STOCHASTIC_DIFFUSION
Stochastic Diffusivity


STOCHASTIC_DIFFUSION is a FORTRAN90 library which implement several versions of a stochastic diffusivity coefficient, using GNUPLOT to create graphic images of sample realizations of the diffusivity field.

The 1D diffusion equation has the form

        - d/dx ( DC(X) d/dx U(X) ) = F(X).
      
where DC(X) is a function called the diffusivity and F(X) is called the source term or forcing term.

In the 1D stochastic version of the problem, the diffusivity function includes the influence of stochastic parameters:

        - d/dx ( DC(X;OMEGA) d/dx U(X;OMEGA) ) = F(X).
      

The 2D diffusion equation has the form

        - Del ( DC(X,Y) Del U(X,Y) ) = F(X,Y).
      

In the 2D 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).
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

STOCHASTIC_DIFFUSION is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BLACK_SCHOLES, a FORTRAN90 library which implements some simple approaches to the Black-Scholes option valuation theory;

CORRELATION, a FORTRAN90 library which contains examples of statistical correlation functions.

GNUPLOT, FORTRAN90 programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

ORNSTEIN_UHLENBECK, a FORTRAN90 library which approximates solutions of the Ornstein-Uhlenbeck stochastic differential equation (SDE) using the Euler method and the Euler-Maruyama method.

PCE_ODE_HERMITE, a FORTRAN90 program which sets up a simple scalar ODE for exponential decay with an uncertain decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.

SDE, a FORTRAN90 library which illustrates the properties of stochastic differential equations, and common algorithms for their analysis, by Desmond Higham;

Reference:

  1. Ivo Babuska, Fabio Nobile, Raul Tempone,
    A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data,
    SIAM Journal on Numerical Analysis,
    Volume 45, Number 3, 2007, pages 1005-1034.
  2. Howard Elman, Darran Furnaval,
    Solving the stochastic steady-state diffusion problem using multigrid,
    IMA Journal on Numerical Analysis,
    Volume 27, Number 4, 2007, pages 675-688.
  3. Roger Ghanem, Pol Spanos,
    Stochastic Finite Elements: A Spectral Approach,
    Revised Edition,
    Dover, 2003,
    ISBN: 0486428184,
    LC: TA347.F5.G56.
  4. Xiang Ma, Nicholas Zabaras,
    An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations,
    Journal of Computational Physics,
    Volume 228, pages 3084-3113, 2009.
  5. Fabio Nobile, Raul Tempone, Clayton Webster,
    A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data,
    SIAM Journal on Numerical Analysis,
    Volume 46, Number 5, 2008, pages 2309-2345.
  6. Dongbin Xiu, George Karniadakis,
    Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos,
    Computer Methods in Applied Mechanics and Engineering,
    Volume 191, 2002, pages 4927-4948.

Examples and Tests:

The test program makes a set of command and data files that can be used by GNUPLOT to create graphic images.

List of Routines:

You can go up one level to the FORTRAN90 source codes.


Last modified on 22 July 2013.