function dc = diffusivity_2d_bnt ( dc0, omega, n, x, y ) %*****************************************************************************80 % %% DIFFUSIVITY_2D_BNT evaluates a 2D stochastic diffusivity function. % % Discussion: % % The 2D diffusion equation has the form % % - Del ( DC(X,Y) Del U(X,Y) ) = F(X,Y) % % where DC(X,Y) is a function called the diffusivity. % % In the stochastic version of the problem, the diffusivity function % includes the influence of stochastic parameters: % % - Del ( DC(X,Y;OMEGA) Del U(X,Y;OMEGA) ) = F(X,Y). % % In this function, the domain is the rectangle [-1.5,0]x[-0.4,0.8]. % % The four stochastic parameters OMEGA(1:4) are assumed to be independent % identically distributed random variables with mean value zero and % variance 1. The distribution is typically taken to be Gaussian or % uniform. % % A collocation approach to this problem would then use the roots of % Hermite or Legendre polynomials. % % Licensing: % % This code is distributed under the GNU LGPL license. % % Modified: % % 19 December 2009 % % Author: % % John Burkardt % % Reference: % % Ivo Babuska, Fabio Nobile, Raul Tempone, % A stochastic collocation method for elliptic partial differential equations % with random input data, % SIAM Journal on Numerical Analysis, % Volum 45, Number 3, 2007, pages 1005-1034. % % Parameters: % % Input, real DC0, the constant term in the expansion of the % diffusion coefficient. Take DC0 = 10. % % Input, real OMEGA(4), the stochastic parameters. % % Input, integer N, the number of evaluation points. % % Input, real X(N), Y(N), the points where the diffusion coefficient is to % be evaluated. % % Output, real DC(N), the value of the diffusion coefficient at (X,Y). % arg = omega(1) * cos ( pi * x ) ... + omega(2) * sin ( pi * x ) ... + omega(3) * cos ( pi * y ) ... + omega(4) * sin ( pi * y ); arg = exp ( - 0.125 ) * arg; dc = dc0 + exp ( arg ); return end