# Should reaction be taken into account in the CFL condition when solving advection-diffusion-reaction equations numerically?

I'm dealing with the numerical resolution of advection-diffusion-reaction equations.

I've encountered references on CFL conditions for conservation laws as well as advection-diffusion equations.
I'm wondering if any references exist on CFL conditions involving reaction as well. I don't have any preferences on the numerical method - my curiosity is general.

CFL is a necessary requirement for stability in transport-dominated systems, but in practice, is an intuitive way to interpret linear stability. A 1D advection-diffusion (or somewhat upwinded advection) operator gives the marked eigenvalues $$\lambda \Delta t$$, while the Forward Euler method gives the stability region (stable in the blue region, unstable in red).

When you change the time step $$\Delta t$$, you scale the marked points (relative to the origin). To make the method stable, you need to choose $$\Delta t$$ such that each $$\lambda \Delta t$$ falls inside the stable (blue) region. In general, you can also choose a different integrator (RK4, etc.) such that its stable region encloses the eigenvalues.

When you add reaction, you're going to move these eigenvalues. You can examine the Jacobian of the reaction mechanism (in general, these are multi-species) to see what shift is being applied. In practice, some reaction mechanism are very stiff (leading to orders of magnitude smaller time step than the advection-diffusion system).

You should define your accuracy metrics and plot the spectrum (or an approximation, such as Ritz (Krylov) estimates) for each part (advection, diffusion, and reaction). If the max stable time step is near your target accuracy, consider the problem as non-stiff and use an explicit method. By plotting the spectrum, you can see if you need to choose a different explicit integrator (e.g., Forward Euler has no stability on the imaginary axis). If the max stable time step is orders of magnitude smaller than would be needed for accuracy, consider using an implicit method.

The CFL condition refers strictly to purely hyperbolic problems where there is a well-defined domain of dependence for the analytical solution. It state that a necessary condition for stability is that the discrete domain of dependence defined by the stencil must include the exact domain of dependence. This is scarcely a mathematical criterion but a common-sense criterion; you cannot obtain a true solution without access to all of the data that it depends on. When you add other effects, like diffusion or reaction, there usually is not a straightforward geometrical interpretation. And since, even for hyperbolic problems the CFL rule is merely a necessary condition, you always have to do the algebra.

The CFL criterion is derived for convective problems usually. You may also get a similar criterion for diffusion problems (although the grid size is squared in the formula).

For the reaction however, there is no direct relation between the reaction process and your space discretisation. Stability of the numerical integration of the reaction terms is usually only dependent on the time step and on the integration method used. You may have a look into the definition of stiffness for ODEs if you want to dig further (for instance with the Robertson problem). The space discretisation (e.g. mesh size) does not have any influence. You may have a look at splitting approaches, where the advection and diffusion terms are integrated with an explicit method with a time step dictated by a CFL-criterion, whereas reactions are integrated with higly-stable implicit time schemes.