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Remove default scaling of the system. The scaling doens't seem to help actual solutions. One example that fails in the flexural code. With large timesteps the solutions seems to blow up. One particular example is profilePresentation on the tdb_presentation_figures branch with k = 0.0005
author Jonatan Werpers <jonatan@werpers.com>
date Wed, 18 Jul 2018 15:42:52 -0700
parents f7ac3cd6eeaa
children
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function [H, HI, D1, D2, D3, D4, e_1, e_m, M, ...
    M4,Q, Q3, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = d4_4(m,h)


    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %%% 4:de ordn. SBP Finita differens         %%%
    %%% operatorer framtagna av Ken Mattsson    %%%
    %%%                                         %%%
    %%% 6 randpunkter, diagonal norm            %%%
    %%%                                         %%%
    %%% Datum: 2013-11-11                       %%%
    %%%                                         %%%
    %%%                                         %%%
    %%% H           (Normen)                    %%%
    %%% D1          (approx f?rsta derivatan)   %%%
    %%% D2          (approx andra derivatan)    %%%
    %%% D3          (approx tredje derivatan)   %%%
    %%% D2          (approx fj?rde derivatan)   %%%
    %%%                                         %%%
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    % M?ste ange antal punkter (m) och stegl?ngd (h)
    % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r
    % vi har 3de och 4de derivator i v?r PDE
    % I annat fall anv?nd de "traditionella" som har noggrannare
    % randsplutningar f?r D1 och D2

    % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm,
    % vilket ?r n?dv?ndigt f?r stabilitet
    
    BP = 6;
    if(m<2*BP)
        error(['Operator requires at least ' num2str(2*BP) ' grid points']);
    end

    H=speye(m,m);
    H_U=[0.35809e5 / 0.100800e6 0 0 0 0 0; 0 0.13297e5 / 0.11200e5 0 0 0 0; 0 0 0.5701e4 / 0.5600e4 0 0 0; 0 0 0 0.45109e5 / 0.50400e5 0 0; 0 0 0 0 0.35191e5 / 0.33600e5 0; 0 0 0 0 0 0.33503e5 / 0.33600e5;];

    H(1:6,1:6)=H_U;
    H(m-5:m,m-5:m)=rot90(H_U,2);
    H=H*h;
    HI=inv(H);


    % First derivative SBP operator, 1st order accurate at first 6 boundary points

%     q2=-1/12;q1=8/12;
%     Q=q2*(diag(ones(m-2,1),2) - diag(ones(m-2,1),-2))+q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
    e=ones(m,1);
    Q=spdiags([e -8*e 0*e 8*e -e], -2:2, m, m)/12;

    %Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2));

    Q_U = [0 0.526249e6 / 0.907200e6 -0.10819e5 / 0.777600e6 -0.50767e5 / 0.907200e6 -0.631e3 / 0.28800e5 0.91e2 / 0.7776e4; -0.526249e6 / 0.907200e6 0 0.1421209e7 / 0.2721600e7 0.16657e5 / 0.201600e6 -0.8467e4 / 0.453600e6 -0.33059e5 / 0.5443200e7; 0.10819e5 / 0.777600e6 -0.1421209e7 / 0.2721600e7 0 0.631187e6 / 0.1360800e7 0.400139e6 / 0.5443200e7 -0.8789e4 / 0.302400e6; 0.50767e5 / 0.907200e6 -0.16657e5 / 0.201600e6 -0.631187e6 / 0.1360800e7 0 0.496403e6 / 0.907200e6 -0.308533e6 / 0.5443200e7; 0.631e3 / 0.28800e5 0.8467e4 / 0.453600e6 -0.400139e6 / 0.5443200e7 -0.496403e6 / 0.907200e6 0 0.1805647e7 / 0.2721600e7; -0.91e2 / 0.7776e4 0.33059e5 / 0.5443200e7 0.8789e4 / 0.302400e6 0.308533e6 / 0.5443200e7 -0.1805647e7 / 0.2721600e7 0;];
    Q(1:6,1:6)=Q_U;
    Q(m-5:m,m-5:m)=rot90(  -Q_U ,2 );

