How to Solve a Highly Non-Linear PDE in Matlab

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Valeri Vardanyan
Valeri Vardanyan on 30 Jul 2017
Hi all,
coming from the theoretical physics community I am not very much familiar with the numerical PDE and ODE solving capabilities of MATLAB.
I have a heavily non-linear elliptic PDE of the following form:
In the left hand side, it's the Laplacian in spherical coordinates; I am also interested in solving this equation in the case of beyond spherical symmetry.
In the right hand side, the prefactor outside the parenthesis is a constant number, n>=1, rho_c is another constant, rho(x) is a given function of x.
I need to impose boundary conditions in the center (the first derivative = 0), and very far away from the center (phi = some given number).
I have three questions:
1. how realistic is it for MATLAB to solve the given equation in 1D? Which function is it the best to use.
2. If I have a stiff system, namely rho_c is very-very much larger than the typical value of rho, is there any chance that MATLAB can handle this?
3. Moving to more complicated geometries, is this a difficult problem for methods like FEM?
I mainly want to know if the given equation is an easy one for engineering standards and whether it can be solved with MATLAB.
Thanks in advance.

Answers (2)

Precise Simulation
Precise Simulation on 4 Aug 2017
You should fairly easily be able to enter this into the FEATool Multiphysics FEM toolbox as a custom PDE , for example the following code
fea.sdim = { 'x' };
fea.grid = linegrid( 10, 0, 1 );
fea.dvar = { 'phi' };
fea.sfun = { 'sflag1' };
fea.eqn = parseeqn( 'phix_x + 2/x*phix_t = mc^2*Rc^2/(n+1)*(rho/rhoc-1/phi^(n+1))', fea.dvar, fea.sdim );
fea.bdr.d = { [] 1 };
fea.bdr.n = { 0 [] };
fea.const = { 'rho' '1+x' ;
'rhoc' 1 ;
'n' 1 ;
'mc' 1 ;
'Rc' 1 };
fea = parseprob( fea );
fea.sol.u = solvestat( fea, 'init', {1} );
postplot( fea, 'surfexpr', 'phi' )
should set up your problem with arbitrary test coefficients. Whether your actual problem is too nonlinear to converge is another issue though.

Torsten
Torsten on 31 Jul 2017
Edited: Torsten on 31 Jul 2017
Try "pdepe" with m=2.
It uses the method-of-lines approach based on a FEM discretization with a stiff ODE integrator in the background.
Best wishes
Torsten.

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