The Asymptotic Eigenvalues of First-Order Spectral Differentiation Matrices

We complete and extend the asymptotic analysis of the spectrum of Jacobi Tau approximations that were first considered by Dubiner. The asymptotic formulas for Jacobi polynomials N P ( , ) , , 1 > − α β α β are derived and confirmed by numerical approximations. More accurate results for the slowest decaying mode are obtained. We explain where the large negative eigenvalues come from. Furthermore, we show that a large negative eigenvalue of order NN22 appears for 1 0 − < < α ; there are no large negative eigenvalues for collocations at Gauss-Lobatto points. The asymptotic results indicate unstable eigenvalues for 1 > α . The eigenvalues for Legendre polynomials are directly related to the roots of the spherical Bessel and Hankel functions that are involved in solving Helmholtz equation inspherical coordinates.


Introduction
Pseudospectral methods were developed to solve differential equations, where derivatives are computed numerically by multiplying a spectral differentiation matrix [1].Compared to finite difference methods that use local information, pseudospectral methods are global, and have exponential rate of convergence and low dissipation and dispersion errors.However in boundary value problems, they are often subject to stability restrictions [2].If the grid is not periodic, the spectral differentiation matrices are typically nonnormal [3], and the nonnormality may have a big effect on the numerical stability and behavior of the methods.On a grid of size N, pseudos-pectral methods require a time step restriction of ) O N − [5] for parabolic prob- lems.
Dubiner [6] carried an asymptotic analysis for theone-dimensional wave equation.He pointed out the ( ) O N boundedness of the eigenvalues of the spectral differentiation matrix with collocation at Legendre points.It was also proposed in [7] that the ) O N − .However, the eigen- values of Chebyshev and Legend respectral differentiation matrices are extremely sensitive to rounding errors and other perturbations [4].On agrid of size N , machine rounding could lead to errors of size The slowest decaying mode and largest wave numbers are often of interest.They affect stabilities and limit time step sizes of pseudospectral approximations.It is reported in [6] [8] that there exists a large negative eigenvalue of order 2 N for Chebyshev spectral differentiation matrix.Where does that eigenvalue come from?When does it appear?And how does it affect the time step size?In this paper, we will consider the first-order spectral differentiation matrix and examine the behavior of its eigenvalues asymptotically and numerically.
Let's consider the first-order eigenvalue problem ( ) The spectrum of the differentiation operator is empty.However, the spectrum of the Tau approximations to the eigenvalue problem affects the stability of the associated wave equation The non-polynomial contribution (the second term) is eliminated by picking We then obtain ( ) ( ) The boundary condition ( ) ( ) In the Tau method [1], the polynomial approximation ( )  3), the boundary condition ( ) It is proved [9] that the eigenvalues lie in the left half-plane for Jacobi polynomial ( ) The eigenvalues are computed numerically using the three term recurrence relation for corresponding Jacobi polynomials [10] (see Figure 1).We will show that the theorem is sharp by obtaining asymptotic results indicating unstable eigenvalues for 1 α > .In order to obtain the asymptotic behavior for large values of λ , we use the method of steepest descents to deform the integration paths to obtain the dominant contribution from saddle points.In general there are two saddle and two boundary points.The balance between the dominant saddle and boundary contributions leads to an asymptotic equation for λ .The two saddle points collide to form a third-order saddle when / N i λ = ± ; nearly merge when 0 α = and / 1 N i λ ±  ; and are too close to the boundary points when ( ) O N λ > .These cases complicate the analysis.).Thick solid: ( ) ( ) , dash-dot: ( ) , where ( ) Thin solid: contours of ( ) B µ .Take dash-dot branch for ( ) The paper is organized as follows.In Section 2, we present the asymptotic analysis and numerical results for Chebyshev polynomials.We generalize the results to Jacobi polynomials in Section 3, and derive the approximations of the slowest decaying mode and largest wave numbers.In Section 4, we show that the eigenvalues for Legendre polynomials are directly related to the roots of spherical Bessel and Hankel functions.The analysis and numerical results for collocation methods are explained in Section 5. Finally we conclude in Section 6.

Chebyshev Polynomials
We use Chebyshev polynomials We will apply the method of steepest descent to deform the integration path without changing the value of the integral, so that it goes through the critical point (saddle point) * φ in such a way that * ( ) R is maximum along the path, and * ( )

R
decreases along either direction away from * φ as rapidly as possible.As N → ∞ , the dominant contributions come from the saddle points and boundary points 0 and 2π .

