Error estimates for Galerkin approximations of the Serre equations

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by  Dimitrios Antonopoulos

ERROR ESTIMATES FOR GALERKIN APPROXIMATIONS OF THE SERRE
EQUATIONS
DIMITRIOSANTONOPOULOS,VASSILIOSDOUGALIS,ANDDIMITRIOSMITSOTAKIS
Abstract. We consider the Serresystem of equations which is anonlinear dispersivesystem that models
7
two-way propagation of long waves of not necessarily small amplitude on the surface of an ideal fluid in a
1
channel. Wediscretizeinspacetheperiodicinitial-valueproblemforthesystemusingthestandardGalerkin
0 finiteelementmethodwithsmoothsplinesonauniformmeshandproveanoptimal-orderL2-errorestimate
2
for the resulting semidiscrete approximation. Using the fourth-order accurate, explicit, ‘classical’ Runge-

n Kuttaschemefortimesteppingweconstructahighlyaccuratefullydiscreteschemeinordertoapproximate
a solutionsofthesystem,inparticularsolitary-wavesolutions,andstudynumericallyphenomenasuchasthe
J resolution of general initial profiles into sequences of solitary waves, and overtaking collisions of pairs of
solitarywaves propagatinginthesamedirectionwithdifferentspeeds.
3

]
A
N 1. Introduction
.
h In this paper we will analyze standard Galerkin-finite element approximations to the periodic initial-
t value problem for the system of Serre equations. The system consists of two pde’s, approximates the two-
a
m dimensional Euler equations of water-wave theory, and models two-way propagation of long waves on the
surface of an ideal fluid in a uniform horizontalchannel of finite depth h . Specifically, if ε=a/h , where a
[ 0 0
is a typical waveamplitude, and σ =h /λ, where λ is a typical wavelength,the system is valid when σ 1
0 ≪
1 and is written in nondimensional, scaled variables in the form:
v
8 ζt+(ηu)x =0, (1)
6 σ2
6 ut+ζx+εuux− 3η η3(uxt+εuuxx−εu2x) x =0. (2)
0 (cid:2) (cid:3)
0 Here x and t are proportional to position along the channel and time, respectively, εζ, where ζ =ζ(x,t), is
1. the elevationof the free surfaceabovea levelofrestatheighty =0 ofthe verticalaxis, η =1+εζ,assumed
0 to be positive, is the water depth (as the horizontal bottom in these variables is located at y = 1), and
−
7 u = u(x,t) is the vertically averaged horizontal velocity of the fluid. (For ε = O(1) the left-hand side of
1 (2) is an O(σ4) asymptotic approximationderived from the equation of conservationof momentum in the x
:
v direction of the 2D-Euler equations; (1) is exact.)
i The system (1)-(2) was first derived by Serre, [30], and subsequently rederived by Su and Gardner, [31],
X
by Green et al., and Green and Naghdi, [17], [18], (who extended it to the case of two spatial variables and
r
a variable bottom), and others. It is also known as Green – Naghdi or fully nonlinear Boussinesq system. For
its formal derivation from the Euler equations and the derivation of related systems, cf. [21]; regarding its
rigorousjustificationasanapproximationoftheEulerequationswereferthereadertotherecentmonograph
by Lannes, [20], and its references.
In case one considers long waves of small amplitude, specifically in the Boussinesq regime ε = O(σ2),
σ 1, it is straightforwardto see that the Serre system becomes
≪
ζ +(ηu) =0,
t x
σ2
u +ζ +εuu u =O(σ4),
t x x xxt
− 3
i.e. reduces (if the right-hand side of the second equation is replaced by zero), to the ‘classical’ Boussinesq
system, [34], which has a linear dispersive term in contrast to the nonlinear dispersive terms present in (2).
2010 Mathematics Subject Classification. 65M60,35Q53.
Key words and phrases. Surface water waves, Serre equations, error estimates, standard Galerkin finite element methods,
solitarywaves.
1
(If the dispersive terms are omitted altogether,the system reduces to the shallow water equations.) Since it
isvalidforε=O(1),theSerresystem,whenwritteninitsvariable-bottomtopographyform,hasbeenfound
suitable for the description of nonlinear dispersive waves even of larger amplitude, such as water waves in
the near-shore zone before they break.
