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view +sbp/+implementations/d4_lonely_8_higher_boundary_order.m @ 1031:2ef20d00b386 feature/advectionRV

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For easier comparison, return both the first order and residual viscosity when evaluating the residual. Add the first order and residual viscosity to the state of the RungekuttaRV time steppers
author Vidar Stiernström <vidar.stiernstrom@it.uu.se>
date Thu, 17 Jan 2019 10:25:06 +0100
parents bf801c3709be
children
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function [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_8_higher_boundary_order(m,h)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %%% 8:te ordn. SBP Finita differens         %%%
    %%% operatorer med diagonal norm            %%%
    %%%                                         %%%
    %%%                                         %%%
    %%% H           (Normen)                    %%%
    %%% D1=H^(-1)Q  (approx f?rsta derivatan)   %%%
    %%% D2          (approx andra derivatan)    %%%
    %%% D2=HI*(R+C*D*S                          %%%
    %%%                                         %%%
    %%% R=-D1'*H*C*D1-RR                        %%%
    %%%                                         %%%
    %%% RR ?r dissipation)                      %%%
    %%% Dissipationen uppbyggd av D4:           %%%
    %%% DI=D4*B*H*D4                            %%%
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %This is 3rd order accurate at the boundary. Not same norm as D1 operator

    BP = 8;
    if(m<2*BP)
        error(['Operator requires at least ' num2str(2*BP) ' grid points']);
    end

    % Norm
    Hv = ones(m,1);
    Hv(1:8) = [0.7488203e7/0.25401600e8, 0.5539027e7/0.3628800e7, 0.308923e6/0.1209600e7, 0.1307491e7/0.725760e6, 0.59407e5/0.145152e6, 0.1548947e7/0.1209600e7, 0.3347963e7/0.3628800e7, 0.25641187e8/0.25401600e8];
    Hv(m-7:m) = rot90(Hv(1:8),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(1:7) = [-0.49e2/0.20e2 6 -0.15e2/0.2e1 0.20e2/0.3e1 -0.15e2/0.4e1 0.6e1/0.5e1 -0.1e1/0.6e1]/h;
    d1_r = -rot90(d1_l, 2);

    d2_l = sparse(m,1);
    d2_l(1:7) = [0.203e3/0.45e2 -0.87e2/0.5e1 0.117e3/0.4e1 -0.254e3/0.9e1 0.33e2/0.2e1 -0.27e2/0.5e1 0.137e3/0.180e3]/h^2;
    d2_r = rot90(d2_l, 2);

    d3_l = sparse(m,1);
    d3_l(1:7) = [-0.49e2/0.8e1 29 -0.461e3/0.8e1 62 -0.307e3/0.8e1 13 -0.15e2/0.8e1]/h^3;
    d3_r = -rot90(d3_l, 2);



    % Fourth derivative, 1th order accurate at first 8 boundary points (still
    % yield 5th order convergence if stable: for example u_tt = -u_xxxx

    stencil = [-0.41e2/0.7560e4, 0.1261e4/0.15120e5, -0.541e3/0.840e3, 0.4369e4/0.1260e4, -0.1669e4/0.180e3, 0.1529e4/0.120e3, -0.1669e4/0.180e3, 0.4369e4/0.1260e4, -0.541e3/0.840e3, 0.1261e4/0.15120e5,-0.41e2/0.7560e4];
    diags = -5:5;
    M4 = stripeMatrix(stencil, diags, m);

    M4_U = [
        0.1031569831e10/0.155675520e9 -0.32874237931e11/0.1452971520e10 0.3069551773e10/0.90810720e8 -0.658395212131e12/0.21794572800e11 0.31068454007e11/0.1816214400e10 -0.39244130657e11/0.7264857600e10 0.1857767503e10/0.2724321600e10 0.1009939e7/0.49420800e8;
        -0.32874237931e11/0.1452971520e10 0.12799022387e11/0.155675520e9 -0.134456503627e12/0.1037836800e10 0.15366749479e11/0.129729600e9 -0.207640325549e12/0.3113510400e10 0.5396424073e10/0.259459200e9 -0.858079351e9/0.345945600e9 -0.19806607e8/0.170270100e9;
        0.3069551773e10/0.90810720e8 -0.134456503627e12/0.1037836800e10 0.6202056779e10/0.28828800e8 -0.210970327081e12/0.1037836800e10 0.2127730129e10/0.18532800e8 -0.4048692749e10/0.115315200e9 0.1025943959e10/0.259459200e9 0.71054663e8/0.290594304e9;
        -0.658395212131e12/0.21794572800e11 0.15366749479e11/0.129729600e9 -0.210970327081e12/0.1037836800e10 0.31025293213e11/0.155675520e9 -0.1147729001e10/0.9884160e7 0.1178067773e10/0.32432400e8 -0.13487255581e11/0.3113510400e10 -0.231082547e9/0.1816214400e10;
        0.31068454007e11/0.1816214400e10 -0.207640325549e12/0.3113510400e10 0.2127730129e10/0.18532800e8 -0.1147729001e10/0.9884160e7 0.11524865123e11/0.155675520e9 -0.29754506009e11/0.1037836800e10 0.14231221e8/0.2316600e7 -0.15030629699e11/0.21794572800e11;
        -0.39244130657e11/0.7264857600e10 0.5396424073e10/0.259459200e9 -0.4048692749e10/0.115315200e9 0.1178067773e10/0.32432400e8 -0.29754506009e11/0.1037836800e10 0.572247737e9/0.28828800e8 -0.11322059051e11/0.1037836800e10 0.3345834083e10/0.908107200e9;
        0.1857767503e10/0.2724321600e10 -0.858079351e9/0.345945600e9 0.1025943959e10/0.259459200e9 -0.13487255581e11/0.3113510400e10 0.14231221e8/0.2316600e7 -0.11322059051e11/0.1037836800e10 0.10478882597e11/0.778377600e9 -0.68446325191e11/0.7264857600e10;
        0.1009939e7/0.49420800e8 -0.19806607e8/0.170270100e9 0.71054663e8/0.290594304e9 -0.231082547e9/0.1816214400e10 -0.15030629699e11/0.21794572800e11 0.3345834083e10/0.908107200e9 -0.68446325191e11/0.7264857600e10 0.9944747557e10/0.778377600e9;
    ];

    M4(1:8,1:8) = M4_U;
    M4(m-7:m,m-7: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