Spectral methods based on Hermite functions for linear hyperbolic equations

Spectral Methods Based on Hermite Functions forLinear Hyperbolic EquationsJulián Aguirre, Judith RivasDepartamento de Matemáticas, Universidad del País Vasco, Aptdo. 644,48080 Bilbao, Spain

Received 31 August 2009; accepted 14 June 2011Published online in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/num.20699

We consider the approximation by spectral and pseudo-spectral methods of the solution of the Cauchy prob-lem for a scalar linear hyperbolic equation in one space dimension posed in the whole real line. To dealwith the fact that the domain of the equation is unbounded, we use Hermite functions as orthogonal basis.Under certain conditions on the coefficients of the equation, we prove the spectral convergence rate of theapproximate solutions for regular initial data in a weighted space related to the Hermite functions. Numericalevidence of this convergence is also presented. © 2011 Wiley Periodicals, Inc. Numer Methods Partial DifferentialEq 000: 000–000, 2011

Keywords: Hermite functions; spectral approximations; linear hyperbolic equations

I. INTRODUCTION

The aim of this article is to define spectral methods based on Hermite functions for the approxi-mation of solutions of the Cauchy problem for the scalar linear hyperbolic equations in one spacedimension

∂u

∂t+ a(x)

∂u

∂x+ b(x) u = 0, x ∈ R, t > 0, (1)

where a and b are regular functions depending only on the space variable, and to prove theirconvergence.

Some basic references on the theory of spectral methods are [1–3] and [15]. The ideaof a spectral method is to approximate the solution of (1) by means of a truncated seriesuN(x, t) = ∑N

k=1 uk(t)φk(x), where {φk}∞k=1 is an orthogonal basis of a functional Hilbert space

Correspondence to: Judith Rivas, Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain (e-mail:[email protected])Contract grant sponsor: Spanish Ministry of Education and Science; contract grant number: MTM2007-62186Contract grant sponsor: Basque Government; contract grant number: IT-305-07

© 2011 Wiley Periodicals, Inc.

2 AGUIRRE AND RIVAS

H , chosen according to the features of the problem. In order to obtain the coefficients uk(t), uN

is forced to be the solution of an approximate equation

∂uN

∂t= LNuN ,

where LN is an approximation of the differential operator Lu = −a ux − b u involving a pro-jection operator from H onto the N -dimensional subspace generated by the first N elementsof the orthogonal basis. Different projection operators lead to different spectral methods, themost commonly studied being spectral or Galerkin methods and pseudo-spectral or collocationmethods.

Classical spectral methods for the solution of partial differential equations are based on trigono-metric, Chebyshev and Legendre polynomials, as can be seen in [4, 6–8], among others. In all ofthese methods, the domain of computation must be bounded. However, as mentioned in [9], manyproblems in science and engineering lead to partial differential equations posed in unboundeddomains, for example, fluid flows in exterior domains, nonlinear wave equations in quantummechanics, electromagnetic fields, and so on. When applying Fourier, Chebyshev, or Legendrespectral methods to such problems, some domain transformation is necessary (truncation of thedomain, change of variables, etc).

On the other hand, when Hermite functions are chosen as orthogonal basis, the domain ofcomputation is the whole real line. Hermite spectral methods where first analyzed for the heatequation in [10] and later for some other parabolic problems in [11], for the Fokker–Planck equa-tion in [5], for the Dirac equation in [9] and for the Vlasov–Poisson system in [13]. Also spectralmethods based on Hermite polynomials have been defined in [12, 14, 16] and [17] for parabolicproblems. Nonlinear hyperbolic problems have been analyzed in [18], where convergence of aspectral viscosity method based on Hermite functions is proved. However, the methods in thatarticle neither give a rate of convergence of the approximations to the exact solution nor can beapplied to linear hyperbolic problems, for which nothing has been done, as far as we know.

This article is organized as follows. In section II, we define and list the properties of theelements involved in our spectral method:

• the Hermite functions and the weighted Hilbert space L2w of which they are an orthogonal

basis;• the projection operators πN , defined as an orthogonal projection, and IN , the interpolation

operator over the zeros of Hermite polynomials.

Since the spatial domain of definition of our approximations is the whole real line, the solutionof (1) must be defined on R × [ 0, ∞). This is verified for instance if a is bounded, but we donot want to impose such a restriction. Besides, we shall need the solution to remain in the spaceof reference L2

w for all t ≥ 0. In section III, we impose conditions on the coefficients a and b toensure that the Cauchy problem for Eq. (1) is well posed in L2

w.The definition of the Hermite spectral method and the proof of its convergence is given in

section IV. We will show the spectral convergence rate of our approximation for regular solu-tions, relying on the stability of the scheme and the known error estimates for the projectionoperator πN .

In section V, the Hermite pseudo-spectral method is considered. Again, we show spectral con-vergence of the approximation with the aid of a stability result, but the proof is more involvedthan in the spectral case. A skew-symmetric decomposition of the term a ux is needed to definethe approximate operator LN .

Numerical Methods for Partial Differential Equations DOI 10.1002/num

HERMITE SPECTRAL METHODS FOR LINEAR HYPERBOLIC EQUATIONS 3

Numerical results providing evidence of the theoretical convergence results obtained insections IV and V are presented in section VI.

Finally, in section VII, we recall the main conclusions and state some open problems relatedto the subject of this article.

II. HERMITE SPECTRAL METHODS

In this section, we describe the elements of our spectral methods as well as the basic propertiesthat will be used in the proof of convergence.