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


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

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



    % Second derivative, 1st order accurate at first 6 boundary points
%     m2=1/12;m1=-16/12;m0=30/12;
%     M=m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);
    %M=(1/12*diag(ones(m-2,1),2)-16/12*diag(ones(m-1,1),1)-16/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2)+30/12*diag(ones(m,1),0));
    M=-spdiags([-e 16*e -30*e 16*e -e], -2:2, m, m)/12;
    M_U=[0.2386127e7 / 0.2177280e7 -0.515449e6 / 0.453600e6 -0.10781e5 / 0.777600e6 0.61567e5 / 0.1360800e7 0.6817e4 / 0.403200e6 -0.1069e4 / 0.136080e6; -0.515449e6 / 0.453600e6 0.4756039e7 / 0.2177280e7 -0.1270009e7 / 0.1360800e7 -0.3751e4 / 0.28800e5 0.3067e4 / 0.680400e6 0.119459e6 / 0.10886400e8; -0.10781e5 / 0.777600e6 -0.1270009e7 / 0.1360800e7 0.111623e6 / 0.60480e5 -0.555587e6 / 0.680400e6 -0.551339e6 / 0.5443200e7 0.8789e4 / 0.453600e6; 0.61567e5 / 0.1360800e7 -0.3751e4 / 0.28800e5 -0.555587e6 / 0.680400e6 0.1025327e7 / 0.544320e6 -0.464003e6 / 0.453600e6 0.222133e6 / 0.5443200e7; 0.6817e4 / 0.403200e6 0.3067e4 / 0.680400e6 -0.551339e6 / 0.5443200e7 -0.464003e6 / 0.453600e6 0.5074159e7 / 0.2177280e7 -0.1784047e7 / 0.1360800e7; -0.1069e4 / 0.136080e6 0.119459e6 / 0.10886400e8 0.8789e4 / 0.453600e6 0.222133e6 / 0.5443200e7 -0.1784047e7 / 0.1360800e7 0.1812749e7 / 0.725760e6;];

    M(1:6,1:6)=M_U;

    M(m-5:m,m-5:m)=rot90(  M_U ,2 );
    M=M/h;

    S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h;
    S_1=sparse(1,m);
    S_1(1:4)=S_U;
    S_m=sparse(1,m);

    S_m(m-3:m)=fliplr(-S_U);

    D2=H\(-M-e_1*S_1+e_m*S_m);


    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



    % Third derivative, 1st order accurate at first 6 boundary points

    q3=-1/8;q2=1;q1=-13/8;
%     Q3=q3*(diag(ones(m-3,1),3)-diag(ones(m-3,1),-3))+q2*(diag(ones(m-2,1),2)-diag(ones(m-2,1),-2))+q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1));
    diags   = -3:3;
    stencil = [-q3,-q2,-q1,0,q1,q2,q3];
    Q3 = stripeMatrix(stencil, diags, m);

    %QQ3=(-1/8*diag(ones(m-3,1),3) + 1*diag(ones(m-2,1),2) - 13/8*diag(ones(m-1,1),1) +13/8*diag(ones(m-1,1),-1) -1*diag(ones(m-2,1),-2) + 1/8*diag(ones(m-3,1),-3));


    Q3_U = [0 -0.88471e5 / 0.67200e5 0.58139e5 / 0.33600e5 -0.1167e4 / 0.2800e4 -0.89e2 / 0.11200e5 0.7e1 / 0.640e3; 0.88471e5 / 0.67200e5 0 -0.43723e5 / 0.16800e5 0.46783e5 / 0.33600e5 -0.191e3 / 0.3200e4 -0.1567e4 / 0.33600e5; -0.58139e5 / 0.33600e5 0.43723e5 / 0.16800e5 0 -0.4049e4 / 0.2400e4 0.29083e5 / 0.33600e5 -0.71e2 / 0.1400e4; 0.1167e4 / 0.2800e4 -0.46783e5 / 0.33600e5 0.4049e4 / 0.2400e4 0 -0.8591e4 / 0.5600e4 0.10613e5 / 0.11200e5; 0.89e2 / 0.11200e5 0.191e3 / 0.3200e4 -0.29083e5 / 0.33600e5 0.8591e4 / 0.5600e4 0 -0.108271e6 / 0.67200e5; -0.7e1 / 0.640e3 0.1567e4 / 0.33600e5 0.71e2 / 0.1400e4 -0.10613e5 / 0.11200e5 0.108271e6 / 0.67200e5 0;];

