The simplest example of hyperbolic PDE is the wave equation,
This is one-dimensional equation (we consider only one space variable x, let’s say, on the interval [0,1]),
but it is not stationary (we consider dependence on time t).
As initial conditions let’s give, for instance, 
and as a boundary condition let’s give 
The wave equation is useful as a test problem, as the exact solution is known for this equation (for each initial and boundary condition).
For initial and boundary conditions given above the exact solution is the following
The convection equations may be considered as a sample of linear hyperbolic PDE. The equations governing unsteady inviscid flow are also hyperbolic (but nonlinear), as well as equations governing steady supersonic inviscid flow.
In general, hyperbolic equations are typical for the problems of transferring, when nothing appears or disappears in the calculation region (just redistributed there).
While solving hyperbolic equations using numeric methods, usually iterations on time t are applied.
As soon as initial condition (at t=0) is given, each next iteration at tj = tj-1 + ∆t uses solution at tj-1 as an initial condition for the next iteration step. Thus, the problem is reduced to the sequence of simpler problems defined only for space coordinates.
Such an approach has its own advantages and disadvantages. Advantage is in lower dimension of the problem (this simplifies calculations). Disadvantage is that the calculations are becoming principally consecutive. This interferes parallelization of calculations.
Were you had a strong solver, ready for parallelization, you would prefer to do calculation on a grid including both space variables and time variable. This is where our Marlin Solver might demonstrate its strength.