As mentioned in the introduction, we will consider an orthogonal basis consisting of Hermitefunctions, defined as

hk(x) = e−x2Hk(x), k = 0, 1, 2 . . . ,

where Hk is the Hermite polynomial of degree k and leading coefficient 2k . It follows from theproperties of Hermite polynomials, [19], that

h′k(x) = −hk+1(x), k = 0, 1, 2 . . . (2)

and

hk+1(x) = 2 x hk(x) − 2 k hk−1(x), k = 0, 1, 2 . . . (3)

Hermite functions verify an orthogonality relation with respect to the weight function w(x) = ex2:∫ ∞

−∞hm(x) hn(x) w(x) dx = δm,n 2n n! √π , m, n = 0, 1, 2, . . . (4)

Hence, they form an orthogonal basis of the weighted L2 space

L2w =

{ϕ : R → R measurable

∣∣∣∣∫ ∞

−∞|ϕ(x)|2 w(x) dx < ∞

},

with norm

‖ϕ‖0,w =(∫ ∞

−∞|ϕ(x)|2 w(x) dx

)1/2

.

For all k ≥ 0, hk is an eigenfunction of the second-order differential operator

Lϕ = −(ϕ′′ + 2 x ϕ′)

with corresponding eigenvalue λk = 2 (k + 1). L is a self-adjoint positive definite operator withcompact inverse in L2

w that generates an analytic semigroup of contractions {e−tL}t≥0.The weighted Sobolev spaces associated to L2

w are

Hkw = {

ϕ : R → R | ϕ, ϕ′, . . . , ϕ(k) ∈ L2w

}, k = 0, 1, 2, . . . .


4 AGUIRRE AND RIVAS

In the following proposition, we give a characterization of H 1w that can be found in [10], as

well as some inequalities regarding the norms of functions in this space.

Proposition 1. Let ϕ ∈ L2w. Then ϕ ∈ H 1

w if and only if w1/2ϕ ∈ H 1(R) and x ϕ ∈ L2w.

Moreover, the following inequalities hold:

‖(w1/2 ϕ)′‖L2 ≤ ∥∥ϕ′∥∥0,w

, (5)

‖ϕ‖0,w ≤ ∥∥ϕ′∥∥0,w

, (6)

‖x ϕ‖0,w ≤ ∥∥ϕ′∥∥0,w

. (7)

Inequality (6) is analogous to Poincare’s inequality for bounded domains. It implies that∥∥ϕ(k)∥∥

0,wis a norm in Hk

w equivalent to∑k

j=0

∥∥ϕ(j)∥∥

0,won Hk

w. For noninteger s > 0, Hsw is

defined by interpolation and for s < 0, Hsw is the dual of H−s

w .As Hermite functions form an orthogonal basis of L2

w, any ϕ ∈ L2w can be expressed in a unique

way as a Fourier–Hermite series of the form

ϕ(x) =∞∑

k=0

ϕkhk(x), where ϕk = 1√π 2k k!

∫ ∞

−∞ϕ(x) hk(x) w(x) dx.

Let VN be the subspace of L2w generated by the first N + 1 Hermite functions {hk}N

k=0. Givenϕ ∈ L2

w, its best approximation in the L2w norm by functions of VN is the orthogonal or spectral

projection

πNϕ =N∑

k=0

ϕkhk .

It is clear that

‖πNϕ‖0,w ≤ ‖ϕ‖0,w ∀ϕ ∈ L2w. (8)

The following estimate of the error of the approximation of ϕ by πNϕ is given in [10].

Proposition 2. Let 0 ≤ µ ≤ σ . There exists a positive constant C, independent of N , such thatfor all ϕ ∈ Hσ

w ,

‖ϕ − πNϕ‖µ,w ≤ CNµ−σ

2 ‖ϕ‖σ ,w . (9)

In practice, the spectral projection πNϕ is of limited value since it is costly to compute accu-rately. There is another way of approximating a continuous function ϕ ∈ L2

w by an element ofVN . Let z0, z1, . . . , zN be the zeros of HN+1, which are all real and distinct. Given ϕ ∈ C(R)∩L2

w,its N -th pseudo-spectral projection is the unique function INϕ ∈ VN such that

INϕ(zj ) = ϕ(zj ), j = 0, 1, . . . , N .



If INϕ(x) = ∑N

k=0 ϕkhk(x), the k-th coefficient ϕk is the result of approximating the exactFourier–Hermite coefficient ϕk by means of a Gaussian quadrature formula with nodes the zerosof HN+1. More precisely,

ϕk =N∑

j=0

wk,jϕ(zj ), where wk,j = 2N−kN !ez2j Hk(zj )

(N + 1)k!(HN(zj ))2. (10)

The following estimate for the weighted Sobolev norms of the error of the pseudo-spectralprojection is given in [20].

Proposition 3. Let σ ≥ 1 and 0 ≤ µ ≤ σ . Then, there exists a positive constant C, independentof N , such that for all ϕ ∈ Hσ

w ,

‖ϕ − INϕ‖µ,w ≤ C N16 + µ−σ

2 ‖ϕ‖σ ,w . (11)

As a consequence of this estimate, the norm of the pseudo-spectral projection is bounded interms of Sobolev norms of the function being approximated.

Proposition 4. There exists a constant C > 0, independent of N , such that for all ϕ ∈ H 1w

‖INϕ‖0,w ≤ ‖ϕ‖0,w + CN−1/3 ‖ϕ‖1,w . (12)

III. WELL-POSEDNESS IN L2w

The method of characteristics, see [21], guarantees the existence and uniqueness of solution ofthe Cauchy problem for (1) in a neighbourhood of R × {0}. However, as we will approximateu(·, t) by functions of VN , we need it to be defined in R × [ 0, T ] for some T > 0. For this, wemust impose some conditions on the coefficients a and b.

Definition. Let a ∈ C1(R). We say that a is an admissible function if all its zeros are isolatedand if in any interval I where a does not vanish, the function 1/a has a primitive A such thatinfx∈I A(x) = −∞.

It is easily proved that if a is an admissible function, then the solution u of (1) is defined inR × [ 0, ∞).

To prove the convergence of the spectral approximations to be defined in next sections we shallneed the well-posedness of the problem in L2

w. We prove this in the next theorem.

