# frozen coefficient vs. constant coefficient

This is a follow up to the question about the method of frozen coefficients. I think it deserves to be a separate question. The frozen coefficient problems are obtained by fixing the coefficients of the variable coefficient problem. For example, consider pde $$u_t=a(x)u_{xx}, x \in \mathbb{R}, u(0,x)=u_0(x)$$, then at every grid point $x_i$ we can have a frozen problem $$u_t=a(x_i)u_{xx}$$. However, even though it is a frozen problem, in a sense that if I write a finite difference scheme, then at every grid point the diffusion coefficient is constant, but this is still quite different from considering the problem $$u_t=au_{xx}, x \in \mathbb{R}, u(0,x)=u_0(x)$$ where $a$ is just a constant number. So, for the latter case, to compute the amplification factor of a finite-difference scheme, I need to write the equation in terms of Fourier transform only once at some grid point $x_i$, but if I freeze the coefficients, I have to consider that at every single node and recompute the amplification factor at each of them. So, if this is the way, a problem with frozen coefficients and a problem with constant coefficient are not equivalent, the constant coefficient problem is rather a subproblem of the frozen coefficient problem, is that correct?

And another related question: if I complicate things more and let $a=a(t,x)$, then by the definition I can't use the Fourier analysis, however, if I freeze the coefficients, it is perfectly applicable? Do I have to take care in a special way of the time-dependent coefficients? I am looking for some intuition behind the method of frozen coefficient. It seems to make everything much simpler but I don't want to abuse that and know my limitations.

• This is also what I want to know about Local Fourier Analysis for analysis of PDE (e.g. pseudo-differential operator in Absorbing Boundary Condition), multigrid and domain decomposition. In many cases, it just works and I really do not know why. Once I saw some slides that some researchers aimed to rigorous LFA for multigrid but still not heard news up to now. – Hui Zhang Apr 15 '13 at 9:37