STOCHASTIC_DIFFUSION
Stochastic Diffusivity

STOCHASTIC_DIFFUSION is a FORTRAN77 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).
      

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 FORTRAN77 library which implements some simple approaches to the Black-Scholes option valuation theory;

CORRELATION, a FORTRAN77 library which contains examples of statistical correlation functions.

GNUPLOT, FORTRAN77 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 FORTRAN77 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 FORTRAN77 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 FORTRAN77 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.

• stochastic_diffusion.f, the source code.
• stochastic_diffusion.sh, BASH commands to compile the source code.

Examples and Tests:

• stochastic_diffusion_prb.f, a sample calling program.
• stochastic_diffusion_prb.sh, BASH commands to compile and run the sample program.
• stochastic_diffusion_prb_output.txt, the output file.

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

• bnt_data.txt, the graphics data file.
• bnt_commands.txt, the GNUPLOT command file.
• bnt_contour.png, a plot of the diffusivity as a function of X and Y.
• elman_data.txt, the graphics data file.
• elman_commands.txt, the GNUPLOT command file.
• elman_contour.png, a plot of the diffusivity as a function of X and Y.
• ntw_data.txt, the graphics data file.
• ntw_commands.txt, the GNUPLOT command file.
• ntw_contour.png, a plot of the diffusivity as a function of X and Y.
• xk_data.txt, the graphics data file.
• xk_commands.txt, the GNUPLOT command file.
• xk_contour.png, a plot of the diffusivity as a function of X.

List of Routines:

• DIFFUSIVITY_1D_XK evaluates a 1D stochastic diffusivity function.
• DIFFUSIVITY_2D_BNT evaluates a 2D stochastic diffusivity function.
• DIFFUSIVITY_2D_ELMAN evaluates a 2D stochastic diffusivity function.
• DIFFUSIVITY_2D_NTW evaluates a 2D stochastic diffusivity function.
• GET_UNIT returns a free FORTRAN unit number.
• R8_EPSILON returns the R8 roundoff unit.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_MAX returns the maximum entry of an R8MAT.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• R8VEC_MAX returns the maximum value in an R8VEC.
• R8VEC_MESH_2D creates a 2D mesh from X and Y vectors.
• R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• THETA_SOLVE solves a pair of transcendental equations.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN77 source codes.