art%3a10.1007%2fs10444-008-9067-6

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

1/32

Adv Comput Math (2009) 30:249280DOI 10.1007/s10444-008-9067-6

LegendreGauss collocation methods for ordinary

differential equations

Ben-yu Guo Zhong-qing Wang

Received: 7 December 2006 / Accepted: 29 November 2007 /

Published online: 16 May 2008 Springer Science + Business Media, LLC 2008

Abstract In this paper, we propose two efficient numerical integration processes forinitial value problems of ordinary differential equations. The first algorithm is theLegendreGauss collocation method, which is easy to be implemented and possessesthe spectral accuracy. The second algorithm is a mixture of the collocation methodcoupled with domain decomposition, which can be regarded as a specific implicitLegendreGauss RungeKutta method, with the global convergence and the spectral

accuracy. Numerical results demonstrate the spectral accuracy of these approachesand coincide well with theoretical analysis.

Keywords LegendreGauss collocation methods Initial value problems of ordinary differential equations Spectral accuracy

Mathematics Subject Classifications (2000) 65L60 65L06 41A10 41A29

1 Introduction

There have been fruitful results on numerical methods for initial value problemsof ordinary differential equations, see, e.g., Butcher [6, 8], Hairer, Norsett andWanner [22], Hairer and Wanner [23], Higham [24], Lambert [25] and Stuart and

Communicated by Zhongying Chen.

B.-y. Guo (B) Z.-q. WangDepartment of Mathematics, Shanghai Normal University,200234 Shanghai, Peoples Republic of Chinae-mail: [email protected]

B.-y. Guo Z.-q. WangDivision of Computational Science of E-institute of Shanghai Universities,200234 Shanghai, Peoples Republic of China

B.-y. Guo Z.-q. WangScientific Computing Key Laboratory of Shanghai Universities,200234 Shanghai, Peoples Republic of China

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

2/32

250 B.-y. Guo, Z.-q. Wang

Humphries [27]. For Hamiltonian systems, we refer to the efficient symplectic differ-ence method, see Feng [11], Feng and Qin [12], Hairer, Lubich,and Wanner [21],and Sanz-Serna and Calvo [26]. As examples, the implicit RungeKutta methodsusually provide the numerical results with high accuracy. We often designed this

kind of numerical schemes based on Taylors expansions or quadrature formulas,see [7, 8, 22, 23, 25] and the references therein. Babuska and Janik [2] also used suchtrick for p version in time of parabolic equations.

As we know, spectral method has been widely used for numerical solutions ofvarious differential equations, see, e.g., Gottlieb and Orsag [14], Canuto, Hussaini,Quarteroni and Zang [9], Funaro [13], Boyd [4], Bernardi and Maday [3], and Guo[15]. Its main merit is the spectral accuracy. However, so far, there is very few methodwith such property, for solving initial value problems of ordinary differential equa-tions. Indeed, it is not easy to design proper algorithms with the spectral accuracy.Recently, Guo [16], and Guo and Wang [17, 18] developed the Jacobi orthogonalapproximation and the Jacobi interpolation, and obtained very sharp results. For thesymmetrical Jacobi approximation, we also refer to the work of Babu ska and Guo[1]. This fact produces the possibility of designing proper collocation methods forinitial value problems of ordinary differential equations, and analyze their numericalerrors precisely.

This paper is devoted to two numerical integration processes. In the next section,we propose the first process by using the collocation with N+ 1 nodes of theLegendreGauss interpolation. It was also described in Lambert [25]. But we derivea new algorithm. Then we use the recent results on the Jacobi interpolation in [18],

to analyze the numerical errors in Section 3. This process has several advantages.Firstly, it is easier to be implemented for nonlinear problems. Next, it possessesthe spectral accuracy. In other words, for any fixed mode N, the smoother theexact solution, the more accurate the numerical result. Moreover, even if the exactsolution has certain singularity, this algorithm still keeps high accuracy. However,the usual implicit RungeKutta schemes do not have such merits. In Section 4,we provide the second process by the LegendreGauss collocation coupled withdomain decomposition. It can be regarded as a specific implicit LegendreGaussRungeKutta method. We also derive a new algorithm with which, we can usemoderate mode N to evaluate numerical solution on all subintervals, step by step.