Theorem 1. Let a ∈ C1(R) be an admissible function, b ∈ C(R) and ϕ ∈ L2w. If there exists a

constant C ∈ R such that

2 x a(x) + a′(x) − 2 b(x) ≤ C ∀x ∈ R, (13)

then the solution of (1) with initial condition u(x, 0) = ϕ(x) verifies

‖u(·, t)‖20,w ≤ eCt ‖ϕ‖2

0,w ∀t > 0.


6 AGUIRRE AND RIVAS

Proof. Suppose that a does not vanish. Then, we may assume without loss of generality thata(x) > 0 for all x ∈ R. Let A be a primitive of 1/a and ξ(x, t) = A−1(A(x) − t). Using themethod of characteristics, u is found to be

u(x, t) = exp

(−

∫ x

ξ(x,t)

b(s)

a(s)ds

)ϕ(ξ(x, t)).

Hence,

‖u(·, t)‖20,w =

∫ ∞

−∞exp

(−2

∫ x

ξ(x,t)

b(s)

a(s)ds

)|ϕ(ξ(x, t))|2 w(x) dx.

Making the change of variable x = A−1(A(ξ) + t) = h(ξ , t) we get

‖u(·, t)‖20,w =

∫ ∞

−∞exp

(−2

∫ h(ξ ,t)

ξ

b(s)

a(s)ds

)|ϕ(ξ)|2 w(h(ξ , t))

a(h(ξ , t))

a(ξ)dξ

=∫ ∞

−∞f (ξ , t) |ϕ(ξ)|2 w(ξ) dξ ,

where

f (ξ , t) = exp

(−2

∫ h(ξ ,t)

ξ

b(s)

a(s)ds

)w(h(ξ , t)) a(h(ξ , t))

w(ξ) a(ξ)≥ 0.

Since A is a primitive of 1/a,

∂f

∂t= (−2 b(h(ξ , t)) + 2 h(ξ , t) a(h(ξ , t)) + a′(h(ξ , t))) f (ξ , t),

and using hypothesis (13) with x = h(ξ , t), we get

∂f

∂t≤ C f (ξ , t).

Integrating this differential inequality, we obtain

f (ξ , t) ≤ f (ξ , 0) eCt = eCt ∀ξ ∈ R.

Therefore,

‖u(·, t)‖20,w =

∫ ∞

−∞f (ξ , t) |ϕ(ξ)|2 w(ξ) dξ

≤ eCt

∫ ∞

−∞|ϕ(ξ)|2 w(ξ) dξ

= eCt ‖ϕ‖20,w .

In the general case in which a has isolated zeros, we divide the real line in intervals where a doesnot vanish and perform the integration over each of these intervals. The corresponding function



f (ξ , t) is positive over all of these intervals so that the previous argument can be applied, yieldingthe result.

Remark 1. Condition (13) establishes a balance between the speed with which the solutionmoves toward infinity, given by a, and the dissipation introduced by b. Since u(·, t) must besquare-integrable with respect to the weight w, its mass cannot move toward infinity too quickly,unless some dissipation is applied. The relation of (13) with the weight w materializes in thefactor 2x = w′(x)/w(x) that appears multiplying a.

As an example of what can happen when (13) is not satisfied, consider the equation ut +ux = 0.It is easy to find initial data ϕ ∈ L2

w such that the corresponding solution u(·, t) �∈ L2w for any

t > 0. However, the Cauchy problem for the equation ut + ux + x u = 0, which has the same a

coefficient, is well posed in L2w. Although the speed of the solution is still constant, the dissipation

introduced by b(x) = x is enough to make the solution square-integrable with respect to w forall t > 0.

If we integrate (13) in the particular case b ≡ 0, we get

x a(x) ≤ x e−x2(

a(0) + C

∫ x

0eξ2

dξ

)∀x ∈ R.

As limx→±∞ x e−x2 ∫ x

0 eξ2dξ = 1/2, there exists a constant C > 0 such that a(x) ≤ C/x if

x > 0, while a(x) ≥ −C/x if x < 0. This means that the solution tends to concentrate aroundthe origin. If b is not identically zero and has the right sign, some dissipation appears and therange of values that a can take is larger.

Remark 2. A few words should be added in order to clarify why we pose the Cauchy problemfor (1) in L2

w instead of working in the unweighted L2 setting that may seem more natural. Anorthogonal basis is L2(R) is obtained by defining φk(x) = e−x2/2Hk(x), k = 0, 1, . . . . However,no error estimates similar to (9) and (11) are known for the corresponding projection operators.These estimates are the key point in obtaining the error bounds to be given in Theorems 2 and 3for our spectral methods. The estimates (9) and (11) rely heavily on the properties of the operatorL, properties that do not hold for the second order operator whose eigenfunctions are φk .

On the other hand, an assumption similar to (13) is also necessary when the Cauchy problemis studied in L2(R). Consider the problem

∂v

∂t+ A(x)

∂v

∂x+ B(x)v = 0, x ∈ R, t > 0,

v(x, 0) = φ(x) ∈ L2(R), x ∈ R,(14)

Reasoning as in the proof of Theorem 1, it is possible to prove that if there exists a constant C

such that

A′(x) − 2 B(x) ≤ C ∀x ∈ R, (15)

then the solution v(·, t) remains in L2(R) for all t > 0. The equation vt + vx + x v = 0 doesnot satisfy this condition, and it is easily seen that the corresponding Cauchy problem is not wellposed in L2(R). The change of variable u(x, t) = e−x2/2 v(x, t) transforms it into a new linearequation of the form (1) with a(x) = A(x) and b(x) = x A(x)+B(x). Assumption (15) for (14)is therefore equivalent to assumption (13) for (1).


8 AGUIRRE AND RIVAS

IV. THE HERMITE SPECTRAL APPROXIMATION

We define the Hermite spectral approximation of the solution of Eq. (1) with initial datumu(x, 0) = ϕ(x) as the solution uN : [ 0, ∞) → VN of the approximated problem

∂uN

∂t+ πN

(a(x)

∂uN

∂x

)+ πN(b(x)uN) = 0, x ∈ R, t > 0,

uN(x, 0) = πNϕ(x), x ∈ R.(16)

Substituting uN(x, t) = ∑N

k=0 uk(t)hk(x) in (16) we get a linear system of ordinary differentialequations for uk(t) with initial conditions uk(0) = ϕk , 0 ≤ k ≤ N . Since such a system has aunique solution, the Hermite spectral approximation is well defined. We next establish the stabilityof the method.

