Matlab: How to write the nonlinear term using pdenonlin solver

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Fehaid Alshammari 2015-2-13

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I'm solving a nonlinear diffusion-advection equation in 2D spatial domain using pdenonlin . It seems straightforward to write the nonlinear term if it's of the form u^2, u^3, .. . The nonlinear term in my equation is of the form 2u/(4+u) . When I use char to pass this term to the f cofficient, the solutions don't seem right! I think I'm doing something wrong here. The equation I'm solving is u_xx+u_yy-2u/(4+u)=0 . What's wrong in my code? Thanks

    c = 1;
    a = 0;
    f = char('-2*u./(4+u)')
    d = 1; xmin=0;xmax=0.575;ymin=0;ymax=0.05315;ymax2=0.066; 
    gdm = [3;4;xmin;xmax;xmax;xmin;ymax;ymax2;ymin;ymin]; 
    g = decsg(gdm, 'S1', ('S1')');
    hmax = .1; % element size
    [p, e, t] = initmesh(g, 'Hmax', hmax);
    [p,e,t] = refinemesh(g,p,e,t);
    [p,e,t] = refinemesh(g,p,e,t);
    [p,e,t] = refinemesh(g,p,e,t);
    [p,e,t] = refinemesh(g,p,e,t);
    [p,e,t] = refinemesh(g,p,e,t);
     numberOfPDE = 1;
    pb = pde(numberOfPDE);
    % Create a geometry entity
    pg = pdeGeometryFromEdges(g);
    bc1 = pdeBoundaryConditions(pg.Edges(1),'u',100);
    bc2 = pdeBoundaryConditions(pg.Edges(2),'u',66);
    bc3 = pdeBoundaryConditions(pg.Edges(3),'u',11);
    bc4 = pdeBoundaryConditions(pg.Edges(4),'g',0);
    pb.BoundaryConditions = [bc1,bc2,bc3,bc4]; 
    u = pdenonlin(pb,p,e,t,c,a,f, 'jacobian', 'lumped');
    figure;
    hold on
    pdeplot(p, e, t, 'xydata', u, 'contour', 'off', 'colormap', 'jet(99)');
    title 'chemical Diffusion, Steady State Solution'
    xlabel 'X-coordinate, cm'
    ylabel 'Y-coordinate, cm'

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Richard Garner 2015-2-14

I think you want to treat your nonlinear term as an "acoeff*u" term. That is, you use a_coeff=1/(4+u). fcoeff should be zero. Now, that having been said, I don't know what happens if 4+u happens to be zero during the course of the calculation. But that would be an algorithm issue. Or perhaps an inherent mathematical, or even numerical, issue.

Fehaid Alshammari 2015-2-16