view +sbp/+implementations/d4_lonely_4_min_boundary_points.m @ 1198:2924b3a9b921 feature/d2_compatible

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 b19e142fcae1
children
line wrap: on
line source

function [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_4_min_boundary_points(m,h)
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %%% 4:de ordn. SBP Finita differens         %%%
    %%% operatorer framtagna av Mark Carpenter  %%%
    %%%                                         %%%
    %%% H           (Normen)                    %%%
    %%% D1=H^(-1)Q  (approx f?rsta derivatan)   %%%
    %%% D2          (approx andra derivatan)    %%%
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %H?r med endast 4 randpunkter


    BP = 4;
    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, 2);

    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
    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');

    % 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=[
        0.8e1/0.3e1 -0.37e2/0.6e1 0.13e2/0.3e1 -0.5e1/0.6e1;
        -0.37e2/0.6e1 0.47e2/0.3e1 -13 0.11e2/0.3e1;
        0.13e2/0.3e1 -13 0.44e2/0.3e1 -0.47e2/0.6e1;
        -0.5e1/0.6e1 0.11e2/0.3e1 -0.47e2/0.6e1 0.29e2/0.3e1;
    ];


    M4(1:4,1:4) = M4_U;
    M4(m-3:m,m-3: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