Lemma 1. Let a ∈ C1(R) be an admissible function that has at most polynomial growth atinfinity and b ∈ C(R) such that condition (13) is satisfied. Let

LNu = −πN

(a

∂

∂x(πNu)

)− πN(b πNu). (17)

Then the evolution operator eLN t is bounded in VN . More precisely, there is a constant C > 0such that

∥∥eLN t ϕ∥∥2

0,w≤ eCt ‖ϕ‖2

0,w ∀ϕ ∈ VN , ∀t > 0. (18)

Proof. Let uN : [ 0, ∞) → VN be such that

∂uN

∂t= LNuN . (19)

Multiplying (19) by uN , integrating with respect to the weight w and taking into account theorthogonality of the Hermite functions, we get

1

2

d

dt‖uN(·, t)‖2

0,w = −∫ ∞

−∞a(x)

∂

∂x

(u2

N

2

)w dx −

∫ ∞

−∞b(x) u2

N w dx.

When we integrate by parts in the first integral of the right-hand side, the boundary terms vanishdue to the growth rate of a at infinity, leading to

1

2

d

dt‖u(·, t)‖2

0,w = 1

2

∫ ∞

−∞(2 x a(x) + a′(x))u2

N w dx −∫ ∞

−∞b(x) u2

N w dx.

Assumption (13) implies

d

dt‖uN(·, t)‖2

0,w ≤ C ‖uN(·, t)‖20,w ,

and the result follows from integration of this differential inequality.



Since (1) is well posed in L2w, the convergence of the Hermite spectral approximation is a

consequence of the stability just proved. We will also give a bound for the order of convergencewhen the exact solution is smooth.

Theorem 2. Let a ∈ C1(R) be an admissible function and b ∈ C(R) such that (13) is satisfiedand a, b have at most polynomial growth of order k at infinity.

Let u be the exact solution of (1) with initial condition u(·, 0) = ϕ ∈ L2w, and let uN be its

Hermite spectral approximation, solution of (16). Then, for all t ≥ 0 there exists a constantC(k, t) > 0, independent of N , such that if u(·, t) ∈ Hµ

w with µ ≥ k + 1, then

‖u(·, t) − uN(·, t)‖0,w ≤ C(k, t) Nk+1−µ

2 supτ∈[0,t]

‖u(·, τ)‖µ,w .

Proof. Let UN = πNu. Defining LN as in Lemma 1,

∂UN

∂t= LNUN + πN

(a(x)

∂

∂x(UN − u)

)+ πN(b(x)(UN − u)),

UN(x, 0) = πNϕ(x),

As LN is a linear operator, wN = UN − uN satisfies

∂wN

∂t= LNwN + πN

(a

∂

∂x(UN − u)

)+ πN(b(UN − u)),

wN(x, 0) = 0,

and hence,

wN(x, t) =∫ t

0e(t−s)LN πN

(a

∂

∂x(UN − u) + b(UN − u)

)(x, s) ds.

Taking norms and applying (18) yields

‖wN(·, t)‖0,w ≤∫ t

0e

C2 (t−s)

∥∥∥∥πN

(a

∂

∂x(UN − u) + b (UN − u)

)(·, s)

∥∥∥∥0,w

ds. (20)

From (8) followed by (7), which can be applied because of the polynomial growth of a at infinity,we get

∥∥∥∥πN

(a

∂

∂x(UN − u)

)(·, s)

∥∥∥∥0,w

≤ C(k)

∥∥∥∥ ∂

∂x(UN − u)(·, s)

∥∥∥∥k,w

.

Now we take into account the definition of the weighted Sobolev norms and the estimate (9),obtaining

∥∥∥∥πN

(a

∂

∂x(UN − u)

)(·, s)

∥∥∥∥0,w

≤ C(k)Nk+1−µ

2 ‖u(·, s)‖µ,w . (21)


10 AGUIRRE AND RIVAS

Arguing in a similar fashion, we see that

‖πN(b (UN − u))(·, s)‖0,w ≤ C(k)Nk−µ

2 ‖u(·, s)‖µ,w . (22)

Substituting (21) and (22) into (20),

‖UN(·, t) − uN(·, t)‖0,w ≤ C(k) Nk+1−µ

2

∫ t

0e

C2 (t−s) ‖u(·, s)‖µ,w ds

≤ C(k, t) Nk+1−µ

2 supτ∈[0,t]

‖u(·, τ)‖µ,w .

Finally, as UN is a good approximation of u, this bound gives us the convergence rate of thespectral approximation. Indeed,

‖uN(·, t) − u(·, t)‖0,w ≤ ‖u(·, t) − UN(·, t)‖0,w + ‖UN(·, t) − uN(·, t)‖0,w

≤ C N− µ2 ‖u(·, t)‖µ,w + C(k, t) N

k+1−µ2 sup

τ∈[0,t]‖u(·, τ)‖µ,w

≤ C(k, t) Nk+1−µ

2 supτ∈[0,t]

‖u(·, τ)‖µ,w .

This error estimate shows the spectral convergence of the approximation uN . If u ∈ L∞(Hµw )

for all µ ≥ k + 1, then the L2w norm of the error tends to 0 faster than any negative power of N .

The growth rate of a and b influences the convergence rate of the approximation. The biggerthis growth is, the more regular the exact solution u must be to ensure convergence.

V. THE HERMITE PSEUDO-SPECTRAL APPROXIMATION

The implementation of the spectral method defined in section IV implies the computation of theFourier–Hermite coefficients of the terms a (uN)x and b uN , which in general cannot be calculatedexactly and must be approximated.

