9.7 The collocation approach

When solving a quasilinear evolution equation
ut = P(t,x,u, ∂∕∂x )u + F(t,x,u ) (9.86 )
using spectral expansions in space and some time evolution scheme, one could proceed in the following way: Work with the truncated expansion of u (t,x ),
∑N 𝒫N [u ](t,x) = bi(t)pi(x ), (9.87 ) i=0
and write the PDE (9.86View Equation) as a system of (N + 1) coupled evolution ordinary differential equations for the bi(t) coefficients, subject to the initial condition
b (t = 0) = ⟨u (t = 0,⋅),p ⟩ , (9.88 ) i i ω
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. (9.86View Equation) 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 N {xi}i=0, 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,

3 ∑ j -∂-- ut(t,x ) = A (t,x, u)∂xj u + F(t,x,u ), (9.89 ) j=1
see Section 3.3. We approximate u by its discrete truncated expansion
uN := 𝒫dN[u]. (9.90 )
Then the system is solved at the collocation points,
∑3 -duN (t,xi) = Aj (t,xi,uN )-∂--uN (t,xi) + F (t, xi,uN ), (9.91 ) dt j=1 ∂xj
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 (9.90View Equation) never needs to be computed for actually solving the system (9.91View Equation): given initial data u(t = 0,x), by construction the interpolant coincides with it at the nodal points,

uN(t = 0,xi) = uN (t = 0,xi), (9.92 )
and the system (9.91View Equation) for uN (t,xi) is directly numerically evolved subject to the initial condition (9.92View Equation).

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,

ut = Aux, A = AT , (9.93 )
and a collocation approach
-d u (t,x ) = A ∂--u (t,x ), (9.94 ) dt N i ∂x N i
at Legendre–Gauss-type nodes. Then, defining
Ed = ⟨uN ,uN ⟩dω=1, (9.95 )
and taking a time derivative, as in Section 8.4,
⟨ ⟩d ⟨ ⟩d ⟨ ⟩d ⟨ ⟩d dEd- = d-u ,u + u ,-d u = -∂-Au ,u + Au , ∂-u . (9.96 ) dt dt N N ω=1 N dt N ω=1 ∂x N N ω=1 N ∂x N ω=1
Now, uN ∂uN ∕∂x is a polynomial of degree (2N − 1), so the above discrete scalar product is exact for Gauss, Gauss–Lobatto and Gauss–Radau collocation points [cf. Eq. (9.75View Equation)]. Therefore, we obtain the energy equality
dEd [ T ]b ---- = (AuN ) uN a, (9.97 ) dt
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. (8.50View Equation), and appropriate boundary conditions.

The weighted norm case ω ⁄= 1 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

[ ] d- s-∂s- ∂suN- dtuN = (...) − 𝜖N(− 1) ∂xs Qm (t,x) ∂xs , s ≥ 1, (9.98 )
where s is the (fixed) order of viscosity, the viscosity amplitude scales as
Cs 𝜖N = N-2s−1, Cs > 0, (9.99 )
and the smoothing functions Qm effectively apply the viscosity to only the upper portion of the spectrum. In more detail, if ˆQj(t) are the Fourier coefficients [cf. Eq. (9.6View Equation)] of Qm (t,x ), then they are only applied to frequencies j > mN in such a way that they satisfy
( ) (2s− 1)∕𝜃 1 − mN-- ≤ ˆQj(t) ≤ 1 (9.100 ) |j|
𝜃 2s − 1 mN ∼ N 𝜃 < --2s--. (9.101 )
The case s = 1 corresponds to a dissipation term involving a second derivative and s > 1 is referred to as super (or hyper) viscosity. Higher values of s (up to √ --- s ∼ N) 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.

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