# Finite Difference solution to nonlinear diffusion equation (Scripts) 1.0

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## Finite Difference solution to nonlinear diffusion equation (Scripts) Publisher's description

### Solves nonlinear diffusion equation which can be linearised as shown for the general nonlinear diffusion equation in Richtmyer & Morton [1]

Solves nonlinear diffusion equation which can be linearised as shown for the general nonlinear diffusion equation in Richtmyer & Morton [1]. The approach is to linearise the pde and apply a Crank-Nicolson implicit finite difference scheme to solve the equation numerically.

Matlab run command
--------------------------

type:
IsoFreeSurfaceSolver

solves the pde:
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frac{partial h}{partial t}=frac{1}{12}frac{partial^2 h^4}{partial x^2}

The pde has the application to isothermal viscous fluid flow spreading on a horizonatal substrate under gravity - Huppert [2]. Note that PDE has been nondimensionlised.

Initial conditions:

t=0: h = (1 - x^2)_{+} + 10^-6 (has prewetting film)

The spatial domain is discretised taking account of symmetry at x = 0:

frac{partial h}{partial x} = 0

and is also used to close the free surface model at the end of the discretised domain (x = xend) - assumes fluid won't reach this boundary.

The pde can be solved analytically using similarity equations as discussed by Barenblatt [3]; used below to validate numerical solution.

References
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[1] "Difference Methods for initial value problems," R. D. Richtmyer and K. W. Morton, John Wiley and Sons, 1967.

[2] "The Propagation of two-dimensional and axisymmetric visous gravity currents over a rigid horizontal surface," H. E. Huppert, Journal of Fluid Mechanics, 1982.

[3] "On some unsteady motions of a liquid or gas in a porous medium, Russian journal Prikladnaya Matematika i Mekhanika, 1952.

#### System Requirements:

MATLAB 7.9 (2009b)
Program Release Status: New Release
Program Install Support: Install and Uninstall

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