The pseudo-spectral method is more easily implemented. However, the difficulty in this caseappears when trying to obtain the stability for the corresponding operator LN = INLIN withsimilar arguments as those used in the spectral case.

Following the ideas of [4], where Chebyshev spectral methods for linear hyperbolic equationsare considered, we will decompose the term a ux in a skew-symmetric fashion before definingthe approximate operator LN . Using the identity

a ux = 1

2(a ux + (a u)x − a′ u), (23)

we define the operator LN : L2w → VN as

LNu = −1

2IN

(a

∂u

∂x+ ∂

∂x(IN(a u)) + (2 b − a′)u

), (24)



and the Hermite pseudo-spectral approximation of the solution of (1) as the solutionvN : [ 0, ∞) → VN of the approximate problem

∂vN

∂t= LNvN , x ∈ R, t > 0,

vN(x, 0) = INϕ(x), x ∈ R.(25)

As for the spectral method, this leads to an initial value problem for a linear system of ordinarydifferential equations whose unknowns are the Fourier–Hermite coefficients of vN . Such a systemhas a unique solution and therefore, the pseudo-spectral approximation is well defined.

To prove the stability of the method, we will exploit the properties of the Gaussian quadratureformula based on the zeros of the Hermite polynomials. Let z0, . . . , zN be the zeros of HN+1.Rewritting the classical Gaussian formula, see [22] integrals of functions with respect to theweight function w can be approximated by a sum of the form

∫ ∞

−∞g(x) ex2

dx ∼N∑

k=0

wk g(zk), (26)

where

wk = 2NN ! √π

(N + 1)(hN(zk))2, k = 0, . . . , N . (27)

This quadrature formula is exact for regular functions g such that w2 g is a polynomial of degreeat most 2 N + 1.

Given φ, ψ ∈ L2w ∩ C(R), we define the discrete scalar product

(φ, ψ)N ,w =N∑

k=0

φ(zk)ψ(zk)wk ,

and the associated discrete norm

‖φ‖N ,0,w = (φ, φ)1/2N ,w.

If φ ψ w2 is a polynomial of degree at most 2 N + 1, then

(φ, ψ)N ,w = (φ, ψ)w,

and if φ ∈ VN , then

‖φ‖N ,0,w = ‖φ‖0,w .

Lemma 2. Let a ∈ C1(R) be and admissible function and b ∈ C(R) such that (13) is verified.Then

∥∥eLN tϕ∥∥2

0,w≤ eCt ‖ϕ‖2

0,w ∀ϕ ∈ VN , ∀t ≥ 0. (28)



Proof. Let ϕ ∈ VN and vN = eLN tϕ. Then, vN verifies the equation ∂tvN = LNv. Multiplyingthe equation by vN and taking the discrete scalar product gives(

∂vN

∂t, vN

)N ,w

= −1

2

(IN

(a

∂vN

∂x+ ∂

∂x(IN(a vN) + (2 b − a′)vN)

), vN

)N ,w

As the value of the pseudo-spectral approximation coincides with the value of the function itselfat the zeros of HN+1, this equality can be rewritten as

d

dt‖vN(·, t)‖2

N ,0,w = −(

a∂vN

∂x+ ∂

∂x(IN(a vN)), vN

)N ,w

− ((2 b − a′)vN , vN)N ,w. (29)

Consider now the first summand on the right hand side. Using the exactness of the quadratureformula for functions whose product with w2 is a polynomial of degree at most 2 N + 1, we seethat (

a∂vN

∂x, vN

)N ,w

=(

∂vN

∂x, IN(avN)

)N ,w

=∫ ∞

−∞

∂vN

∂xIN(avN) w dx,

because the derivative of a function of the subspace VN is in VN+1. Integrating by parts andrewriting the result in terms of the discrete product, we get(

a∂vN

∂x, vN

)N ,w

= −∫ ∞

−∞vN

(∂

∂x(IN(avN)) + 2 x IN(a vN)

)w dx

= −(

vN ,∂

∂x(IN(a vN))

)N ,w

− (vN , 2 x IN(a vN))N ,w

= −(

vN ,∂

∂x(IN(a vN))

)N ,w

− (vN , 2 x a vN)N ,w.

Hence, (a

∂vN

∂x+ ∂

∂x(IN(a vN)), vN

)N ,w

= −(vN , 2 x a vN)N ,w,

and substituting in (29), we get that

d

dt‖vN(·, t)‖2

N ,0,w = (vN , 2 x a vN)N ,w − ((2 b − a′)vN , vN)N ,w.

By assumption (13) on a and b,

d

dt‖vN(·, t)‖2

N ,0,w ≤ C (vN , vN)N ,w = C ‖vN(·, t)‖2N ,0,w .

Integrating this inequality,

‖vN(·, t)‖2N ,0,w ≤ eCt ‖vN(·, 0)‖2

N ,0,w .

Finally, vN(·, t) ∈ VN for all t ≥ 0, so that ‖vN(·, t)‖N ,0,w = ‖vN(·, t)‖0,w and the result follows.

The convergence of the pseudo-spectral method is again based on this stability result.



Theorem 3. Let a ∈ C1(R) be an admissible function and b ∈ C(R) such that (13) is satisfiedand assume that a, a′ and b have at most a polynomial growth of order k at infinity. Let u be thesolution of (1) with initial condition u(·, 0) = ϕ ∈ H 1

w and let vN be its pseudo-spectral approx-imation, solution of (25). Then, for each t ≥ 0 there exists a constant C(k, t) > 0, independentof N , such that if u(·, t) ∈ Hµ

w with µ ≥ k + 2, then

‖u(·, t) − vN(·, t)‖0,w ≤ C(k, t) Nk+2−µ

2 + 16 sup

τ∈[0,t]‖u(·, τ)‖µ,w .

