(m);
         ];

        M = M/h;
        D2 = HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m');
    end
    D2 = @D2_fun;


    % Fourth derivative
    stencil = [-1/6,2,-13/2, 28/3,-13/2,2,-1/6];
    diags = -3:3;
    M4 = stripeMatrix(stencil, diags, m);

    M4_U = [
        0.5762947e7/0.2316384e7 -0.6374287e7/0.1158192e7 0.573947e6/0.165456e6 -0.124637e6/0.289548e6 0.67979e5/0.2316384e7 -0.60257e5/0.1158192e7;
        -0.6374287e7/0.1158192e7 0.30392389e8/0.2316384e7 -0.2735053e7/0.289548e6 0.273109e6/0.165456e6 0.83767e5/0.1158192e7 0.245549e6/0.2316384e7;
        0.573947e6/0.165456e6 -0.2735053e7/0.289548e6 0.5266855e7/0.579096e6 -0.1099715e7/0.289548e6 0.869293e6/0.1158192e7 -0.10195e5/0.144774e6;
        -0.124637e6/0.289548e6 0.273109e6/0.165456e6 -0.1099715e7/0.289548e6 0.3259225e7/0.579096e6 -0.324229e6/0.72387e5 0.1847891e7/0.1158192e7;
        0.67979e5/0.2316384e7 0.83767e5/0.1158192e7 0.869293e6/0.1158192e7 -0.324229e6/0.72387e5 0.2626501e7/0.330912e6 -0.7115491e7/0.1158192e7;
        -0.60257e5/0.1158192e7 0.245549e6/0.2316384e7 -0.10195e5/0.144774e6 0.1847891e7/0.1158192e7 -0.7115491e7/0.1158192e7 0.21383077e8/0.2316384e7;
    ];

    M4(1:6,1:6) = M4_U;
    M4(m-5:m,m-5:m) = rot90(M4_U, 2);
    M4 = 1/h^3*M4;

    D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r');
end