The Cauchy problem for the Serre system in nondimensional variables, that we still denote by x, t, u,
η =1+ζ, is written for x R, t 0 as
∈ ≥
η +(ηu) =0, (3)
t x
1
u +η +uu η3(u +uu u2) =0, (4)
t x x− 3η xt xx− x x
(cid:2) (cid:3)
with given initial conditions
η(x,0)=η (x), u(x,0)=u (x), x R. (5)
0 0
∈
In [24] Li proved that the initial-value problem (3)-(5) is well posed locally in time for (η,u) Hs Hs+1,
∈ ×
for s > 3/2, provided minx∈Rη0(x) > 0, and that the property minx∈Rη(x,t) > 0 is preserved while the
solution exists. (Here Hs =Hs(R), for s real,is the subspace of L2(R) consisting of (classes of) functions f
for which ∞ (1+ξ2)s fˆ(ξ)2dξ < , where fˆis the Fourier transform of f.) Li also provided a rigorous
−∞ | | ∞
justificatioRn for the Serreequations as anapproximationofthe Euler equations. Localwell-posednessof the
system in 1D in its variable bottom formulation was proved in [19]. For results on the well-posedness and
justification of the general 2D Green–Naghdi equations with bottom topography,we refer the reader to [20]
and its references. It should be noted that local temporal existence of the Cauchy problem for the scaled
equations (1)-(2) may be established in intervals of the form [0,T ], where T =O(1/ε).
ε ε
It is not hard to see, cf. [17], [23], that suitably smooth and decaying solutions of (3)-(5) preserve, over
∞ ∞
their temporal interval of existence, the mass ηdx, momentum ηudx, and energy integrals. The
−∞ −∞
latter invariant (Hamiltonian) is given by R R
1 ∞ 1
E = ηu2+ η3u2 +(η 1)2 dx. (6)
2Z 3 x −
−∞(cid:2) (cid:3)
In addition, as Serre had already noted in the second part of his paper, [30], the system (3)-(4) possesses
solitary-wavesolutionsandafamilyofperiodic(cnoidal)travellingwavesolutions;cf. also[11]forthelatter.
Closed-form formulas are known for both of these families of solutions.
In recent years many papers dealing with the numerical solution of the Serre system and its enhanced
dispersion and variable bottom topography variants have appeared. In these works the reader may find,
amongother,numericalstudiesofthegeneration,propagation,andinteractionofsolitaryandcnoidalwaves,
of the interaction of waves with boundaries, and of the effects of bottom topography on the propagation of
the waves. The numerical methods used include spectral schemes, cf. e.g. [25], [15], finite difference and
finitevolumemethods,cf. e.g. theearlypaper[27],and[32],[7],[9],[8]anditsreferences,[10],[15],standard
Galerkinmethods, cf. e.g. [29], [28], et al. In some of these papers the results ofnumericalsimulations with
theSerresystemshavebeencomparedwithexperimentaldataandalsowithnumericalsolutionsoftheEuler
equations. These comparisons bear out the effectiveness of the Serre systems in approximating the Euler
equations in a variety of variable-bottom-topography test problems, cf.e.g. [32],[7],[10], especially when the
equations are solved with hybrid numerical techniques, wherein the advective terms of the equations are
discretizedbyshock-capturingtechniques while the dispersiveterms aretreatede.g. by finite differences, cf.
e.g. [8], [9].
In the paper at hand we consider the periodic initial-value problem for the Serre equations (3)-(4) with
periodic initial data on the spatial interval [0,1], assuming that it has smooth solutions over a temporal
interval [0,T] that satisfy min η(x,t)>0.
(x,t)∈[0,1]×[0,T]
Insection2wediscretizetheprobleminspacebythestandardGalerkinmethodusingthesmoothperiodic
splines of order r 3 (i.e. piecewise polynomials of degree r 1 2) on a uniform mesh of meshlength
≥ − ≥
h. We compare the Galerkin semidiscrete approximation with a suitable spline quasiinterpolant, [33], and,
using the high order of accuracy of the truncation error(due to cancellations resulting from periodicity and
theuniformmesh),andanenergystabilityandconvergenceargument,weprovea priori optimal-ordererror
estimates in L2, i.e. of O(hr), for both components of the semidiscrete solution. This is the first error
estimate for a numerical method for the Serre system that we are aware of. As expected, the presence of
2
the nonlinear dispersive terms complicates the error analysis that is now considerably more technical than
in analogous proofs of convergence in the case of Boussinesq systems, [6], and the shallow water equations,
[2].
In section 3 we present the results of numerical experiments that we performed in order to approximate
solutions of the periodic initial-value problem for the Serre equations using mainly cubic splines in space
and the fourth-order accurate, explicit, ‘classical’ Runge-Kutta scheme for time stepping. We check first
that the resulting fully discrete scheme is stable under a Courant number restriction, enjoys optimal order
of accuracyin various norms,and approximates to high accuracyvarious types of solutions of the equations
including solitary-wave solutions. We then use this scheme to illustrate properties of the solitary waves.
In the preliminary section 3.1 we compare by analytical and numerical means the amplitudes A , A ,
S CB
A of the solitary waves of, respectively, the Serre equations, the CB system, and the Euler equations,
Euler
corresponding to the same speed c > 1, for small values of c2 1. Our study complements the analogous
−
numericalcomputations ofLi et al., [25], andour conclusionis that alwaysA <A andthat upto about
S CB
c = 1.2, A < A <A and A A < A A . For larger speeds the solitary waves of
S Euler CB Euler S Euler CB
| − | | − |
both long-wave models are no longer accurate approximations of the solitary wave of the Euler equations.
