## 2 The Theory of Linear Constant Coefficients Evolution Equations and Generalizations to
Quasi-linear Systems

In this section, I summarize the main results of the theory of first order evolutionary partial differential
equation systems. I do this by first developing the theory of linear constant coefficients evolution equation
systems in , that is, equations of the type:

where indicates a “vector” valued function of dimension in , its time
derivative, , a matrix whose components smoothly depend on: . For most of
the results, no particular form for the dependence of on is needed, as long as it is continuous. But
for simplicity one can think of as given by:
We
shall focus on the Cauchy (or initial value) Problem for the above system, namely under what conditions it
is true that given the value of at , , say, there exists a unique solution, , to the
above system with . Later we shall mention a related problem which is important on most
numerical schemes used in relativity, namely the initial-boundary value problem, where one also prescribes
some data on time-like boundaries.
What follows is a short account of Chapter II of [48], see also Chapter IV of [36]. After this, I indicate
what aspects of the theory generalize to quasi-linear systems, and under which further assumptions this is
so. I also give some indications of the relation of this theory to the stability issues of numerical simulations.
This section can be skipped by those not interested in the mathematical theory itself or those who already
know it.