Numerical solution of partial differential equations

Numerical solution of partial differential equations is the computational task of finding approximate answers to partial differential equations, the equations that relate a quantity to its rates of change across several variables such as space and time, when an exact formula is not practical. It is studied as a branch of numerical analysis, with specialized methods for hyperbolic, parabolic and elliptic equations. The usual approach replaces the continuous problem with a finite one. The finite difference method stores values at grid points and approximates derivatives by differences between them. The method of lines discretizes every dimension but one, so that solvers built for ordinary differential equations can be reused. Finite element, finite volume and spectral methods instead represent the solution through small elements, fluxes across small volumes, or global basis functions. Domain decomposition and multigrid methods then speed up solving the very large systems these produce. The source regards finite difference as the simplest to learn, finite element and finite volume as well suited to complicated geometries, and spectral methods as the most accurate when the solution is smooth.