Proof. et VN = INu. Decomposing the term a ux of (1) as in (23) and applying the operatorIN , we get

∂VN

∂t+ 1

2IN

(a

∂u

∂x

)+ 1

2IN

∂

∂x(a u) + 1

2IN((2 b − a′)u) = 0,

VN(x, 0) = INϕ(x).

Then, if wN = VN − vN , since LN is linear, we have that

∂wN

∂t= LNwN + 1

2IN

(a

∂

∂x(VN − u)

)+ 1

2IN

∂

∂x(IN(a (VN − u)))

− 1

2IN

∂

∂x((I − IN)(aVN)) + 1

2IN((2 b − a′)(VN − u)),

wN(0) = 0, and therefore

wN(x, t) =∫ t

0e(t−s)LN AN(x, s) ds,

where

AN = 1

2IN

(a

∂

∂x(VN − u)

)+ 1

2IN

∂

∂x(a(VN − u))

− 1

2IN

∂

∂x((I − IN)(aVN)) + 1

2IN((2 b − a′)(VN − u)).

Taking into account (28),

‖wN(·, t)‖0,w ≤∫ t

0e

C2 (t−s) ‖AN(·, s)‖0,w ds. (30)

Next we bound the L2w-norm of each of the summands appearing in AN . Only the details for the

first one will be given, the rest being obtained in a similar way. From (12), we obtain∥∥∥∥IN

(a

∂

∂x(VN − u)

)∥∥∥∥0,w

≤∥∥∥∥a

∂

∂x((I − IN)u)

∥∥∥∥0,w

+ C N−1/3

∥∥∥∥a∂

∂x((I − IN)u)

∥∥∥∥1,w

.

Now, recalling the behavior of a at infinity as well as property (7), we deduce that∥∥∥∥IN

(a

∂

∂x(u − VN)

)∥∥∥∥k+1,w

≤ C(k)

∥∥∥∥ ∂

∂x((I − IN)u)

∥∥∥∥k,w

+ C(k) N−1/3

∥∥∥∥ ∂

∂x((I − IN)u)

∥∥∥∥k+1,w

.



Finally, we take into account the definition of the weighted Sobolev norms and the estimate (11)for the interpolation error to conclude that, if µ ≥ k + 2, then

∥∥∥∥IN

(a

∂

∂x(u − VN)

)∥∥∥∥0,w

≤ C(k)(N16 + k+1−µ

2 + N16 + k+2−µ

2 − 13 ) ‖u(·, s)‖µ,w

≤ C(k)Nk+2−µ

2 − 16 ‖u(·, s)‖µ,w .

Similarly, it is possible to get the same inequality for the L2w-norm of

IN

∂

∂x(a(VN − u)), IN

∂

∂x((I − IN)(a VN)), and IN((2 b − a′)(u − VN)).

Substitution of these last estimates in (30) yields

‖wN(·, t)‖0,w ≤ C(k)Nk+2−µ

2 + 16

∫ t

0e

C2 (t−s) ‖u(·, s)‖µ,w ds

≤ C(k, t)Nk+2−µ

2 + 16 sup

τ∈[0,t]‖u(·, τ)‖µ,w .

Finally,

‖u(·, t) − vN(·, t)‖0,w ≤ ‖u(·, t) − VN(·, t)‖0,w + ‖VN(·, t) − vN(·, t)‖0,w

≤ C(k, t)Nk+2−µ

2 + 16 sup

τ∈[0,t]‖u(·, τ)‖µ,w .

As in the Galerkin method, spectral convergence is achieved. However, because of the slightlypoorer approximation properties of IN compared with those of πN , more regularity of the solutionis needed to ensure convergence.

VI. NUMERICAL EXPERIMENTS

In this section, we present some numerical experiments that show the convergence of the spectralmethods analyzed in the previous sections. Although convergence in L2

w of the spectral approxi-mation uN to the exact solution u has been proved, the norm of the error, ‖u(·, t) − uN(·, t)‖0,w,can only be approximated. We compute instead (an approximation of) the L∞ norm of the errormultiplied by the square root of w, given by

EN∞,w(t) = max

0≤i≤1000exi

2/2|u(xi , t) − uN(xi , t)| ∼ ‖w1/2(u(·, t) − uN(·, t))‖∞,

where {xi}1000i=0 are points that in each experiment are chosen uniformly on an interval of interest

(which may vary with t).



Example 1. Our first example corresponds to the case of constant a. As mentioned in section III,the problem,

∂v

∂t+ ∂v

∂x= 0, x ∈ R, t > 0,

v(x, 0) = φ(x), x ∈ R,(31)

is not well-posed in L2w, the natural setting being L2(R). In order to apply our spectral methods,

we make the change of dependent variable u(x, t) = e−x2/2v(x, t), that leads to

∂u

∂t+ ∂u

∂x+ x u = 0, x ∈ R, t > 0,

u(x, 0) = ϕ(x) = e−x2/2φ(x), x ∈ R.

(32)

The exact solution of (32) is u(x, t) = et2/2e−xtϕ(x − t).Because of (2) and (3), in this particular case, it is possible to write explicitly the sys-

tem of ordinary differential equations for the coefficients of the spectral approximation. IfuN(x, t) = ∑N

k=0 uk(t)hk(x) is the solution of

∂uN

∂t+ πN

∂uN

∂x+ πN(x uN) = 0, x ∈ R, t > 0,

uN(x, 0) = πNϕ(x) = ∑N

k=0 ϕkhk(x), x ∈ R,

(33)

then the coefficients uk(t) are the solution of

u′0(t) = −u1(t),

u′k(t) = uk−1(t)/2 − (k + 1)uk+1(t), k = 1, . . . , N − 1,

u′N(t) = uN1(t)/2,

uk(0) = ϕk , k = 0, . . . , N .

(34)

To avoid time discretization errors and to show evidence of spectral convergence, the computa-tions have been carried out in Mathematica� with 300 digits of precision. For the calculation ofEN

∞,w(t), we have taken equidistributed points in the interval [ −5 + t , 5 + t ].The values of EN

∞,w(t) for t = 1 and different values of N are presented in Table I. The first

column corresponds to an initial datum ϕ1(x) = e−x2that is in Hm

w for all m ≥ 0, and hence,

TABLE I. EN∞,w(1) error of the spectral approximation of the solution of equation ut + ux + xu = 0 fordifferent initial data.

