I'm describing some work on my website, and I'm wondering if my math modeling and computer simulation work is described ok: I say
math modeling and numerical simulation.
Should I say "simulation" and leave out the "numerical" part since it's understood?
What about "math modeling"? I see some people use the term "computational modeling" or "numerical modeling".
The obvious answer is "it depends". However, it's not helpful.
I would certainly separate the work in mathematical modeling and actual numerical simulation. Sometimes it might be a bit tough to draw the line in between, but I think it's usually possible. Thus, by using work in mathematical modeling and numerical simulation does not seem to be redundant and actually captures the right keywords.
However, if you need a very short sentence, using a "blanket" phrase, the actual name of this site computational sciences might work. Since it is pretty much what we, computational scientists, do: we perform numerical simulations of mathematical models, usually describing some practical phenomena.
In more detail:
this involves my own interpretation and sepration of the terms, as well as a particular descriptive example that I am familiar with.
mathematical modeling, I would use this term to address the part of your work that was focused on the analysis, derivation, and improvement of the mathematical description of the model describing some system or a process. This would be also aligned with a Wikipedia definition
numerical simulation, performing a simulation using certain numerical methods, usually (nowadays, all the time) using a computer system. That starts from taking a certain mathematical model, evaluating its feasibility for the numerical simulation (problem setup, problem size, computational resources).
For example, you start from Maxwell's equations (which describe a physical phenomenon on some level) and apply them to specific conditions. In particular, say, a surface electric-field integral-equation formulation for magneto-quasi-statics. The formulation derivation (under certain assumption) is the part of the work corresponding to the mathematical modeling.
Now, to validate this formulation and show its advantages on a practical example (inductance extraction), you solve it numerically (finite differences, finite element method, boundary element method, etc). This part would correspond to the numerical simulation aspect.
For me, there is a clear hierarchy going from reality to a simulation. The first step is to understand reality as much as you can and propose a model for this reality, typically without formally writing down equations. You define what physical/chemical/biological/... processes are involved. Already in this step, you introduce an error: you can never model the whole reality (you set boundaries to your model) and never to the tiniest details. The outcome of this first step is what I would call a conceptual model.
In the second step, you try to derive or apply existing mathematical models (i.e. equations) that govern the physics/chemistry/biology/... of your conceptual model. This again introduces errors because the mathematical model will not always capture the complete physics/chemistry/biology.
In a third step, you apply algorithms to solve your mathematical model (which can be any combination of PDEs, ODEs, non-linear equations, optimization, ...). And, of course, also here you introduce errors since you typically discretize time, space, energy, ...
A comparison between reality and the conceptual model is sometimes called "model qualification". This is typically done in language and by means of expert judgement: "Do we need to take this effect into account?" "Is this parameter important?". A technique used for this is the PIRT (Phenomena Identification and Ranking Table) approach.
A comparison between the mathematical equations and standard solutions (often called manufactured solutions) is called verification (you verify that the algorithms solve the mathematical equations adequately). Verification tells you nothing about how to solutions of your mathematical model represent reality but it tells you if your numerical algorithms are up to the job (for example, solving a stiff set of ODEs with an explicit method would be a bad choice).
In order to claim that your model has predictive capacity (i.e. it is capable of predicting reality), you need validation experiments where you compare experimental data to predictions from your simulations. This is called "model validation".
A very good report to read about this topic is Verification, Validation, and Predictive Capability in Computational Engineering and Physics by Oberkampf, Trucano and Hirsch.
So yes, for me there is a clear distinction between "math modeling" and "numerical simulation".
