# 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.

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### 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.