Saddle Contributions
The saddle points of ( ) The integration paths follow the constant-phase contours and redefine σ to be sign ( ) Following the standard saddle point approximation ([11] §7.3), we evaluate the integrals near the saddle points and obtain the dominant contribution As N → ∞ when ( ) 0 The two saddles points collide and form athird-order saddle point at ( ) 3) is a third-order saddle point.A similar saddle point approximation leads to

Boundary Contributions
We approximate the contour at 0 φ = by the straight line and obtain the dominant contribution at 0 as N → ∞ .The approximation is the same near 2

Balance between Saddle and Boundary Contributions
There are four possible balances from Section 2.1: , k ∈  , the balance between ( 10) and ( 12) leads to ( ) As N → ∞ , this implies the limit curve In case (2), from Section 2.1.2,the balance leads to the limit curve 0,| | 1 The balances in cases ( 3) and ( 4) lead to inconsistent results.The limit curve ( 14) is in agreement with Dubiner's result ( [9] Equation (8.5)).It can be divided into three parts: the interval [ , )  i i ± ∞ and a curve connecting i ± in the left half-plane.The number of eigenvalues distributed around each part is given by the number of intersections of Equation ( 14) and eigenvalues respectively.There are left about eigenvalues distributed around the curve connecting i ± in the lefthalf-plane.Figure 1 shows the limit curve and eigenvalue approximations for Cheybshev polynomials.The numerical eigenvalues are computed using 128-bit or 34 decimal digits of precision.They distribute near the limit curve except a large negative real eigenvalue ~( ) O N µ , which is addressed in the next section.The asymptotic results are accurate even for small N 's.
From Equation (13) we derive that the slowest decaying eigenmode has wave number where The eigenvalues are plotted in Figure 3.They demonstrate better approximations than Dubiner's results ([6] Equation (8.6)).
If the boundary condition ( ) , and the limit curve is a reflection by the imaginary axis.

The Large Negative Eigenvalue
The largest wave number limits the time step size of the numerical approximation.When ( ) O N λ > , the saddle points ( ) are too close to the boundaries 0 0, 2 φ π = .
Thus we have to take the integration path of I 1 from 0 up to ( ) sin The balance between saddle and boundary contributions implies a polynomial equation for The real root of (18) approaches a constant as the degree of the equation increases.Solving the equation of degree 5 gives

Jacobi Polynomials
We now generalize the asymptotic analysis in Section 2 to Jacobipolynomials

Approximation of Jacobi Polynomials
We approximate Jacobi polynomials in two regions, i.e. near 0 and away 0.

Limit Curve
Using approximation (19), the integral in ( 4) becomes with the same ( ) A similar analysis gives the saddle contribution This is in agreement with the results for Chebyshev polynomials in Section 2, where ( ) ( ) Using the asymptotic formula for Bessel function of the first kind [13], ( ) ( ) . From (20), we have Plug (23) to the integral in (4) and we obtain the boundary contribution (0) ~0 b I when 0 α = .

0 ≠ α
The dominant balance is between saddle and boundary contributions.This gives the same limit curve as (14).The slowest decaying eigenmodes are plotted in Figure 3 for 0.3, 0.5 The theorem in [9] proves that the eigenvalues lie in the left half-plane for Jacobi polynomial ( ) We have derived asymptotically that the eigenvalue becomes unstable for 1 α > .

0 = α
The dominant balance is between two saddle contributions from ( ) 1 * φ ± , and this leads to a new limit curve The intervals [ , ) i i ± ∞ are excluded because only one saddle point contributes.The limit curve and eigenvalue approximations for Chebyshevpolynomials of the 2 nd kind ( 1/ 2 ) and Legendre polynomials ( 0 α β = = ) are plotted in Figures 4 and 5.The eigenvalues huddle around the limit curve.Note that even at a small N = 10, the eigenvalue approximations for Legendre polynomials lie exactly on the dashed curve, which is true for all the other figures.).Thick solid: ( ) , dash-dot: ( ) ). Thick solid:
Apply the cubic transformation, ( ) ( ) , to map the saddle points of ( ) ρ φ in the φ -plane to the saddle points η ± in the ζ -plane.It is shown [15] that the mapping is uniformly regular and one-to-one for all µ in a neighborhood of i ± .The integral (21) becomes , where Ai is the Airy function.The eigenvalues are related to the zeros of the Bessel and Hankel functions of half-integer order 1/ 2 N + [16].Further discussions are presented in Section 4. The slowest decaying eigenmode's wave number satisfies ( ) Ai (z) has a maximal zero 0 0 , 0 c c − > [17].Then Equation (28) gives This is also the largest wave number (see Section 3.4).Hence when 0 α = , the largest wave number is of or- der N instead of 2 N .

