Finite Difference solution to nonlinear diffusion equation 1.0 
Date Added: July 11, 2013  Visits: 267 


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 CrankNicolson implicit finite difference scheme to solve the equation numerically.Matlab run commandtype:IsoFreeSurfaceSolversolves the pde: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[1] "Difference Methods for initial value problems," R. D. Richtmyer and K. W. Morton, John Wiley and Sons, 1967.[2] "The Propagation of twodimensional 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.

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