Partial Differential Equations

A characteristic of partial differential equations (PDEs) is that the solution changes as a function of more than one independent variable. Usually these variables are time and one or more spatial coordinates. The numerical solution of a PDE therefore often requires the solution to be approximated not only in time as in ODEs, but in space as well. If all derivatives are approximated by finite differences at a finite number of points, a set of algebraic equations is obtained whose solution can be found using root solving algorithms. This is the common approach for solving time-independent PDEs. In contrast, PDEs which involve time as one of the independent variables are usually solved with the method of lines. In this case only spatial derivatives are discretised, while the time derivative is left as a continuous function. The result is a system of ODEs in time that can be solved with the initial value problem solvers from previous chapters. Typically, the dimension of the ODE or algebraic system is much larger than the number of components in the original partial differential equation.