I'm having a bit of trouble understanding the following, so it'd be great if anyone has any nice explanations! Thanks in advance!
Consider the hyperbolic system $$u_t = Au_x + Bu$$ where $A$ and $B$ are matrices. Why can't we write a stable and convergent finite difference scheme for this PDE if $A$ isn't diagonalizable?
As an aside, I know that, in order for this to be considered a hyperbolic system, $A$ needs to be diagonalizable with real eigenvalues. If $A$ doesn't meet one of those criteria, then the system is called weakly hyperbolic.
The system is weakly hyperbolic if the matrix $\xi_1 A + \xi_2 B$ has real eigenvalues for all $\xi = (\xi_1, \xi_2)$, but the matrix doesn't need to be diagonalizable. It is known that such systems may be ill-posed (1). Let us consider a trivial example with $u$ in $\Bbb R^2$, and with the matrices $$ A = \begin{pmatrix} 1& 1\\ 0& 1 \end{pmatrix} $$ and $B = 0$. We have: $$ \begin{aligned} {u_1}_t &= {u_1}_x + {u_2}_x \\ {u_2}_t &= {u_2}_x \end{aligned} $$ The Cauchy problem could be solved with the method of characteristics. Here, let us apply spatial Fourier transforms instead, which give \begin{aligned} {\hat u_1}_t &= \text i k\, ({\hat u_1} + {\hat u_2}) \\ {\hat u_2}_t &= \text i k\, {\hat u_2} \end{aligned} in Fourier domain ($\hat u = \int u\, e^{-\text i kx}\text d x$). Integrating these evolution equations, we have \begin{aligned} \hat u_1(k,t) &= \big(U_1(k)+ \text i k t\, U_2(k)\big)\, e^{\text i k t} \\ \hat u_2(k,t) &= U_2(k)\, e^{\text i k t} \end{aligned} where $U_i$ are the spatial Fourier transforms of $u_i$ at $t=0$. Now, the computation of the inverse transforms gives \begin{aligned} u_1(x,t) &= u_1(x+t, 0) + t {u_2}_{x}(x+t, 0)\\ u_2(x,t) &= u_2(x+t, 0) \end{aligned} which is unbounded as $t$ increases if $u_2$ isn't initially constant. One notes that the Cauchy problem is ill-posed, since an infinitesimal perturbation of the initial data may entail arbitrarily large error at time $t>0$.
(1) H.-O. Kreiss, O.E. Ortiz. "Some mathematical and numerical questions connected with first and second order time-dependent systems of partial differential equations." In: J. Frauendiener, H. Friedrich (eds) The Conformal Structure of Space-Time, Springer, 2002, p. 359-370. doi:10.1007/3-540-45818-2_19 arXiv:gr-qc/0106085