This simplifies actual calculation and saves work. This process not only has thesame advantages as the first process, but also the global convergence. Thereby, itis more appropriate for long time calculation of dynamical systems. In particular,for any smooth solution and fixed (mesh size in time), the numerical error decaysexponentially as N increases, while for any fixed mode N= s 1, the numericalerror decays at least as rapid as the usual s-stage implicit LegendreGauss RungeKutta method. In fact, the numerical error of this method is of the order 2ss2s+1.Thus, for raising the numerical accuracy, increasing the mode N suitably is muchmore effective than decreasing the mesh size . Section 5 is for the LegendreGauss

collocation methods of systems of ordinary differential equations. As an example, weconsider a model Hamilton system. The suggested scheme keeps the symplecticityand the spectral accuracy. We present numerical results in Section 6, which indicatethe spectral accuracy of our algorithms and coincide well with theoretical analysis. Inparticular, our new approaches provide much more accurate numerical results and

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

3/32

LegendreGauss collocation methods 251

cost much less computational time than the usual implicit LegendreGauss RungeKutta schemes. The final section is for some concluding discussions.

2 LegendreGauss collocation method

In this section, we derive the numerical integration process by using the LegendreGauss interpolation and describe its implementation.

Let Ll(t) be the standard Legendre polynomial of degree l. The shifted Legendrepolynomials to [0, T], LT,l(t) are defined by

LT,l(t) = Ll

2t

T 1

= (1)

l

l!dl

dtl

tl

1 tT

l, l= 0, 1, 2, .

In particular,

LT,0(t) = 1, LT,1(t) =2t

T 1, LT,2(t) =

6t2

T2 6t

T+ 1.

According to the properties of the standard Legendre polynomials, we have

(l+ 1)LT,l+1(t) (2l+ 1)

2t

T 1

LT,l(t) + lLT,l1(t) = 0, l 1, (2.1)

d

dtLT,l+1(t)

d

dtLT,l1(t) = 2

T(2l+ 1)LT,l(t), l 1. (2.2)

The set of LT,l(t) is a complete L2(0, T)orthogonal system, namely,T

0

LT,l(t)LT,m(t)dt=T

2l+ 1 l,m (2.3)

where l,m is the Kronecker symbol. Thus for any v L2(0, T),

v(t) =

l=0

vT,lLT,l(t),

vT,l = 2

l+ 1T

T0

v(t)LT,l(t)dt. (2.4)

We now turn to the LegendreGauss interpolation. We denote by tNj , 0 j N,the nodes of the standard LegendreGauss interpolation on the interval (1, 1).The corresponding Christoffel numbers are Nj , 0 j N. The nodes of the shiftedLegendreGauss interpolation on the interval (0, T) are the zeros of LT,N+1(t), de-

noted by tNT,j, 0 j N. Clearly, tNT,j =T

2

tNj + 1

. The corresponding Christoffel

numbers are NT,j =T

2Nj , 0 j N.

Let PN(0, T) be the set of polynomials of degree at most N. Due to the propertyof the standard LegendreGauss quadrature, it follows that for any

P2N+1(0, T),

T0

(t)dt= T2

11

T

2(t+ 1)

dt= T

2

Nj=0

Nj

T

2

tNj + 1

=

Nj=0

NT,j

tNT,j

.

(2.5)

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

4/32


Let (u, v)T and vT be the inner product and the norm of space L2(0, T), respec-tively. We also introduce the following discrete inner product and norm,

(u, v)T,N=

N

j=0 u tNT,j v tNT,jNT,j, vT,N = (v,v)12

T,N

.

Thanks to (2.5), for any P2N+1(0, T) and PN(0, T),(,)T = (,)T,N, T = T,N. (2.6)

The shifted LegendreGauss interpolation IT,Nv(t) PN(0, T) is determined by

IT,Nv

tNT,j

= v

tNT,j

, 0 j N.