In section 3.2 we study numerically the resolution of general initial profiles into sequences of solitary waves
whenthe evolutionoccurs accordingto the Serreorthe CB equations. The number ofthe emergingsolitary
waves seems to be the same for both systems and agrees with the prediction of the asymptotic analysis of
[16]. However, the emerging solitary waves of the CB equations are faster and of larger amplitude than
their Serre counterparts. Finally, in section 3.3 we make a careful numerical study of overtaking collisions
of two solitarywaves of the Serre equations, as the ratio of their amplitudes is varied. We observedtypes of
interactionthataresimilartothecases(a),(b),and(c)ofLax’sLemma2.3in[22]fortheKdVequation. In
addition, for the Serre system, there is apparently another type of interaction, intermediate between Lax’s
cases (a) and (b).
In this paper we denote, for integer k 0, by Hk = Hk (0,1) the usual L2-based Sobolev spaces of
≥ per per
periodic functions on [0,1] and their norms by . We let Ck = Ck [0,1] be the k-times continuously
k·kk per per
differentiable 1-periodic functions. The inner product on L2 = L2(0,1) is denoted by (, ) and the corre-
· ·
sponding norm simply by . The norms on Wk =Wk(0,1) and L∞ =L∞(0,1) are denoted by
and , respectively. Pk·akre the polynomials o∞f degre∞e at most r. k·kk,∞
∞ r
k·k
The paper is dedicated to Jerry Bona, long-time friend, teacher and mentor, on the occasion of his 70th
birthday.
Acknowledgement: This work was partially supported by the programmatic agreement between Re-
search Centers-GSRT 2015-2017in the framework of the Hellenic Republic - Siemens agreement.
D.E.MitsotakiswassupportedbytheMarsdenFundadministeredbythe RoyalSocietyofNewZealand.
2. Galerkin semidiscretization
We shall analyze the Galerkin semidiscrete approximation of the periodic initial-value problem for the
Serre system in the following form. Assuming that η is positive, we multiply the pde (4) by η and consider
the periodic initial-value problem for the resulting system. Specifically, given T > 0, for t [0,T] we seek
∈
1-periodic functions η(,t) and u(,t) satisfying
· ·
η +(ηu) =0,
t x
(x,t) [0,1] [0,T],
ηu +ηη +ηuu 1 η3(u +uu u2) =0, ∈ × (S)
t x x− 3 xt xx− x x
η(x,0)=η (x), u(x(cid:2),0)=u (x), 0 x(cid:3) 1,
0 0
≤ ≤
where η , u are given 1-periodic functions. For the purposes of the error estimation we shall assume that
0 0
η and u are smooth enough with min η (x) c >0 for some constant c and that (S) has a unique
0 0 0≤x≤1 0 0 0
≥
sufficiently smooth solution (η,u) which is 1-periodic in x for all t [0,T] and is such that η(x,t) c for
0
∈ ≥
(x,t) [0,1] [0,T].
∈ ×
2.1. Smooth periodic splines and the quasiinterpolant. Let N be a positive integer and h = 1/N,
x =ih, i=0,1,...,N. Forintegerr 2considerthe associatedN-dimensionalspaceofsmooth1-periodic
i
≥
3
splines
S = φ Cr−2[0,1]:φ P ,1 i N .
h { ∈ per [xi−1,xi] ∈ r−1 ≤ ≤ }
(cid:12)
It is well known that S has the(cid:12)following approximationproperties: Given a sufficiently smooth 1-periodic
h
function v, there exists χ S such that
h
∈
s−1
hj v χ Chs v , 1 s r,
j s
k − k ≤ k k ≤ ≤
Xj=0
and
s−1
hj v χ Chs v , 1 s r,
j,∞ s,∞
k − k ≤ k k ≤ ≤
Xj=0
forsomeconstantC independentofh andv. Moreoverthereexists aconstantC independent ofhsuchthat
the inverse properties
χ Ch−(β−α) χ , 0 α β r 1,
β α
k k ≤ k k ≤ ≤ ≤ −
χ Ch−(s+1/2) χ , 0 s r 1,
s,∞
k k ≤ k k ≤ ≤ −
hold for all χ S . (In the sequel we shall denote by C generic constants independent of h.)
