Algorithmic Parameters Should Be Seen And Not Heard
A mathematical model that describes a physical process has parameters that measure the qualities of that process. These physical parameters take a range of values, the extent of which is set by the complexity and intention of the model. The model may be designed to reproduce acute behaviour or a broad variety of phenomena.
A solution of the model often requires an algorithm which can give an approximate solution while introducing non-physical parameters that I'll call algorithmic parameters. One example is the convergence criterion often denoted tolerance which measures how approximate the solution can be. Others might be smoothness, spatial resolution and maximum time-step.
While varying the physical parameters should lead to different solutions each of which demonstrates some plausible behaviour of the physical process; the values of the algorithmic parameters should be beyond some critical set of values such that they have no effect on the solution. In my experience this is not as easy as it sounds. Sometimes practical reasons, such as computational time and storage, prevent you from running the algorithm beyond the critical boundaries, in the safe space of meaningful physical solutions.
I have no stunning resolution to this predicament except the kind written on mugs and cereal boxes. That is try and keep trying but at some point compromise and interpret the solutions to the degree they enable you to.
Alastair Clarke
10 August, 2019