Some time integrators, notably the Runge-Kutta-Chebyshev method, implemented in the RKC code from Sommeijer & Verwer, gives the user an option to provide a callback with an estimate of the spectral radius of the Jacobian matrix, $J = dF/dy$.
(I'm not aware if the Julia version of RKC in DifferentialEquations.jl also provides this option.)
The description of this option is,
c INFO(2): RKC needs an estimate of the spectral radius of the Jacobian. c You must provide a function that must be called SPCRAD and c have the form c c double precision function spcrad(neqn,t,y) c integer neqn c double precision t,y(neqn) c c spcrad = < expression depending on info(2) > c c return c end c c You can provide a dummy function and let RKC compute the c estimate. Sometimes it is convenient for you to compute in c SPCRAD a reasonably close upper bound on the spectral radius, c using, e.g., Gershgorin's theorem. This may be faster and/or c more reliable than having RKC compute one.
When using the method of lines (MOL) with a finite-difference or finite-volume discretization of a PDE with constant coefficients, estimating the spectral radius using Gershgorin's theorem is feasible. However for meshes with unequal intervals, time- or space-varying coefficients, or even problems with cylindrical or spherical symmetry, this becomes difficult in practice.
Are there any tricks or other properties of the PDE (i.e. the maximum principle) that one can apply to find an estimate of the spectral radius?