h
∈
Thom´ee and Wendroff, [33], proved that there exists a basis φ N of S with supp(φ ) = O(h), such
{ j}j=1 h j
that if v a sufficiently smooth 1-periodic function, the associated quasiinterpolant Q v = N v(x )φ
h j=1 j j
satisfies P
Q v v Chr v(r) . (7)
h
k − k≤ k k
Inaddition, itwasshownin[33]thatthe basis φ N maybe chosensothat the followingpropertieshold:
{ j}j=1
(i) If ψ S , then
h
∈
ψ Ch−1 max (ψ,φ ). (8)
i
k k≤ 1≤i≤N| |
(It follows from (8) that if ψ S , f L2 are such that
h
∈ ∈
(ψ,φ )=(f,φ )+O(hα), for 1 i N,
i i
≤ ≤
i.e. if (ψ P f,φ ) Chα, 1 i N, where P is the L2-projection operator on S , then ψ
h i h h
| − | ≤ ≤ ≤ k k ≤
ψ P f + P f Chα−1+ f .)
h h
k − k k k≤ k k
(ii) Let w be a sufficiently smooth 1-periodic function and ν, κ integers such that 0 ν,κ r 1. Then
≤ ≤ −
(Q w)(ν),φ(κ) =( 1)κhw(ν+κ)(x )+O(h2r+j−ν−κ), 1 i N, (9)
h i − i ≤ ≤
(cid:0) (cid:1)
where j =1 if ν+κ is even and j =2 if ν+κ is odd.
(iii) Let f, g be sufficiently smooth 1-periodic functions and ν and κ as in (ii) above. Let
β = f(Q g)(ν),φ(κ) ( 1)κ Q (fg(ν))(κ) ,φ , 1 i N.
i h i − − h i ≤ ≤
(cid:0) (cid:1) (cid:0) (cid:2) (cid:3) (cid:1)
Then
max β =O(h2r+j−ν−κ), (10)
i
1≤i≤N| |
where j as in (ii).
It is straightforwardto see that the following result also holds for the quasiinterpolant:
Lemma 2.1. Let r 3, v Hr (0,1) Wr (0,1) and put V =Q v. Then
≥ ∈ per ∩ ∞ h
V v Chr−j v , j =0,1,2, (11)
j r
k − k ≤ k k
V v Chr−j−1/2 v , j =0,1,2, (12)
j,∞ r,∞
k − k ≤ k k
V C, and V C, j =0,1,2. (13)
j j,∞
k k ≤ k k ≤
If in addition min v(x) c >0, then there exists h such that
0≤x≤1 0 0
≥
min V(x) c /2, for h h . (14)
0 0
0≤x≤1 ≥ ≤
4
Proof. Theestimates(11),(12)follow,aswasremarkedin[13],fromtheapproximationandinverseproperties
of S and (7), and imply the bounds in (13). To prove (14) note that by (12)
h
V(x)= V(x) v(x) +v(x) Chr−1/2 v +c c0,
− ≥− k kr,∞ 0 ≥ 2
(cid:0) (cid:1)
for h h , with h such that Chr−1/2 v c /2. (cid:3)
≤ 0 0 0 k kr,∞ ≤ 0
2.2. Consistency of the semidiscrete approximation. The standardGalerkinsemidiscretizationof(S)
is defined as follows. We seek (η ,u ):[0,T] S , satisfying for t [0,T] the equations
h h h
→ ∈
(η ,φ)+((η u ) ,φ)=0, φ S ,
ht h h x h
∀ ∈
(ηhuht,χ)+ 13(ηh3uhtx,χ′)+(ηhηhx,χ)+(ηhuhuhx,χ) (15)
+ 1 η3(u u u2 ,χ′)=0, χ S ,
3 h h hxx− hx ∀ ∈ h
(cid:0) (cid:1)
with initial conditions
η (0)=Q η , u (0)=Q u . (16)
h h 0 h h 0
We first establish the consistency of this semidiscretization to the p.d.e. system in (S) by proving an
optimal-order L2 estimate of a suitable truncation error of (15).
Proposition 2.1. Let (η,u) be the solution of (S) and let η, u be sufficiently smooth, 1-periodic in x. Let
r 3, H =Q η, U =Q u, and define ψ,δ :[0,T] S by the equations
h h h
≥ →
(H ,φ)+((HU) ,φ)=(ψ,φ), φ S ,
t x h
∀ ∈
(HUt,χ)+ 13(H3Utx,χ′)+(HHx,χ)+(HUUx,χ) (17)
+ 1 H3(UU U2 ,χ′)=A(δ,χ), χ S ,
3 xx− x ∀ ∈ h
(cid:0) (cid:1)
where A(v,w)=(v,w)+(v′,w′) denotes the H1 inner product. Then, there exists a constant C independent
of h, such that
max ( ψ(t) + δ(t) ) Chr. (18)
1
0≤t≤T k k k k ≤
Proof. Let ρ = Q η η = H η and σ = Q u u = U u. From the first pde in (S) and from (17) we
h h
− − − −
obtain
(ψ,φ)=(ρ ,φ)+((HU) (ηu) ,φ), φ S .