Because of(2.6), for any PN+1(0, T),(IT,Nv,)T = (IT,Nv,)T,N = (v,)T,N. (2.7)

We can expand IT,Nv(t) as

IT,Nv(t) =N

l=0v NT,lLT,l(t). (2.8)

Using (2.4) and (2.7) yields

v NT,l = 2l+ 1T (IT,Nv, LT,l)T = 2l+ 1T (v, LT,l)T,N, 0 l N. (2.9)Furthermore, we have that for any PN+1(0, T),

(t) =N+1l=0

NT,lLT,l(t), IT,N(t) = Nl=0

NT,lLT,l(t).With the aid of(2.9), (2.6) and (2.4), we deduce that

NT,l = 2l+ 1T (, LT,l)T,N = 2l+ 1T (, LT,l)T = NT,l, 0 l N.The above result, along with (2.3) and (2.6), gives that for any PN+1(0, T),

2T,N = IT,N2T,N = IT,N2T = TN

l=0

1

2l+ 1 NT,l2

=T

N

l=01

2l+ 1 NT,l2

T

N+1

l=01

2l+ 1 NT,l2

=

2T. (2.10)

We now consider the model problemd

dtU(t) = f(U(t), t), 0 < t T,

U(0) = U0.(2.11)

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

5/32


The collocation method for solving (2.11) is to seek uN(t) PN+1(0, T), such that

d

dtuN

tNT,k

= f

uN

tNT,k

, tNT,k

, 0 k N,

uN

(0) = U0.(2.12)

This is an implicit scheme. If f(z, t) satisfies certain conditions, then (2.12) has aunique solution, see Appendix of this paper.

Lambert [25] designed an algorithm to resolve the discrete system (2.12). The keypoint is to solve a system with the unknown d

dtuN

tNT,k

, and then calculate uN

tNT,k

.

To this end, one expanded the unknown ddt

uN(t) by the Lagrange interpolationand derived the coefficients in the algorithm. But, as is well-known, the Lagrangeinterpolation is not stable for large N. We now derive a new implementation, inwhich one calculates the unknown uNtNT,k directly. In particular, we expand uN(t)by the shifted Legendre orthogonal polynomials, which leads to a stable algorithmeven for large N. The numerical results presented in Section 6 of this paper confirmthis assertion. For this purpose, let

uN(t) =N+1l=0

uNT,lLT,l(t), 0 < t T. (2.13)Clearly, uN(t)LT,l(t) P2N+1(0, T) for 0 l N. Therefore, multiplying (2.13) byLT,l(t), integrating the result over the interval (0, T), and using (2.3) and (2.6), we

verify that

uNT,l = 2l+ 1T uN, LT,lT = 2l+ 1T uN, LT,lT,N= 2l+ 1

T

Nj=0

uN

tNT,j

LT,l

tNT,j

NT,j, 0 l N. (2.14)

On the other hand, LT,l(0) = (1)l. Thereby, we obtain from (2.12)(2.14) that

uNT,N+1 = (1)N+1U0 + Nl=0

(1)N+luNT,l= (1)N+1U0+ 1

T

Nl=0

Nj=0

(1)N+l(2l+1)uN

tNT,j

LT,l

tNT,j

NT,j. (2.15)

Furthermore, let [l] be the integer part ofl. By virtue of (2.2),d

dt

LT,l(t)

=d

dt

LT,l

2(t)

+2

T

(2l

1)LT,l

1(t).

= ddt

LT,l4(t) +2

T(2l 1)LT,l1(t) +

2

T(2l 5)LT,l3(t)

= = 2T

l1

2

m=0

(2l 4m 1)LT,l2m1(t).