The Large Negative Eigenvalue
The formulas for saddle points contributions are not valid when the saddle points are too close to the boundaries at ( ) O N λ > . The approximation of Jacobi polynomials has to be replaced by Equation (20).If λ is not real, we can take the same integration paths as in Section 3.2 and obtain the saddle contribution.The boundary contribution remains the same with λ used first and then replaced by N µ .The balance between saddle and boundary contributions when 0 λ ≠ , or between two saddle contributions when 0 λ = gives the same limit curve as before.
If λ is real negative, we take the integration paths as in Section 2.4.The balance leads to a polynomial equ- where ( ) when n is odd   α + ≤ , i.e. 0 α ≤ , unstable when 0 α > .There is no large negative eigenvalue with Chebyshev extreme points used.This can be derived asymptotically using the same method as before, or simply follows from col 1 α α = + .

Conclusions
We have presented pseudospectral approximations of the first-order spectral differentiation matrices with zero boundary condition.We use the method of steepest descent to deform the integration path without changing the value of the integral, in order to obtain the dominant from saddle and boundary points.This approach leads to the asymptotic formulas for the eigenvalues of Jacobi Tau method.Numerical examples with Chebyshev of the 1 st and 2 nd kind and Legendre polynomials are presented.They agree well with the asymptotic analysis, even at small sizes.
The approximations for the slowest decaying modes give more accuracy than those obtained in [6].The largest wave numbers have raised interest in stabilities in pseudospectral approximations.We show that a large negative eigenvalue of order 2 N appears for 1 0 − < < .The collocation methods are also examined.There are no large negative eigenvalues for collocations at Gauss-Lobatto points.
For Jacobi polynomials, the eigenvalues lie in the left half-plane if 1 1 α − < ≤ .We show that the theorem is sharp by obtaining asymptotic results that indicate unstable eigenvalues for 1 α > .The eigenvalues for Legen- dre polynomials are related to the roots of Bessel and Hankel functions of half-integer order, spherical and modified spherical Bessel and Hankel functions.These results complete Dubiner's earlier analysis [6].
eigenvalues could be shrunk to ( ) O N byreplacing Chebyshev Tau method with Legendre Tau method.That would mean a time step size increase from time step restriction in theory, is subject to an ( )

u x is determined from the boundary
orthogonal to all polynomials of degree 1 n − with respect to the weight function ( ) 0 w x ≥ in the interval ( 1,1) − .For Jacobi weight function

Figure 1 .
Figure 1.The limit curve and eigenvalue approximations for Chebyshev polynomials ( 1 2 α β − = =).Thick solid: s ≡ to illustrate the approach and derive the asymptotic formulas.They correspond to the class of Jacobi polynomials with 1 e. the steepest-descent curves, from 0 to i π + ∞ for I 1 and from 2π to i π − ∞ for I 2 , passing through the saddle points ( ) * σ φ .The steepest curves corresponding to the opposite signs of µ I are mirror images about φ π = R axis.Define

Figure 6 shows the ratio 2 /
N λ obtained from (31) and by solving Equation (30) of degree 4, respectively, confirmed by the numerical eigenvalues at 40 N = for different values of α .The real root of the polynomial of degree > 2 is negative for 0 1/ 2 α < <and 1/ x approaches −∞ as α approaches 1/2 from the left.Thus, there is a large negative eigenvalue of Legendre extreme points, the balance is now between saddle and boundary contributions since col 1 α = , and the limit curve is the same as the other two.

Figure 7
shows the eigenvalues obtained using the Tau and collocation methods, together with their limit curves for Chebyshev ( polynomials.Collocations at Chebyshev and Legendre extreme points are stable.Collocation at Chebyshev extreme points of the 2 nd kind becomes unstable.In general, collocation at Gauss-Lobatto points is stable when 1 1