# Monotonicity of Errors with Respect to Step Sizes in Numerical Methods for PDEs

Consider a non-linear partial differential equation (PDE) (e.g., Burgers' equation) that is solved numerically using a finite difference method (or a similar approach). Suppose a grid search is performed on the spatial and temporal step sizes.

Are there scenarios where a larger step size can produce a more accurate and stable solution compared to a smaller step size? Under what conditions, if any, can we expect the errors in the numerical solution to be non-monotonically increasing with larger step sizes?

I would appreciate any references, theoretical results, or practical experiences related to this question.

• How do you measure the "error"? Apr 10 at 0:58
• At some point, reducing the grid step size will have a detrimental effect, e.g., in evaluating $(f_{i+1}-f_{i-1})/\Delta$ you'd be subtracting one number from another, very close to it, so the loss of accuracy will be inevitable. So eventually the approximation error will start growing with reducing the grid step size. Apr 15 at 20:28

Based on practical experience, it's extremely rare that the error is a strictly monotone function of mesh size. Sure, if you pick a few points you are likely to see monotone behavior, but if you keep zooming in eventually you will find an exception.

If you want to specifically construct schemes with non-monotone error behavior, you could do that by basing them on approximation bases that are more accurate in certain regions than others, rather than the traditional approach of basing them on matching terms in a Taylor series.

• Thank you for your answer. 1) If we perform a dense study on error vs mesh size and observe a non-monotone error behavior, then plot whether the errors exceed some threshold, do you think we will get some fractal plot? Something like this: arxiv.org/html/2402.06184v1. 2) Why would one want to specifically construct a scheme with non-monotone error behavior? Apr 17 at 11:25
• Comments aren't the best place for further questions, but here goes: 1. I have no idea; but I can't see an obvious reason why it would be fractal. 2. You are the only person I know who is interested in strict monotonicity of the error, so if you don't want to do that then perhaps nobody does. Apr 17 at 15:55

More stable is in my opinion not a very clear term with regards to your question. For the time integration of a semi-discrete system, less spatially refined discretisation tends to be more stable (see e.g. the largest eigenvalue of the discrete Laplacian which scales with the square of the number of points) when integrated with an explicit scheme, but this may not affect well chosen implicit schemes.

Regarding the error, I know that some differential-algebraic equations may produce a "hump" phenomena where the error temporarily increases for some variables as the time step is decreased. See e.g. Hairer & Wanner "Solving Ordinary Differential Equations II". For ODEs producing a marginally unstable dynamics, it is also possible that there exists a threshold step size above which an implict time integration scheme stabilises the solution, thus the error would be relatively independent of the step size in that range.

For PDEs, it may also depend on how you measure the error. I often have seen that if I measure a local error(e.g. at one point in space), it may change sign at one point in the convergence study, thus actually crossing 0 for a specific discretisation, and later converging to 0 for finer discretisations. In that case, computing a global space-error or an average error on multiple points usually restores the correct monotonous behaviour.

• I definitely second that last paragraph. It's quite a common observation. Apr 10 at 0:59