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

6/32


Hence, we use (2.13)(2.15) to obtain

d

dtuN(t) =

N+1

l=1 uNT,l

d

dtLT,l(t) = 2

T

N+1

l=1 uNT,l

l1

2

m=0(2l 4m 1)LT,l2m1(t)

= 2T2

Nl=1

(2l+ 1) N

j=0uN

tNT,j

LT,l

tNT,j

NT,j

l12

m=0(2l 4m 1)LT,l2m1(t)

+2

T2

N

l=0 (1)N+l(2l+ 1)N

j=0 uNtNT,j LT,ltNT,jNT,j N2

m=0(2N 4m + 1)LT,N2m(t)

+(1)N+1 2U0

T

N2

m=0

(2N 4m + 1)LT,N2m(t). (2.16)

For simplicity of statements, we shall use the following notations,

aNT,k,j = 2NT,jN

l=1(2l+ 1)LT,l

tNT,j

l12 m=0

(2l 4m 1)LT,l2m1

tNT,k

+2NT,jN

l=0(1)N+l(2l+ 1)LT,l

tNT,j

N2 m=0

(2N 4m + 1)LT,N2m

tNT,k ,

b NT,k = (1)N+12 N2

m=0(2N 4m + 1)LT,N2m tNT,k , 0 j N.

Then (2.16) reads

d

dtuN

tNT,k

= 1T2

Nj=0

aNT,k,juN

tNT,j

+ U0

Tb NT,k, 0 k N. (2.17)

Further, let

uN

= uNtNT,0 , uNtNT,1 , , uNtNT,NT , bNT = b NT,0, b NT,1, , b NT,NT ,FNT

uN = fuNtNT,0 , tNT,0 , fuNtNT,1 , tNT,1 , , fuNtNT,N , tNT,NT ,

and ANT be the matrix with the entries aNT,k,j, 0 j, k N. Then we can rewrite

(2.17) as the following matrix form,

ANTu

N = T2FNT

uN TU0bNT. (2.18)

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

7/32


In actual computation, we first use (2.18) to evaluate uN

tNT,k

, 0 k N, and thenuse (2.13)(2.15) to obtain

uN(T)=

N+1

l=0 uNT,l =1

T

N

j=0 N

l=0 1 + (1)N+l (2l+ 1)LT,ltNT,j uN

tNT,j

NT,j+ (1)N+1U0. (2.19)

Remark 2.1 Lambert [25] also derived a collocation method based on (2.12). Wright[28] explored the relationship between that method and the implicit RungeKuttamethod. In [25], one approximates the derivative d

dtU(t) by wN(t) PN+1(0, T),

which is expanded by the Lagrange interpolation, namely,

wN(t) = Nk=0

w N

tNT,kLk(t), Lk(t) =

0jN,j=k

t tNT,jtNT,k tNT,j

. (2.20)

Then one evaluates wN

tNT,k

by

wN

tNT,k = ftNT,k

0

w N(t)dt+ U0, tNT,k

. (2.21)

Finally, the numerical solution is determined by

uN

tNT,k = tNT,k

0

wN(t)dt+ U0. (2.22)

By (2.20), we could replace the integral involved in (2.21) and (2.22) by a linearcombination ofwN

tNT,k

, see pages 194195 of [25].

In our algorithm (2.18), we calculate U

tNT,k

directly. In particular, for strongly

nonlinear function f(z, t), it is easier to resolve (2.18) than (2.21). Next, as we know,

the Lagrange interpolation (2.20) is not stable for large N. However, we used theshifted Legendre interpolation in this work, which is stable for large N. Furthermore,in the estimates of numerical errors of our method, there exist certain additionalfactors like Nr, see Sections 35 of this paper. Therefore, our method might bemore stable and more accurate for large mode N. The numerical results presented inSection 6 of this paper confirm the above assertion.

3 Error analysis of LegendreGauss collocation method

In this section, we analyze the convergence of scheme (2.12). In particular, we shallprove the spectral accuracy of numerical solution uN(t). As usual, we shall firstcompare uN(t) with the interpolation IT,NU(t). For this purpose, let

GNT,1(t) = IT,Nd

dtU(t) d

dtIT,NU(t).