    Q3(1:6,1:6)=Q3_U;
    Q3(m-5:m,m-5:m)=rot90(  -Q3_U ,2 );
    Q3=Q3/h^2;



    S2_U=[2 -5 4 -1;]/h^2;
    S2_1=sparse(1,m);
    S2_1(1:4)=S2_U;
    S2_m=sparse(1,m);
    S2_m(m-3:m)=fliplr(S2_U);



    D3=H\(Q3 - e_1*S2_1 + e_m*S2_m +1/2*(S_1'*S_1) -1/2*(S_m'*S_m) ) ;

    % Fourth derivative, 0th order accurate at first 6 boundary points (still
    % yield 4th order convergence if stable: for example u_tt=-u_xxxx

    m3=-1/6;m2=2;m1=-13/2;m0=28/3;
%     M4=m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);
    diags   = -3:3;
    left_stencil = [m3,m2,m1];
    stencil = [left_stencil,m0,fliplr(left_stencil)];
    M4 = stripeMatrix(stencil, diags, m);

    %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0));

    M4_U=[0.4596181e7 / 0.1814400e7 -0.10307743e8 / 0.1814400e7 0.160961e6 / 0.43200e5 -0.535019e6 / 0.907200e6 0.109057e6 / 0.1814400e7 -0.29273e5 / 0.604800e6; -0.10307743e8 / 0.1814400e7 0.8368543e7 / 0.604800e6 -0.9558943e7 / 0.907200e6 0.2177057e7 / 0.907200e6 -0.11351e5 / 0.86400e5 0.204257e6 / 0.1814400e7; 0.160961e6 / 0.43200e5 -0.9558943e7 / 0.907200e6 0.4938581e7 / 0.453600e6 -0.786473e6 / 0.151200e6 0.1141057e7 / 0.907200e6 -0.120619e6 / 0.907200e6; -0.535019e6 / 0.907200e6 0.2177057e7 / 0.907200e6 -0.786473e6 / 0.151200e6 0.3146581e7 / 0.453600e6 -0.4614143e7 / 0.907200e6 0.24587e5 / 0.14400e5; 0.109057e6 / 0.1814400e7 -0.11351e5 / 0.86400e5 0.1141057e7 / 0.907200e6 -0.4614143e7 / 0.907200e6 0.185709e6 / 0.22400e5 -0.11293343e8 / 0.1814400e7; -0.29273e5 / 0.604800e6 0.204257e6 / 0.1814400e7 -0.120619e6 / 0.907200e6 0.24587e5 / 0.14400e5 -0.11293343e8 / 0.1814400e7 0.16787381e8 / 0.1814400e7;];

    M4(1:6,1:6)=M4_U;

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

    S3_U=[-1 3 -3 1;]/h^3;
    S3_1=sparse(1,m);
    S3_1(1:4)=S3_U;
    S3_m=sparse(1,m);
    S3_m(m-3:m)=fliplr(-S3_U);

    D4=H\(M4-e_1*S3_1+e_m*S3_m  + S_1'*S2_1-S_m'*S2_m);


    % L=h*(m-1);
    %
    % x1=linspace(0,L,m)';
    % x2=x1.^2/fac(2);
    % x3=x1.^3/fac(3);
    % x4=x1.^4/fac(4);
    % x5=x1.^5/fac(5);
    %
    % x0=x1.^0/fac(1);

    S_1  = S_1';
    S2_1 = S2_1';
    S3_1 = S3_1';
    S_m  = S_m';
    S2_m = S2_m';
    S3_m = S3_m';



end