We are concerned here with fully-discrete spectral/pseudospectral approximations to initial-boundary value problems associated with hyperbolic equations. In this context, the spectral (and respectively, the pseudospectral) approximations consist of truncation (and, respectively, collocation) of N-term spatial expansions, which are expressed in terms of general Jacobi polynomials; Chebyshev and Legendre expansions are the ones most frequently found in practice. We will show that such N-term approximations are stable, provided their time step, , fulfills the CFL-like condition, .
To clarify the origin of such a CFL-like condition in our case, we recall that the Jacobi polynomials are in fact the eigenfunctions of second-order singular Sturm-Liouville problems. Our arguments show that the main reason for the above CFL limitation is the growth of the Nth eigenvalue associated with these Sturm-Liouville problems.
We start with the scalar constant-coefficient hyperbolic equation,
which is augmented with homogeneous conditions at the inflow boundary,
To approximate (meth_cheb.1), we use forward Euler time-differencing on the left, and either spectral or dospectral differencing on the right. Thus, we seek a temporal sequence of spatial -polynomials, , such that
Here, is a -polynomial which characterizes the specific (pseudo)spectral method we employ, v' denotes spatial differentiation, and is a free scalar multiplier to be determined by the boundary constraint
We shall study the so called spectral tau method associated with
general Jacobi polynomials ,
. The generality of our spectral formulation includes as a special case, the dospectral Jacobi methods which are collocated at the interior extrema of , i.e.,
Indeed, the spectral and dospectral Jacobi methods are closely related since is a scalar multiple of . For example, and correspond to Chebyshev spectral and psidospectral methods, respectively.
Let be the N distinct zeros
of the forcing polynomial .
For Jacobi type methods,
(meth_cheb.4) and (meth_cheb.5), the
nodes are the zeros of Jacobi polynomials associated
with the Gauss and Gauss-Lobatto quadrature rules, with minimal gridsize
The spectral approximation (meth_cheb.3a) restricted to these points reads
and is augmented with the homogeneous boundary conditions
Equations (meth_cheb.7a), (meth_cheb.7b) furnish a complete equivalent formulation of the spectral approximation (meth_cheb.3a), (meth_cheb.3b). An essential ingredient in a stability theory of such approximations lies in the choice of appropriate -weighted norms
We now make the definition of
. We say the approximation (meth_cheb.7a),
(meth_cheb.7b) is stable
if there exist discrete weights,
, and a constant independent of N, such that
and it is strongly stable if (meth_cheb.9) holds with Const = 1 and ,
With this in mind we turn to our main stability result stating
Theorem 4.1 deals with the
stability of both the spectral tau methods
associated with , and the closely related dospectral methods
associated with .
In each case, there are (at least two) different weighted stability results,
different choices of discrete -weighted norms; these discrete weights
are given by
.25in 2. . The CFL condition (meth_cheb.12) places an stability restriction on the time step . Indeed, this stability restriction involves two factors : the eigenvalues associated with Jacobi equation (2.4.9),
and the collocated Gauss nodes, which accumulate within neighborhoods near the boundaries,
Thus, the CFL condition (meth_cheb.12) boils down to
(For the practical range of parameters, , we have ).
3. . The stability statement
asserted in theorem (4.1) is formulated
in terms of discrete
seminorms, , which are -weighted by either
(meth_cheb.14a) or (meth_cheb.14b).
We note that are in fact
well-defined norms on the space of -polynomials satisfying the
vanishing boundary condition (meth_cheb.7b), i.e.,
corresponding to (meth_cheb.14a) or (meth_cheb.14b)
and in view of (2.5.16),
Moreover, in view of (meth_cheb.15b), one may convert the
stability statement (meth_cheb.13)
into the usual -type stability estimate at the expense of possible
algebraic growth which reads
4. . Let us integrate by parts the
differential equation (meth_cheb.1) against (1 + x)u.
Thanks to the homogeneous
boundary condition (meth_cheb.2) we find
This estimate corresponds to the special case of the stability statement (meth_cheb.13) for the spectral Legendre tau method () weighted by (meth_cheb.14b). The exponential time decay indicated in (meth_cheb.20), and more generally in (meth_cheb.13), is due to the special choice of -weighted stability norms. The weights in (meth_cheb.14a), (meth_cheb.14b) involve the essential factors or which amplify the inflow boundary values in comparison to the outflow ones. Since in the current homogeneous case, vanishing inflow data is propagating into the domain, this results in the exponential time decay indicated in (meth_cheb.20) and likewise in the stability statement (meth_cheb.13).
5. . A stability statement similar to theorem 4.1 is valid in the inflow case where a < 0. Assume that the CFL condition (meth_cheb.12) holds with , then (meth_cheb.13) follows with discrete weights or .
As we noted before, there are several variants of theorem 4.1; we quote below two of these variants.
The spectral Jacobi method (meth_cheb.4) satisfies the
stability estimate (meth_cheb.13) with
we proceed as follows. Squaring of (meth_cheb.7a) yields
and we turn to estimate the two expressions, I and II, on the right of (meth_cheb.22).
First let us note that since the -polynomial
vanishes at the inflow boundary, (meth_cheb.3b), we have
Also, a straightforward computation shows that
where is given in (meth_cheb.21b).
Now, since , the Gauss quadrature rule (Gauss.rule) implies
We integrate by parts the right-hand side of I, substitute from (meth_cheb.23), and in view of (meth_cheb.24) we obtain
Next, let us consider the second expression, II, on the right of (meth_cheb.22). As before, we substitute from (meth_cheb.23) and obtain
To proceed we invoke the following
In fact, one more application of Gauss quadrature yields
The inequalities (meth_cheb.29), (meth_cheb.28) together with (meth_cheb.27) imply
and the result (meth_cheb.13) follows.
Since is proportional to , we conclude the stability of the dospectral method (meth_cheb.5), with and .
As mentioned before, alternative variants of theorem 4.1 are possible. For example, one may employ a stable norm weighted by (instead of the weights used before. This yields the
The spectral Jacobi tau method (meth_cheb.4).
satisfies the stability estimate (meth_cheb.13) with
we omit the detailed derivation (-- which as before, hinges on the exactness of Gauss quadrature rule for 2N-polynomials), consult (2.5.4). If we replace the Gauss quadrature rule by the Gauss-Lobatto one, we are led to stability of the dospectral method (meth_cheb.5) with and with the same given in (meth_cheb.32b).