t x x h
− ∀ ∈
Since
(HU ηu) = (ρ+η)(σ+u) (ηu) =(ρσ) +(ησ) +(uρ) ,
− x x− x x x x
(cid:0) (cid:1)
it follows that
(ψ,φ)=(ρ ,φ)+((ρσ) ,φ)+(η σ+u ρ,φ)+(ψ,φ), φ S , (19)
t x x x h
∀ ∈
where ψ :[0,T] S is given by e
h
→
e(ψ,φ)=(ησ ,φ)+(uρ ,φ), φ S .
x x h
∀ ∈
In order teo estimate ψ we take into account (10) and obtain for 1 i N
≤ ≤
(ψ,φ )=(ησe,φ )+(uρ ,φ )
i x i x i
e =(η(Qhu)x ηux,φi)+(u(Qhη)x uηx,φi)
− −
=(Q (ηu ) ηu ,φ )+(Q (uη ) uη ,φ )+γ ,
h x x i h x x i i
− −
where max γ Ch2r+1. Therefore, using the remark following (8) and (7) we conclude that
1≤i≤N i
| |≤
ψ Chr. (20)
k k≤
Taking noew φ=ψ in (19), by (11), (20) we obtain
ψ Chr. (21)
k k≤
5
Proving an analogous estimate for δ is more complicated due to the presence of the nonlinear dispersive
terms. From the second pde in (S) and (17) we see that
A(δ,χ)=(HUt−ηut,χ)+ 31(H3Utx−η3utx,χ′)+(HHx−ηηx,χ)
+(HUUx−ηuux,χ)+ 31(H3UUxx−η3uuxx,χ′) (22)
− 31(H3Ux2−η3u2x,χ′), ∀χ∈Sh.
For the first term in the right-hand side of (22) we have
HU ηu =(ρ+η)(σ +u ) ηu =Hσ +u ρ,
t t t t t t t
− −
and by (7) and (13) we get
HU ηu = Hσ +u ρ Chr. (23)
t t t t
k − k k k≤
To treat the second term in the right-hand side of (22) we write
H3U η3u =(ρ+η)3(σ +u ) η3u
tx tx tx tx tx
− −
=U ρ3+3ηU ρ2+3η2ρσ +3η2u ρ+η3σ ,
tx tx tx tx tx
i.e.
1(H3U η3u )=v +v ,
3 tx− tx 1 1 (24)
v = 1U ρ3+ηU ρ2+η2ρσ , v =η2u ρ+ 1η3σ ,
1 3 tx tx e tx 1 tx 3 tx
Using (13), (12), and (11) we see that e
v C( ρ3 + ρ2 + ρσ )
1 tx
k k≤ k k k k k k
C( ρ 2 ρ + ρ ρ + ρ σ ) Ch2r−3/2,
≤ k k∞k k k k∞k k k k∞k txk ≤
from which it follows that
v Chr. (25)
1
k k≤
For the third term we have
HH ηη =(ρ+η)(ρ +η ) ηη =ρρ +η ρ+ηρ ,
x x x x x x x x
− −
i.e.
HH ηη =v +v ,
x x 2 2
− (26)
v =ρρ +η ρ, v =ηρ ,
2 x x 2e x
while for the fourth term we write
e
HUU ηuu =(ρ+η)(σ+u)(σ +u ) ηuu
x x x x x
− −
=(ρ+η)(σσ +(uσ) +uu ) ηuu
x x x x
−
=Hσσ +ρ(uσ) +uu ρ+ηu σ+ηuσ ,
x x x x x
or
HUU ηuu =v +v ,
x x 3 3
− (27)
v =Hσσ +ρ(uσ) +uu ρ+ηu σ, v =ηuσ ,
3 x x e x x 3 x
Using again (11)-(13) we see, as in the estimatioen of v1 that
v + v Chr. (28)
2 3
k k k k≤
For the fifth term in the right-hand side of (22) we write
H3UU η3uu =(ρ+η)3(σ+u)(σ +u ) η3uu
xx xx xx xx xx
− −
=ρ3UU +3ηρ2UU +3η2ρσσ +3η2ρ(σu +uσ )
xx xx xx xx xx
+3η2ρuu +η3σσ +η3σu +η3uσ ,
xx xx xx xx
6
i.e.
1(H3UU η3uu )=v +v ,
3 xx− xx 4 4
v = 1UU ρ3+ηUU ρ2+η2ρσσ +η2(u ρσ+uρσ )+ 1η3σσ (29)
4 3 xx xx e xx xx xx 3 xx
v =η2uu ρ+ 1η3u σ+ 1η3uσ .
4 xx 3 xx 3 xx
e
Hence, from (11)-(13)
v C( ρ 2 ρ + ρ ρ + ρ σ σ
k 4k≤ k k∞k k k k∞k k k k∞k k∞k xxk
+ ρ σ + ρ σ + σ σ ) Ch2r−5/2.
