Physicists often have a very precise mathematical theory describing exactly how a system will operate. Unfortunately, it is often the case that solving these equations in order to produce a useful prediction is a computationally difficult problem. This is especially true with quantum mechanics, where only a handful of simple models can be solved exactly. Even apparently simple problems, such as calculating the wavefunction of an electron orbiting an atom in a strong electric field, may require great effort to formulate a practical algorithm.
Many other more general numerical problems fall loosely under the domain of computational physics, although they could easily be considered pure mathematics or part of any number of applied areas. For example:
- Solving differential equations
- Evaluating integrals
- Stochastic methods, specifically the Monte Carlo Method
- Specialised partial differential equation methods, for example the finite difference method and the finite element method
- The matrix eigenvalue problem - i.e. the problem of finding eigenvalues of very large matrices.