Just as the motion of a point, namely a trajectory, is modeled by an ODE, since time is the only independent variable involved, more complicated objects such fields, namely multi-variable maps into a vector space or some more complicated space, are modeled by PDEs. The examples are ubiquitous. Temperature and scalar potentials are real-valued fields. Electromagnetic waves and vector potentials are examples of vector-valued fields. The probability amplitude of a quantum particle is a complex-valued field. The gravitational waves generated by accelerated masses are tensor-valued fields. In this section we present several algorithms to solve PDEs numerically. This is a huge field and we can here just briefly present the most elementary ones. See the References for a few suggested texts with a much larger coverage of algorithms.
