conv_flux_brooks_corey_derivative.m

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function [flux, lambda] = conv_flux_brooks_corey_derivative(glob, U, params)
% function [flux, lambda] = conv_flux_brooks_corey_derivative(glob, U, params)
% convective flux for Buckley-Leverett problem with Brooks-Corey functions
%
% function computing the nonlinear convective flux of a Buckley-Leverett with Brooks-Corey approximation
% problem.
% ``\partial_u f(x,u) = \frac{\lambda_w'(u)}{\lambda_w(u)+\lambda_n(u)} + \frac{\lambda_w(u)(\lambda_w'(u)+\lambda_n'(u))}{\left(\lambda_w(u)+\lambda_n(u)\right)^2} \quad 0\leq u \leq 1``.
%
% ``\lambda_w(u) = \frac{u^{\frac{2+3\lambda}{\lambda}}}{\mu_1},``
% ``\lambda_n(u) = \frac{(1-u)^2 (1-u^{\frac{2+\lambda}{\lambda})}}{\mu_2},``
% 
% required fields of params:
%      bl_lambda               : mobility factor `\lambda`
%      bl_mu1                  : viscosity of wetting phase `\mu_1`
%      bl_mu2                  : viscosity of non-wetting phase `\mu_2`
%                                information
%
% See also conv_flux_brooks_corey().
%

% glob column check
if params.debug
  if ~isempty(glob) && size(glob,1) < size(glob,2)
    warning('coordinates in variable glob are given row-wise, but expected them to be column-wise');
    if params.debug > 2
      keyboard;
    end
  end
end

%X = glob(:,1);
%Y = glob(:,2);

ld       = params.bl_lambda;
mu1      = params.bl_mu1;
mu2      = params.bl_mu2;

lambda1  = U.^((2+3*ld)/ld)/mu1;
lambda2  = (1-U).^2.*(1-U.^(2/ld+1))/mu2;

mobility = lambda1 + lambda2;

lambda1d = U.^((2+2*ld)/ld)/mu1*(2/ld+3);
lambda2d = -1/mu2*(2*(1-U).*(1-U.^(2/ld+1)) + (1-U).^2.*(1-U.^(2/ld)*(2/ld+1)));

mobilityd = lambda1d + lambda2d;

flux = lambda1d ./ mobility + lambda1.*mobilityd./mobility.^2;

flux = [ flux, flux ] .* [ones(size(flux)), zeros(size(flux))];

lambda = 1/max(flux(:));

if params.debug && ( max(U)-eps > 1 || min(U) + eps < 0 )
  error('U is outside admissable bounds [0,1]');
end

if params.decomp_mode>0
  error('function is nonlinear and does not support affine decomposition!');
end