### 9.7 The collocation approach

When solving a quasilinear evolution equation
using spectral expansions in space and some time evolution scheme, one could proceed in the following way:
Work with the truncated expansion of ,
and write the PDE (9.86) as a system of coupled evolution ordinary differential equations for the
coefficients, subject to the initial condition
where the quadratures can be approximated using, say, Gauss-type points. One problem with this approach
is that if the equation is nonlinear or already at the linear level with variable coefficients, the right-hand
side of Eq. (9.86) needs to be re-expressed in terms of truncated expansions at each timestep. Besides the
complexity of doing so, accuracy is lost because higher-order modes due to nonlinearities or coupling with
variable coefficients are not represented. Or, worse, inaccuracies from those absent modes move to lower
frequency ones. This is one of the reasons why collocation methods are instead usually preferred for this
class of problems.
In the collocation approach the differential equation is exactly solved, in physical space, at the
collocation points , which are those appearing in Gauss quadratures (Section 9.4). Assume
for definiteness that we are dealing with a symmetric hyperbolic system in three dimensions,

see Section 3.3. We approximate by its discrete truncated expansion
Then the system is solved at the collocation points,
where the spatial derivatives are approximated using spectral differentiation as described in
Section 9.6. The system can then be evolved in time using the preferred time integration method; see
Section 7.3.
From an implementation perspective, there is actually very little difference between a spectral
collocation method and a FD one: the only two being that the grid points need to be Gauss-type ones
and that the derivative is computed using global interpolation at those points. In fact, the
actual projection (9.90) never needs to be computed for actually solving the system (9.91): given
initial data , by construction the interpolant coincides with it at the nodal points,

and the system (9.91) for is directly numerically evolved subject to the initial condition
(9.92).

##### Stability.

As discussed in Section 9.4, in the Legendre case the discrete truncated expansion using
Gauss-type quadratures leads to SBP. In analogy with the FD case (Section 8.3), when the
continuum system can be shown to be well posed through the energy method, a semi-discrete energy
estimate can be shown by using the SBP property, at least for constant coefficient systems, and
modulo boundary conditions (discussed in the following Section 10). Consider the same case
discussed in Section 8.4, a constant coefficient symmetric hyperbolic system in one dimension,
and a collocation approach
at Legendre–Gauss-type nodes. Then, defining
and taking a time derivative, as in Section 8.4,
Now, is a polynomial of degree , so the above discrete scalar product is exact for
Gauss, Gauss–Lobatto and Gauss–Radau collocation points [cf. Eq. (9.75)]. Therefore, we obtain the
energy equality
and numerical stability follows modulo boundary conditions. For the case of a symmetric-hyperbolic system
with variable coefficients and lower-order terms, one obtains an energy estimate using skew-symmetric
differencing; see Eq. (8.50), and appropriate boundary conditions.
The weighted norm case is more involved. In fact, already the advection equation is not well
posed under the Chebyshev norm; see, for example, [237].

##### Spectral viscosity.

In analogy with numerical dissipation (Section 8.5), spectral viscosity (SV) adds a resolution-dependent
dissipation term to the evolution equations without sacrificing spectral convergence. SV was introduced by
Tadmor in [408]. For simplicity, consider the Fourier case. Then spectral viscosity involves adding to the
evolution equations a dissipative term of the form
where is the (fixed) order of viscosity, the viscosity amplitude scales as
and the smoothing functions effectively apply the viscosity to only the upper portion of the spectrum.
In more detail, if are the Fourier coefficients [cf. Eq. (9.6)] of , then they are only
applied to frequencies in such a way that they satisfy
with
The case corresponds to a dissipation term involving a second derivative and is
referred to as super (or hyper) viscosity. Higher values of (up to ) dissipate ‘less
aggressively’.
The Legendre and Chebyshev cases are similar and are discussed in [291, 292]. The webpage [406]
keeps a selected list of publications on spectral viscosity.