# STOCHASTIC_DIFFUSION Stochastic Diffusivity

STOCHASTIC_DIFFUSION, a C 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).
```

### Languages:

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

### Related Data and Programs:

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

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

gnuplot_test, C 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 C 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 C 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 C 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.
Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos,
Computer Methods in Applied Mechanics and Engineering,
Volume 191, 2002, pages 4927-4948.

### 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.
• DIFFUSIVITY_2D_PWC: piecewise constant diffusivity function.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• R8_EPSILON returns the R8 round off unit.
• R8_MAX returns the maximum of two R8's.
• R8_UNIFORM_01 returns a pseudorandom R8 scaled to [0,1].
• R8MAT_MAX returns the maximum entry of an R8MAT.
• R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
• R8VEC_MAX returns the value of the maximum element in a R8VEC.
• R8VEC_MESH_2D creates a 2D mesh from X and Y vectors.
• R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
• R8VEC_NORMAL_01_NEW returns a unit pseudonormal R8VEC.
• R8VEC_PRINT prints an R8VEC.
• R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC.
• R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC.
• THETA_SOLVE solves a pair of transcendental equations.
• TIMESTAMP prints the current YMDHMS date as a time stamp.