PDEPE nonlinear

Question

I would like to use Matlab's pdepe to solve this system:

$$ s_t =(sr)_x + s_{ xx } \\ r_t =(\frac{ A }{ B }r^2+s)_x + \frac{ A }{ -K } r_{ xx } $$

where $A$, $B$ and $K$ are constants. I need to solve this using MATLAB pdepe. The problem is the nonlinear term in my equation: I don't know how to treat the nonlinear part. Boundary and initial conditions are not a problem in that part.

So I've tried to write my PDE in the form that pdepe expects: $$\left( \begin{array}{c} 1 \\ 1 \\ \end{array}\right) \times \frac{ \partial }{ \partial t }\left( \begin{array}{c} s \\ r \\ \end{array}\right)=\left( \begin{array}{c} s\frac{ \partial r }{ \partial x } + r\frac{ \partial s }{ \partial x } \\ 2\frac{A}{B}r\frac{ \partial r }{ \partial x } +\frac{ \partial s }{ \partial x }\\ \end{array}\right)+\frac{ \partial }{ \partial x }\left( \begin{array}{c} \frac{ \partial s }{ \partial x } \\ \frac{A}{-K}\frac{ \partial r }{ \partial x } \\ \end{array}\right)+\left( \begin{array}{c} 0 \\ 0 \\ \end{array}\right) $$

In MATLAB:

function [c,f,s] = eqtn(x,t,u,DuDx)

c = 1;

f = ????

s = 0;

How to code the function $f$?

Answer 1

The most straightforward way to write the eqtn function is to define the nonlinear terms as part of the returned s vector as follows:

f = [DuDx(1); -A/K*DuDx(2)];
s = [u(1)*DuDx(2)+u(2)*DuDx(1); 2*A/B*u(2)*DuDx(2)+DuDx(1)];

The question that immediately comes to mind is which terms are appropriate to include in the f vector compared to s?

The PDE system for many physical problems is derived based on the notion of some quantity being conserved over time. This directly leads to the idea of a "flux" which is what the f vector represents. That is, f has a well-defined meaning in the physical problem.

The definition of f is particularly important in defining boundary conditions. Boundary conditions in pdepe take the form

p + q*f=0

at the right and left ends of the domain. If the boundary condition has a non-zero q, its value depends directly on how f is defined.