∞ ∞ xx ∞ xx
k k k k k k k k k k k k ≤
Therefore, since r 3,
≥
v Chr. (30)
4
k k≤
Finally, for the last term in the right-hand side of (22) we have
H3U2 η3u2 =(ρ+η)3(σ +u )2 η3u2
x − x x x − x
=ρ3U2+3ηρ2U2+3η2ρ(σ2+2u σ +u2)+η3(σ2+2u σ ),
x x x x x x x x x
i.e.
1(H3U2 η3u2)=v +v ,
3 x − x 5 5
v = 1ρ3U2+ηρ2U2+η2ρσ2+2η2ρu σ + 1η3σ2, (31)
5 3 x x e x x x 3 x
v =η2u2ρ+ 2η3u σ .
5 x 3 x x
e
From (11)-(13) we have as before
v C( ρ 2 ρ + ρ ρ + ρ σ σ
k 5k≤ k k∞k k k k∞k k k k∞k xk∞k xk
+ ρ σ + σ σ ) Ch2r−5/2,
∞ x x ∞ x
k k k k k k k k ≤
which gives, since r 3,
≥
v Chr. (32)
5
k k≤
Hence, from (22), (24), (26), (27), (29), and (30) we have for χ S
h
∈
A(δ,χ)=(Hσ +u ρ,χ)+(v ,χ′)+(v +v ,χ)+(v v ,χ′)+(δ,χ), (33)
t t 1 2 3 4 5
−
e
where we have defined δ :[0,T] S by the equation
h
→
e
(δ,χ)=(v ,χ′)+(v ,χ)+(v ,χ)+(v ,χ′) (v ,χ′), χ S .
1 2 3 4 5 h
− ∈
Using theedefinitieons of v ,e1 i 5e, from (24e), (26), (2e7), (29), and (31), we obtain for χ S
i h
≤ ≤ ∈
(δ,χ)=(η2u ρe,χ′)+ 1(η3σ ,χ′)+(ηρ ,χ)+(ηuσ ,χ)+(η2uu ρ,χ′)
tx 3 tx x x xx (34)
e + 13(η3uxxσ,χ′)+ 31(η3uσxx,χ′)−(η2u2xρ,χ′)− 32(η3uxσx,χ′).
The term (δ,χ) consists, like (ψ,φ), of L2 inner products of ρ, σ, σ and their spatial derivatives with χ or
t
χ′ and with smooth periodic functions as weights. To treat these terms we invoke again the cancellation
e e
7
property (10) of the quasiinterpolant and write for 1 i N
≤ ≤
(δ,φ )=(η2u Q η,φ′) (η3u ,φ′)+ 1 η3(Q u ) ,φ′ 1(η3u ,φ′)
i tx h i − tx i 3 h t x i − 3 tx i
(cid:0) (cid:1)
e + η(Qhη)x,φi (ηηx,φi)+(ηu(Qhu)x,φi) (ηuux,φi)
− −
+(cid:0)(η2uu Q η,(cid:1)φ′) (η3uu ,φ′)+ 1(η3u Q u,φ′) 1(η3u u,φ′)
xx h i − xx i 3 xx h i − 3 xx i
+ 1(η3u(Q u) ,φ′) 1(η3uu ,φ′) (η2u2Q η,φ′)+(η3u2,φ′)
3 h xx i − 3 xx i − x h i x i
2(η3u (Q u) ,φ′)+ 2(η3u2,φ′)
− 3 x h x i 3 x i
= Q (η3u ) (η3u ) ,φ +γ(1)
− h tx x − tx x i i
(cid:0)1 Q(cid:2) (η3u )(cid:3) (η3u ) ,φ(cid:1) +γ(2)+ Q (ηη ) ηη ,φ +γ(3)
− 3 h tx x − tx x i i h x − x i i
+ Q(cid:0) (η(cid:2)uu ) η(cid:3)uu ,φ +γ(4) (cid:1) Q [(η3u(cid:0)u ) ] (η3uu )(cid:1),φ +γ(5)
h x − x i i − h xx x − xx x i i
(cid:0)1 Q (η3uu ) (η(cid:1)3uu ) ,φ(cid:0) +γ(6) (cid:1)
− 3 h xx x − xx x i i
1(cid:0)Q (cid:2)(η3uu ) (cid:3) (η3uu ) ,φ (cid:1)+γ(7)
− 3 h xx x − xx x i i
+ Q(cid:0) ((cid:2)η3u2) ((cid:3)η3u2) ,φ +γ(8(cid:1))
h x x − x x i i
+(cid:0)2 Q(cid:2) (η3u2)(cid:3) (η3u2) ,φ(cid:1) +γ(9),
3 h x x − x x i i
(cid:0) (cid:2) (cid:3) (cid:1)
where max γ(1) + γ(3) + γ(4) + γ(5) + γ(6) + γ(8) Ch2r+1, while max γ(2) + γ(7) +
1≤i≤N | i | | i | | i | | i | | i | | i | ≤ 1≤i≤N | i | | i |
γ(9) Ch2r−1.(cid:0)Therefore, by the remark following (8) and b(cid:1)y (7) we conclude (cid:0)
| i | ≤
(cid:1)
δ C(h2r−2+hr) Chr. (35)
k k≤ ≤
Putting neow χ=δ in (33) and using (23), (25), (28), (30), (32), and (35) we obtain finally
δ Chr, (36)
1
k k ≤
which together with (21) gives the desired estimate (18). (cid:3)
2.3. Error estimate. We now prove using an energy technique an optimal-order L2 estimate for the error
of the semidiscrete approximationdefined by the initial-value problem (15)-(16).