N ϕ1 ϕ2

32 1.26 e − 24 8.13 e − 0264 1.52 e − 56 5.70 e − 02

128 1.16 e − 129 4.01 e − 02256 6.75 e − 296 2.83 e − 02



TABLE II. EN∞,w(10) error of the spectral approximation of the solution of equation ut + ux + xu = 0 fora regular initial datum.

N ϕ1

64 1.00 e + 00128 2.67 e − 11256 1.53 e − 48

spectral convergence is achieved, as can be clearly observed. In the second column, we show theerrors obtained when the initial datum is ϕ2(x) = |x|e−x2

, which is only in H 1w. The convergence

is much slower than in the other case.Even for greater values of t , the results are good for regular initial data, as can be seen in

Table II, where the values of EN∞,w(t) are given for t = 10 and initial datum ϕ1(x) = e−x2

.However, greater values of N must be considered to get small errors.

Remark 3. Even if equation ut + ux = 0 is not well posed in L2w, if we choose the initial data

u(·, 0) in the space

H ={ϕ : R → R |

∫ ∞

−∞e2xτ |ϕ(x)|2w(x) dx < ∞ ∀τ ≥ 0

},

then the solution remains in L2w for all t ≥ 0. Moreover, it is easy to prove that the Galerkin

approximation, solution of

∂uN

∂t+ πN

∂uN

∂x= 0, x ∈ R, t > 0,

uN(x, 0) = πNϕ(x) = ∑N

k=0 ϕkhk(x), x ∈ R,

(35)

is the spectral projection of the exact solution, that is, uN(·, t) = πNu(·, t) for all t ≥ 0. Therefore,if u is regular enough, convergence follows from the properties of πN . The coefficients of uN canbe explicitly written in terms of the Fourier–Hermite coefficients of the initial datum:

uk(t) = uk(t) =k∑

j=0

t k−j

(k − j)! ϕj , k = 0, . . . , N . (36)

In Table III, we show the errors EN∞,w(1) for the initial data ϕ1(x) = e−x2

, whose derivatives of any

order are in H, and ϕ2(x) = |x|e−x2, that is in H but is not regular. Here, again the computations

have been done using Mathematica� to avoid time discretization errors.

TABLE III. EN∞,w(1) error of the spectral approximation of the solution of equation ut + ux = 0 fordifferent initial data.

N ϕ1 ϕ2

32 1.62 e − 14 1.22 e − 0164 7.77 e − 37 8.40 e − 02

128 1.06 e − 90 5.67 e − 02256 2.37 e − 217 3.80 e − 02



TABLE IV. Error of the spectral approximation for the solution of ut − xux = 0 with initial datum

ϕ(x) = e−(x−1)2 , t = 1.

N EN∞,w(1)

32 2.95 e − 0264 2.36 e − 03

128 1.54 e − 05256 1.03 e − 09

Example 2. A second example in which the spectral approximation can be computed exactlycomes from taking a(x) = −x and b ≡ 0. Assumption (13) is verified and hence convergenceof both the spectral and pseudo-spectral methods is achieved. However, again in this case, it iseasily checked that the spectral approximation and the spectral projection of the exact solutioncoincide:

uN(·, t) = πNu(·, t) =N∑

k=0

uk(t)hk , ∀t ≥ 0,

where

uk(t) = e−(k+1)t

[k/2]∑j=0

(1 − e2t )j

22j j ! ϕk−2j , k = 0, . . . , N . (37)

Therefore, a better convergence rate estimate can be obtained from the properties of the spectralprojection.

In Table IV, we present the values of EN∞,w(1) for the initial datum ϕ(x) = e−(x−1)2

. Here aswell, the computations have been made using Mathematica�, and the points to approximate theL∞-norm have been taken equidistributed in the interval [−5, 5].

Only in some particular cases, as the ones presented before, the Fourier–Hermite coefficientsof a (uN)x and b uN can be written explicitly in terms of the coefficients of uN . For arbitrary a

and b, πN(a∂uN

∂x) and πN(b uN) cannot be computed exactly, and the pseudo-spectral method is

better suited for these cases. In the following examples, we will always take b ≡ 0. Convergenceof the solution of

∂uN

∂t+ 1

2IN

(a

∂uN

∂x+ ∂

∂x(IN(a uN)) − a′uN

)= 0, x ∈ R, t > 0,

uN(x, 0) = INϕ(x), x ∈ R,(38)

has been proved in section V. Nevertheless, we will also analyze the error for the approximationobtained as the solution of the approximated problem one gets when substituting the spectraloperator by the pseudo-spectral one in (16), that is,

∂uN

∂t+ IN

(a

∂uN

∂x

)= 0, x ∈ R, t > 0,

uN(x, 0) = INϕ(x), x ∈ R.(39)



TABLE V. Errors of the pseudo-spectral approximation of equation ut + 11+x2 ux = 0 with initial datum

ϕ(x) = e−x2sin x, t = 1.

Solution of (38) Solution of (39)

N EN0,w(1) EN∞,w(1) EN

0,w(1) LN∞,w(1)

32 1.93 e − 02 2.00 e − 02 1.86 e − 02 2.16 e − 0264 4.39 e − 03 4.25 e − 03 4.50 e − 03 4.82 e − 03

128 4.63 e − 04 5.33 e − 04 4.99 e − 04 6.29 e − 04256 4.31 e − 05 8.97 e − 05 4.16 e − 05 8.51 e − 05

The values of a function of VN at the nodes of the quadrature formula are obtained by a lineartransformation over its coefficients. Due to (2), derivation is also a linear transformation, as ismultiplication by a. Finally, the coefficients of the pseudo-spectral approximation are given bythe quadrature formula (10), once we have the values of the corresponding function at the nodes.Therefore, both (38) and (39) lead to systems of ordinary differential equations for the coefficientsof uN . We have applied a Runge-Kutta method of order 4 with adaptive step size, [23], to solvethese systems of ODE’s.

