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).
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 FORTRAN77 library which
implements some simple approaches to
the BlackScholes 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 OrnsteinUhlenbeck
stochastic differential equation (SDE) using the Euler method and
the EulerMaruyama 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:

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 10051034.

Howard Elman, Darran Furnaval,
Solving the stochastic steadystate diffusion problem using multigrid,
IMA Journal on Numerical Analysis,
Volume 27, Number 4, 2007, pages 675688.

Roger Ghanem, Pol Spanos,
Stochastic Finite Elements: A Spectral Approach,
Revised Edition,
Dover, 2003,
ISBN: 0486428184,
LC: TA347.F5.G56.

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 30843113, 2009.

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 23092345.

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 49274948.
Examples and Tests:
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.
Last modified on 05 August 2013.