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view +sbp/+implementations/d2_6.m @ 1198:2924b3a9b921 feature/d2_compatible

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Add OpSet for fully compatible D2Variable, created from regular D2Variable by replacing d1 by first row of D1. Formal reduction by one order of accuracy at the boundary point.
author Martin Almquist <malmquist@stanford.edu>
date Fri, 16 Aug 2019 14:30:28 -0700
parents f7ac3cd6eeaa
children
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function [H, HI, D1, D2, e_1, e_m, M, Q, S_1, S_m] = d2_6(m,h)
    
    BP = 6;
    if(m<2*BP)
        error(['Operator requires at least ' num2str(2*BP) ' grid points']);
    end

    e_1=sparse(m,1);e_1(1)=1;
    e_m=sparse(m,1);e_m(m)=1;

    H=speye(m,m);
    H(1:6,1:6)=diag([13649/43200,12013/8640,2711/4320,5359/4320,7877/8640, ...
             43801/43200]);
    H(m-5:m,m-5:m)=rot90(diag([13649/43200,12013/8640, ...
                2711/4320,5359/4320,7877/8640,43801/43200]),2);

    H=H*h;
    HI=inv(H);


    % D1 har en fri parameter x1.
    % Ett optimerat varde ger x1=0.70127127127127 = 331/472
    x1=0.70127127127127;

    diags   = -3:3;
    stencil = [-1/60,3/20,-3/4,0,3/4,-3/20,1/60];
    D1 = stripeMatrix(stencil, diags, m);

    D1(1:6,1:9)=[-21600/13649, 43200/13649*x1-7624/40947, -172800/13649*x1+ ...
             715489/81894, 259200/13649*x1-187917/13649, -172800/13649* ...
             x1+735635/81894, 43200/13649*x1-89387/40947, 0, 0, 0; ...
             -8640/12013*x1+7624/180195, 0, 86400/12013*x1-57139/12013, ...
             -172800/12013*x1+745733/72078, 129600/12013*x1-91715/12013, ...
             -34560/12013*x1+240569/120130, 0, 0, 0; 17280/2711*x1-715489/162660, -43200/2711*x1+57139/5422, 0, 86400/2711*x1-176839/8133, -86400/2711*x1+242111/10844, 25920/2711*x1-182261/27110, 0, 0, 0; -25920/5359*x1+187917/53590, 86400/5359*x1-745733/64308, -86400/5359*x1+176839/16077, 0, 43200/5359*x1-165041/32154, -17280/5359*x1+710473/321540, 72/5359, 0, 0; 34560/7877*x1-147127/47262, -129600/7877*x1+91715/7877, 172800/7877*x1-242111/15754, -86400/7877*x1+165041/23631, 0, 8640/7877*x1, -1296/7877, 144/7877, 0; -43200/43801*x1+89387/131403, 172800/43801*x1-240569/87602, -259200/43801*x1+182261/43801, 172800/43801*x1-710473/262806, -43200/43801*x1, 0, 32400/43801, -6480/43801, 720/43801];
    D1(m-5:m,m-8:m)=rot90( -D1(1:6,1:9),2);
    D1=D1/h;

    Q=H*D1 + 1/2*(e_1*e_1') - 1/2*(e_m*e_m');

    %D2=(2*diag(ones(m-3,1),3)-27*diag(ones(m-2,1),2)+270*diag(ones(m-1,1),1)+270*diag(ones(m-1,1),-1)-27*diag(ones(m-2,1),-2)+2*diag(ones(m-3,1),-3)-490*diag(ones(m,1),0))/180;
    diags   = -3:3;
    stencil = 1/180*[2,-27,270,-490,270,-27,2];
    D2 = stripeMatrix(stencil, diags, m);
    
    D2(1:6,1:9)=[114170/40947, -438107/54596, 336409/40947, -276997/81894, 3747/13649, 21035/163788, 0, 0, 0;6173/5860, -2066/879, 3283/1758, -303/293, 2111/3516, -601/4395, 0, 0, 0;-52391/81330, 134603/32532, -21982/2711, 112915/16266, -46969/16266, 30409/54220, 0, 0, 0;68603/321540, -12423/10718, 112915/32154, -75934/16077, 53369/21436, -54899/160770, 48/5359, 0, 0;-7053/39385, 86551/94524, -46969/23631, 53369/15754, -87904/23631, 820271/472620, -1296/7877, 96/7877, 0;21035/525612, -24641/131403, 30409/87602, -54899/131403, 820271/525612, -117600/43801, 64800/43801, -6480/43801, 480/43801];
    D2(m-5:m,m-8:m)=rot90( D2(1:6,1:9) ,2 );

    D2=D2/h^2;

    S_U=[-25/12, 4, -3, 4/3, -1/4]/h;
    S_1=sparse(1,m);
    S_1(1:5)=S_U;
    S_m=sparse(1,m);
    S_m(m-4:m)=fliplr(-S_U);


    M=-H*D2-e_1*S_1+e_m*S_m;
    S_1 = S_1';
    S_m = S_m';
end