### 4.4 Fully-discrete analysis

Stability issues have been discussed separately for time (Section 4.1.2) and space (Section 4.3)
discretizations. The global picture (fully discrete analysis), taking into account both discretizations is, in
general, very difficult to study. However, it is possible in some particular cases to address the problem and
in the following lines, we shall perform a fully discrete analysis of the advection equation (169), using a
Legendre collocation method in space and a forward Euler scheme in time. Using the notation of
Section 4.1
where the last term imposes the boundary condition . We consider this relation at
the Legendre–Gauss collocation points , which are zeros of ; the square of this
expression taken at these collocation points gives
We
multiply by , where are the Legendre–Gauss weights, and sum over to obtain
For stability we require that a certain discrete energy of be bounded in time:
which means that
With
the exactness of the Legendre–Gauss quadrature rule for polynomials of degree lower than , we
have
and,
with an additional integration by parts,
The stability condition obtained from energy analysis translates into an upper bound for the
timestep, which can be seen as an accurate estimate of the CFL restriction on the timestep:
#### 4.4.1 Strong stability-preserving methods

The above fully-discrete analysis must, in principle, be performed for every time-marching scheme.
Therefore, it is very convenient to have a way of extending the results from the first-order Euler
method to higher-order methods. Strong stability-preserving Runge–Kutta and multistep methods
preserve these kinds of stability properties, including those following from nonlinear stability
analysis. A general review of the subject has been done by Shu [200], and we list some results
here.

If we consider the general time ODE:

arising from the spatial discretization of the PDE (116), we suppose that, after discretization in time using
the first-order forward Euler scheme, the strong stability requirement gives a CFL
restriction of the type (176)
We can then write an -stage Runge–Kutta method in the form
and see that, as long as and , all of the intermediate stages are simply convex
combinations of forward Euler operators. If this method is strongly stable for , under the
condition (178), then the intermediate stages can be bounded and the Runge–Kutta scheme is stable under
the CFL condition
In the same manner, one can devise strong stability-preserving explicit multistep methods of the
form

which can also be cast into convex combinations of forward Euler steps and, therefore, these multistep
methods are also stable, provided that
Examples of useful coefficients for Runge–Kutta and multistep strong stability-preserving methods can be
found in [200, 117]. The best of these methods are those for which the CFL coefficient is as large as
possible.