# STOCHASTIC_HEAT2D 2D Steady Heat Equation with Stochastic Diffusivity

STOCHASTIC_HEAT2D, a MATLAB program which solves the steady state heat equation in a 2D rectangular region with a stochastic heat diffusivity, using the finite difference method (FDM), and stochastic model from Babuska, Nobile and Tempone.

The physical region, and the boundary conditions, are suggested by this diagram:

```                   U = 0, Y = 1.0
+-------------------+
|          ...      |
|         .:*:.     |
U = 0    |          :::      | U = 0
X = 0.0  |                   | X = 1.0
|                   |
+-------------------+
U = 0, Y = 0.0
```

A heat source is applied, which is concentrated at the point X = 0.6, Y = 0.8, whose center is suggested by the "*" in the diagram.

The region is covered with a grid of NX by NY nodes, and an NX by NY array U is used to record the temperature. The correspondence between array indices and locations in the region is suggested by giving the indices of the four corners:

```                  I = NY
+------------------+
|                  |
J = 1  |                  |  J = NX
|                  |
+------------------+
I = 1
```

The form of the steady heat equation is

```        - d/dx K(x,y) du/dx - d/dy K(x,y) du/dy = F(x,y)
```
where K(x,y) is the heat conductivity, and F(x,y) is a heat source term. For this program, the heat conductivity function is assumed to be a function of four stochastic parameters OMEGA(1) through OMEGA(4); the form of the functional dependence is described in a paper by Babusk, Nobile and Tempone.

By using a simple finite difference approximation, this single equation can be replaced by NX * NY linear equations in NX * NY variables; each equation is associated with one of the nodes in the mesh. Nodes long the boundary generate boundary condition equations, while interior nodes generate equations that approximate the steady heat equation.

The linear system is sparse, and can easily be solved directly in MATLAB.

Choosing a particular set of values for OMEGA produces a single "realization" of the temperature field. However, an interesting question is to observe the influence of the OMEGA's on the solution U. As an example, the program fixes the values of OMEGA(3) and OMEGA(4), evaluates the solution U over a grid of values for OMEGA(1) and OMEGA(2), and plots the mean of the solution.

### Languages:

STOCHASTIC_HEAT2D is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

FD2D_HEAT_STEADY, a MATLAB program which uses the finite difference method to solve the 2D Time Independent Heat Equations.

FEM_50_HEAT, a MATLAB program which implements a finite element calculation specifically for the heat equation.

FEM1D_HEAT_STEADY, a MATLAB program which uses the finite element method to solve the 1D Time Independent Heat Equations.

FEM2D_HEAT, a MATLAB program which solves the 2D time dependent heat equation on the unit square.

HEATED_PLATE, a MATLAB program which solves the steady state heat equation in a 2D rectangular region, and is intended as a starting point for a parallel version.

### 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.

### Source Code:

• boundary.m, specifies the portion of the system matrix and right hand side associated with boundary nodes.
• diffusivity_2d_bnt.m, evaluates the stochastic heat diffusivity function.
• interior.m, specifies the portion of the system matrix and right hand side associated with interior nodes.
• r8vec_normal_01.m, returns a vector of normally distributed random numbers.
• r8vec_uniform_01.m, returns a vector of uniformly distributed random numbers.
• stochastic_heat2d.m, sets up and solves the linear system for the stochastic 2D steady heat equation.
• timestamp.m, prints the YMDHMS date as a timestamp.

Last revised on 07 February 2019.