In this book we discuss a class of

In this book we discuss a class of time discretization techniques, which are
termed strong stability preserving (SSP) time discretizations and have been
developed over the past 20 years as an effective tool for solving certain types
of large ordinary differential equation (ODE) systems arising from spatial
discretizations of partial differential equations (PDEs).
The numerical solution of ODEs is of course an established research area.
There are many well-studied methods, such as Runge–Kutta methods and
multistep methods. There are also many excellent books on this subject,
for example [8, 64, 83]. ODE solvers for problems with special stability
properties, such as stiff ODEs, are also well studied, see, e.g., [32, 17].
The SSP methods are also ODE solvers, therefore they can be analyzed
by standard ODE tools. However, these methods were designed specifically
for solving the ODEs coming from a spatial discretization of time-dependent
PDEs, especially hyperbolic PDEs. The analysis of SSP methods can be
facilitated when this background is taken into consideration.
In the one-dimensional scalar case, a hyperbolic conservation law is
given by
ut + f(u)x = 0 (1.1)
where u is a function of x and t, and the subscripts refer to partial derivatives;
for example, ut = ∂u
∂t . Hyperbolic PDEs pose particular difficulties
for numerical methods because their solutions typically contain discontinuities.
We refer the readers to, e.g. the book of Smoller [94] and the lecture
notes of LeVeque [68] for an introduction to hyperbolic conservation laws
and their numerical solutions. We would like to find a numerical approximation,
still denoted by u with an abuse of notation, which discretizes the
spatial derivative (e.g. f(u)x in (1.1)) in the PDE. The result is then a