# Estimating the spectral radius when applying the method of lines

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              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            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?

• What is the PDE in question, and what are you using the spectral radius for? Commented Mar 27 at 7:42
• The PDE is diffusion in cylindrical coordinates, i.e. $\partial_t u = (1/r)\partial_r(r D(u) \partial_r u)$. AFAIK, the spectral radius helps the adaptive RKC integrator to determine the next step size or the number of stages needed. Commented Mar 27 at 11:20
• If the PDE was $\partial_t u = \partial_{xx}u + f(x,t)$ discretized via the standard second-order finite difference, the spectral radius estimate would be $4/h^2$, where $h$ is the spatial step. This can be inferred easily from applying the Gershgorin Circle Theorem to the Jacobian matrix of the semi-discrete system of ODEs that follows from the method of lines. Commented Mar 27 at 15:15
• You can try to derive a step size such that the off-diagonals are non-negative also in your case. But that depends on the discretisation. Some discretisation do not allow for non-negative off diagonals regardless of the step. Commented Mar 27 at 16:04
• Maybe you can use Gershgorin's theorem on one of the line of the Jacobian that involves the smallest $h$. This would roughly give you something like $4/h_{min}^2$. This may however be too conservative. Maybe you can try this, and see how the spectral radius internally determined by RKC behaves compared to this estimate, and may also actually compute the true spectral radius on some specific solution fields ? Commented Mar 28 at 9:30

ROCK4 is similar to RKC. Its authors use a modified power method to estimate the spectral radius automatically when it is not provided : see https://www.math.pku.edu.cn/teachers/litj/notes/numer_anal/SISC_23_2041.pdf

Since ROCK4 is highly efficient for the solution of diffusive problems on non-uniform meshes, I guess this estimate is very well suited.

• That's what RKC does too I believe. I'm looking for more analytical means or rules I can use in combination with the Gershgorin theorem. Commented Sep 26, 2023 at 15:46