Let \[D_r = \{z\in\mathbf{C} : |z+r|\le r\}.\]

The Jeltsch-Nevanlinna theorem (BIT, 1986) states that if \(S\) is the stability region of a one-step method and \(D_r\subset S\), then \(r\le m\) (in fact something a bit stronger is stated, and it was later generalized to multistep methods). The proof is by purely algebraic means.

An alternative, simpler proof is possible by using the CFL theorem. Consider the advection equation \[u_t + u_x = 0\] with periodic boundary conditions semi-discretized by first-order upwind differencing: \[U_i'(t) = -\frac{1}{\Delta x} \left(U_i-U_{i-1}\right).\] This has the form \(U'=L_{\Delta x}U\) where \(L_{\Delta x}\) is a circulant matrix whose eigenvalues all lie on the boundary of the disk \(D_{1/\Delta x}\).

Now discretize in time with a consistent one-step, \(m\)-stage method. Suppose that \(rD_{1/\Delta x} = D_{r/\Delta x} \subset S\), where \(S\) is the stability region of the method. Then the method is stable (hence convergent) for \(\Delta t = r\Delta x\). But the CFL condition states that it cannot be convergent for \(\Delta t > m\Delta x .\) So \(r\le m\).

This is essentially the proof given by Sanz Serna and Spijker (Numerische Mathematik, 1986). Their Theorem 5 is a much stronger generalization.

Other discretizations

Of course, one could apply the reasoning above to any consistent discretization of the advection equation. Letting \(\hat{D}\) denote the region enclosed by the spectrum of a given method, and letting \(q\) denote its stencil width (in the upwind direction only), the result obtained is as follows: If \(r\hat{D}\subset S\), then \(r\le qm\).

Of particular interest are centered schemes with an asymptotically vanishing diffusive term added: \[U_i'(t) = -\frac{U_{i+1}-U_{i-1}}{\Delta x} + \epsilon\Delta x \frac{U_{i-1}-2U_i + U_{i+1}}{\Delta x ^2}.\]

The eigenvalues of this semidiscretization are (see e.g. LeVeque p. 207): \[\lambda = \frac{2\epsilon}{\Delta x}(\cos(\theta)-1) - \frac{i}{\Delta x}\sin(\theta).\]

This leads immediately to Theorem 5 of Sanz Serna & Spijker.

Parabolic problems

For parabolic problems, the CFL condition only implies (for explicit methods) that \(\Delta t \le r\Delta x^\alpha\) where \(\alpha>1\). But all methods I know of have stability restrictions of the form \(\Delta t\le r \Delta x^2\). If one could prove the latter necessary, this would give an alternative route to showing that the optimal polynomials for imaginary axis inclusion include an interval proportional to \(m^2\).