Editing Computational physics (section)

==Challenges in computational physics==

Computational physics problems are in general very difficult to solve exactly. This is due to several (mathematical) reasons: lack of algebraic and/or analytic solvability, [[complexity]], and chaos. For example, even apparently simple problems, such as calculating the [[wavefunction]] of an electron orbiting an atom in a strong [[electric field]] ([[Stark effect]]), may require great effort to formulate a practical algorithm (if one can be found); other cruder or brute-force techniques, such as [[graphical method]]s or [[root finding]], may be required. On the more advanced side, mathematical [[perturbation theory]] is also sometimes used (a working is shown for this particular example [[Perturbation theory#Example of degenerate perturbation theory – Stark effect in resonant rotating wave|here]]). In addition, the [[computational cost]] and [[computational complexity theory|computational complexity]] for [[many-body problem]]s (and their [[n-body problem|classical counterpart]]s) tend to grow quickly. A macroscopic system typically has a size of the order of <math>10^{23}</math> constituent particles, so it is somewhat of a problem. Solving quantum mechanical problems is generally of [[EXP|exponential order]] in the size of the system<ref>{{Cite journal|last=Feynman|first=Richard P.|author-link=Richard Feynman|date=1982|title=Simulating physics with computers|journal=International Journal of Theoretical Physics|language=en|volume=21|issue=6–7|pages=467–488|doi=10.1007/bf02650179|bibcode=1982IJTP...21..467F|s2cid=124545445|issn=0020-7748}} [https://web.archive.org/web/20170812065758/http://www.mrtc.mdh.se/~gdc/work/ARTICLES/2014/3-CiE-journal/Background/SimulatingPhysicsWithComputers.pdf Article PDF]</ref> and for classical N-body it is of order N-squared. Finally, many physical systems are inherently nonlinear at best, and at worst [[chaos theory|chaotic]]: this means it can be difficult to ensure any [[numerical error]]s do not grow to the point of rendering the 'solution' useless.<ref name="Sauer1997" />