Theorem 2.1. Suppose that the solution (η,u) of (S) is sufficiently smooth and satisfies min η(x,t)
0≤x≤1
≥
c for t [0,T] for some positive constant c . Suppose that r 3 and that h is sufficiently small. Then,
0 0
∈ ≥
there is a unique solution (η ,u ) of (15)-(16) on [0,T], which satisfies
h h
max ( η(t) η (t) + u(t) u (t) ) Chr. (37)
h h
0≤t≤T k − k k − k ≤
Proof. Clearly the ode initial-value-problem(15)-(16) has a unique solution locally in t. While this solution
exists we let H =Q η, U =Q u, θ =H η , and ξ =U u . Using (15) and (17) we have
h h h h
− −
(θ ,φ)+((HU) (η u ) ,φ)=(ψ,φ), φ S ,
t x h h x h
− ∀ ∈
(HUt−ηhuht,χ)+ 31(H3Utx−ηh3uhtx,χ′)+(w1+w2,χ) (38)
+ 1(w w ,χ′)=A(δ,χ), χ S ,
3 3− 4 ∀ ∈ h
where
w =HH η η , w =HUU η u u ,
1 x h hx 2 x h h hx
− −
w =H3UU η3u u , w =H3U2 η3u3 .
3 xx− h h hxx 4 x − h hx
Since
HU η u =H(U u )+U(H η ) (H η )(U u ),
h h h h h h
− − − − − −
it follows that
HU η u =Hξ+Uθ θξ, (39)
h h
− −
8
and, consequently, from the first equation in (38)
(θ ,φ)+((Hξ) ,φ)+((Uθ) ,φ) ((θξ) ,φ)=(ψ,φ), φ S . (40)
t x x x h
− ∀ ∈
From (40), putting φ=θ and using integration by parts, we obtain
1 d θ 2+((Hξ) ,θ)+ 1(U θ,θ) ((θξ) ,θ)=(ψ,θ). (41)
2dtk k x 2 x − x
Now, putting χ=ξ in the second equation in (38) we see that
(HUt−ηhuht,ξ)+ 31(H3Utx−ηh3uhtx,ξx)+(w1+w2,ξ)+ 13(w3−w4,ξx)=A(δ,ξ). (42)
For the first term in the left-hand side of (42) we have
HU η u =H(U u ) (H η )(U u )+U (H η ),
t h ht t ht h t ht t h
− − − − − −
that is
HU η u =Hξ θξ +U θ,
t h ht t t t
− −
and therefore
(HU η u ,ξ)=(Hξ ,ξ) (θξ ,ξ)+(U θ,ξ)
t h ht t t t
− −
1
= 1 d H(x,t)ξ2(x,t)dx 1(H ,ξ2)
2dtZ − 2 t (43)
0
1
1 d θ(x,t)ξ2(x,t)dx+ 1(θ ,ξ2)+(U θ,ξ).
− 2dtZ 2 t t
0
For the second term in the left-hand side of (42) it holds that
H3U η3u =H3(U u )+(H3 η3)U (H3 η3)(U u )
tx− h htx tx− htx − h tx− − h tx− htx
=H3ξ +(H3 η3)U (H3 η3)ξ ,
tx − h tx− − h tx
from which
(H3U η3u ,ξ )=(H3ξ ,ξ )+((H3 η3)U ,ξ ) ((H3 η3)ξ ,ξ ).
tx− h htx x tx x − h tx x − − h tx x
Hence,
1
1(H3U η3u ,ξ )= 1 d H3(x,t)ξ2(x,t)dx 1(H2H ,ξ2)
3 tx− h htx x 6dtZ x − 2 t x
0
1 (44)
+ 1((H3 η3)U ,ξ ) 1 d (H3 η3)(x,t)ξ2(x,t)dx
3 − h tx x − 6dtZ − h x
0
+ 1(H2H η2η ,ξ2).
2 t− h ht x
For w we have
1
w =HH η η =H(H η )+H (H η ) (H η )(H η )
1 x h hx x hx x h h x hx
− − − − − −
=(Hθ) θθ ,
x x
−
i.e.