Example 3. As a first example, we choose a(x) = (1+x2)−1, which is strictly positive, bounded,and satisfies condition (13). In this particular case, the equations of the characteristics can beintegrated explicitly and the exact solution computed in terms of the initial datum. Therefore,exact values of u can be used to compute the errors of the approximations.

To obtain the values in Table V, we have chosen the initial datum ϕ(x) = e−x2sin x. The first

column corresponds to the error

EN0,w(t) = ‖INu(·, t) − uN(·, t)‖0,w

at time t = 1. The key point in the proof of convergence is the comparison of the pseudo-spectralapproximation with the pseudo-spectral projection of the exact solution. The norm of this termgives the convergence rate. Therefore, EN

0,w(t) is a good measure of the error of the pseudo-spectral approximation. The values in the second column are those of EN

∞,w(1), uN being thesolution of (38). The third and fourth columns present the same values, but corresponding to theapproximation (39).

Although spectral convergence is not as clear as in the previous examples, the error introducedby the Runge-Kutta method in the time discretization must be taken into account.

Example 4. We consider now a(x) = −x3(1+x2)−1. It has a real zero, linear growth at infinityand satisfies (13).

In Table VI, we give the errors corresponding to taking the initial datum ϕ(x) = e−x2sin x. The

equations of characteristics have been numerically integrated using Mathematica� to evaluatethe exact solution of the Cauchy problem. Approximations (38) and (39) are again considered.As in the previous example, the errors have the same magnitude for both of them.

Example 5. Finally, we have considered an admissible function that has infinitely many isolatedzeros in the real line and verifies condition (13), a(x) = (1 + x2)−1 sin x. Table VII is analogousto the previous ones.



TABLE VI. Errors of the pseudo-spectral approximations of the solution of equation ut − x3

1+x2 ux = 0

with initial datum ϕ(x) = e−x2sin x, t = 1.



0,w(1) EN∞,w(1)

32 3.01 e − 02 3.69 e − 02 4.08 e − 02 3.69 e − 0264 7.99 e − 03 1.02 e − 02 7.52 e − 03 1.06 e − 02

128 9.79 e − 04 1.17 e − 03 1.03 e − 03 1.14 e − 03256 1.96 e − 05 1.64 e − 05 2.92 e − 05 3.09 e − 05

TABLE VII. Errors of the pseudo-spectral approximations of the solution of equation ut + sin x

1+x2 ux = 0

with initial datum ϕ(x) = e−x2sin x, t = 1.



0,w(1) EN∞,w(1)

32 2.39 e − 03 3.02 e − 03 3.16 e − 03 6.37 e − 0364 1.66 e − 04 2.19 e − 04 2.45 e − 04 3.33 e − 04

128 8.68 e − 05 1.11 e − 04 8.70 e − 05 1.10 e − 04256 8.23 e − 05 1.10 e − 04 8.23 e − 05 1.10 e − 04

The error estimates obtained in Theorems 2 and 3 show a dependence on the growth rate of a

and b at infinity. The bigger this growth is the more regular the function must be to ensure conver-gence. In Table VIII, we compare the errors obtained with a1(x) = (1 + x2)−1, that correspondsto k = 0, and a2(x) = −x3(1+x2)−1, where k = 1. The initial datum is ϕ(x) = x5e−x2

χ[ 0,∞)(x),that is only in H 5

w. The dependence of the estimates on k in therefore observed.

Remark 4. Higher values of k may be considered at the expense of taking b not identically zeroin order to fulfill condition (13). However, large polynomial growth of the functions a and b leadto a system of ODE’s for coefficients uk(t) with huge entries and a better suited time discretizationmethod should be considered, but this is out of the scope of this article.

VII. CONCLUSIONS AND OPEN PROBLEMS

We have shown that Hermite spectral approximations are a viable alternative for the approximationof regular solutions of the Cauchy problem for linear hyperbolic equations posed in the whole real

TABLE VIII. EN∞,w(1) error of the spectral approximation of the solution of equation ut + aux = 0 fordifferent growth rate at infinity of functions a.

N a1 a2

32 2.30 e − 02 1.29 e − 0164 4.59 e − 03 3.90 e − 02

128 4.54 e − 04 6.16 e − 03256 4.20 e − 05 1.62 e − 04



line. We have proved that they are convergent in L2w, with spectral accuracy both in the Galerkin

and in the collocation cases, and we have presented numerical evidence of this convergence rate.Although spectral methods for linear hyperbolic problems were developed quite long ago,

unbounded domains of definition of the problem have not so far been considered. One of theadvantages of Hermite spectral methods is that they can be directly applied to problems definedin R, with no need to modify the domain into a bounded one.

As mentioned before, the estimates obtained in Theorems 2 and 3 imply spectral accuracy ofour approximations. Under the assumption of regularity of the exact solution, the spectral andpseudo-spectral approximations converge to the exact solution in L2

w faster than any negativepower of N .

Nothing has been mentioned about the case of nonsmooth solutions. Hyperbolic equationshave no regularizing effect; if the initial data are not regular, the solution will remain nonsmoothfor all t > 0. When nonsmooth functions are approximated by a truncated Fourier series, Gibbsphenomenon appears, whose main features are spurious oscillations in a neighborhood of thesingularities and a dramatic loss in accuracy. For discontinuous functions, the truncated Fourierseries are only first order accurate. The same situation occurs when truncated Fourier–Hermiteseries are used, and numerical evidence can be found comparing the values in the first and secondcolumns of Table III.

[8] and [24] show examples of how to tackle this problem of nonsmooth solutions in the case ofFourier spectral methods. In the case of Hermite spectral approximations, we leave the questionfor a future work.

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Spectral methods based on Hermite functions for linear hyperbolic equations