(w ,ξ)= (Hθ,ξ ) (θθ ,ξ). (45)
1 x x
− −
Moreover
w =HUU η u u =HU(U u )+(HU η u )U (HU η u )(U u ),
2 x h h hx x hx h h x h h x hx
− − − − − −
and in view of (39)
w =HUξ +(Hξ+Uθ θξ)U (HU η u )ξ .
2 x x h h x
− − −
Thus, integrating by parts,
(w ,ξ)= 1((HU) ,ξ2)+(HU ,ξ2)+(UU θ,ξ) (U θξ,ξ)
2 −2 x x x − x (46)
+ 1((HU η u ) ,ξ2).
2 − h h x
9
Now, since
w =H3UU η3u u =H3UU η3u (U ξ )
3 xx− h h hxx xx− h h xx− xx
=η3u ξ +(H3U η3u )U ,
h h xx − h h xx
and since
H3U η3u =H3U η3(U ξ)=(H3 η3)U +η3ξ,
− h h − h − − h h
we get
w =η3u ξ +(H3 η3)UU +η3U ξ,
3 h h xx − h xx h xx
and therefore
(w ,ξ )= 1(3η2η u +η3u ,ξ2)+((H3 η3)UU ,ξ )+(η3U ξ,ξ ). (47)
3 x −2 h hx h h hx x − h xx x h xx x
In addition
w =H3U2 η3u2 =H3U2 η3(U ξ )2 =(H3 η3)U2+2η3U ξ η3ξ2
4 x − h hx x − h x− x − h x h x x− h x
=(H3 η3)U2+η3U ξ +η3u ξ .
− h x h x x h hx x
Hence, from this relation and (47) it follows that
(w w ,ξ )=((H3 η3)(UU U2),ξ ) 3(η2η u +η3u ,ξ2)
3− 4 x − h xx− x x − 2 h hx h h hx x (48)
+(η3U ξ,ξ ) (η3U ,ξ2).
h xx x − h x x
Noting that
((Hξ) ,θ) ((θξ) ,θ) (Hθ,ξ ) (θθ ,ξ)=(H ξ,θ),
x x x x x
− − −
and taking into account (45), (43), (44), (46), and (48), if we add (41) and (42) we obtain
1
1 d θ 2+ 1 d [H(x,t)ξ2(x,t)+ 1H3(x,t)ξ2(x,t)]dx
2dtk k 2dtZ 3 x
0
1 1 (49)
= 1 d θ(x,t)ξ2(x,t)dx+ 1 d (H3 η3)(x,t)ξ2(x,t)dx
2dtZ 6dtZ − h x
0 0
+w +w +w +w ,
1 2 3 4
where
e e e e
w = (H ξ,θ) 1(U θ,θ), w = 1(H ,ξ2) 1(θ ,ξ2) (U θ,ξ),
1 − x − 2 x 2 2 t − 2 t − t
w = 1(H2H ,ξ2) 1((H3 η3)U ,ξ ) 1(H2H η2η ,ξ2),
e3 2 t x − 3 − he tx x − 2 t− h ht x
w = (w ,ξ) 1(w w ,ξ )+(ψ,θ)+A(δ,ξ).
e4 − 2 − 3 3− 4 x
From (13) if follows that
e
w C θ ξ +C θ 2. (50)
1
| |≤ k kk k k k
Taking into account (16), (11), (12), a straightforward estimate for θ that we get from (40) with φ = θ,
e t
and arguing by continuity, we conclude that there is a maximal time t (0,T] such that the solution of
h
∈
(15)-(16) exists for 0 t t and satisfies
h
≤ ≤
max ( θ (s) + θ(s) + ξ(s) ) 1. (Y)
t ∞ 1,∞ 1,∞
0≤s≤th k k k k k k ≤
Then, from (Y) and (13) it follows for 0 t t that
h
≤ ≤
w C ξ 2+C θ ξ , (51)
2
| |≤ k k k kk k
To derive a bound for w , note that H3 η3 =θ(H2+Hη +η2), and therefore that
e 3 − h h h
w = 1(H2U θ,ξ ) 1(HU η θ,ξ ) 1(U η2θ,ξ )+ 1(η2η ,ξ2).
3 −3 tex x − 3 tx h x − 3 tx h x 2 h ht x
Since η θ + H , it follows from (Y) and (13) that η C for 0 t t . Similarly,
k hek1,∞ ≤k k1,∞ k k1,∞ k hk1,∞ ≤ ≤ ≤ h
η C for 0 t t . Therefore, using again (13) we conclude for 0 t t that
ht ∞ h h
k k ≤ ≤ ≤ ≤ ≤
w C( θ ξ + ξ 2). (52)
3 x x
| |≤ k kk k k k
10
e