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

8/32


Then we have from (2.11) that

d

dtIT,NU

tNT,k

= f

U

tNT,k

, tNT,k

GNT,1

tNT,k

, 0 k N. (3.1)

Further, let EN(t) = uN(t) IT,NU(t). Subtracting (3.1) from (2.12) yieldsd

dtEN

tNT,k

= GNT,1 tNT,k+ GNT,2 tNT,k , 0 k N,EN(0) = U0 IT,NU(0)

(3.2)

where

GNT,2

tNT,k

= f

uN

tNT,k

, tNT,k

f

IT,NU

tNT,k

, tNT,k

.

We now multiply the first formula of (3.2) by 2ENtNT,kNT,k, and sum the resultingequation for 0 k N. Then it follows that

2

EN,

d

dtEN

T,N

= ANT,1 + ANT,2 (3.3)

where

ANT,1 = 2

GNT,1, EN

T,N, ANT,2 = 2

GNT,2, E

N

T,N.

Obviously,d

dtEN

(t) PN(0, T). Thus by (2.6),2

EN,

d

dtEN

T,N

= 2

EN,d

dtEN

T

= EN(T)2 (EN(0))2. (3.4)Since GNT,1(t) PN(0, T), we use (2.6) to obtain that for any > 0,

|ANT,1| EN2T,N +1

GNT,12T,N = EN2T,N +

1

GNT,12T.

Inserting the above and (3.4) into (3.3) gives that

(EN(T))2 ANT,2 + EN2T,N +1

GNT,12T + (EN(0))2. (3.5)

We next estimate GNT,1T. Let IN be the standard LegendreGauss interpolationon the interval (1, 1). Denote by c a generic positive constant independent ofT, Nand any function. According to (4.27) of [18] with = = = = 0, for any v Hr

(1t2)r(1, 1) and integer r 0,

INv

v

2

L2

(1,1) cN2r

1

1 1 t2r dr

dtrv(t)

2

dt.

Moreover, by (4.25) of[18] with = = 0, for any v Hr(1t2)r1 (1, 1) and integer

r 2, ddt(INv v)2

L2(1,1) cN42r

11

1 t2r1 dr

dtrv(t)

2dt.

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

9/32


Accordingly,

IT,Nv v2T cN2rT

0

tr(T t)r

dr

dtrv(t)

2dt, (3.6)

ddt(IT,Nv v)2

T

cN42rT

0

tr1(T t)r1

dr

dtrv(t)

2dt. (3.7)

For simplicity of statements, we shall use the following notation

RrT, (v) =

tr+2 (T t) r+2 dr+2vdtr+22

T

, = 1, 2.

Then, by (3.6) with U and r+ 1, instead ofv and r, we haveIT,N ddtU ddtU2

T

cN2r2RrT,1(U).

Moreover, by (3.7) with U and r+ 2, instead ofv and r, we have ddt(IT,NU U)2

T

cN2rRrT,1(U).

Therefore

GNT,12T 2 ddt(IT,NU U)2T + 2 IT,N ddtU ddtU2T c N2r + N2r2RrT,1(U) cN2rRrT,1(U). (3.8)

On the other hand, for any v H1(0, T) (see Appendix of this paper),

maxt[0,T]

|v(t)|2 2T

v2T + 2T

dv

dt

2

T

. (3.9)

For simplicity, let

rT, (U) =

1

+ T

R

rT,1(U) + T1 N4RrT,2(U).

Then a combination of (3.6), (3.7) and (3.9), leads toEN(0)

2 =|IT,NU(0)U0|2 =|IT,NU(0)U(0)|2 2

TIT,NUU2T + 2T

d

dt(IT,NUU)

2

T

cN2rrT,(U). (3.10)

Substituting (3.8) and (3.10) into (3.5), we observe thatEN(T)

2 ANT,2 + EN2T,N + cN2rrT, (U). (3.11)So far, it remains to deal with ANT,2. In the sequel, we shall deal with it for twodifferent cases.

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

10/32


Case I In this case, f(z, t) fulfills the following condition that for any z1 and z2,

|f(z1, t) f(z2, t)| |z1 z2|, > 0. (3.12)

Consequently, we use (2.10) to deduce that

ANT,2 2GNT,2T,NENT,N 2EN2T,N 2EN2T. (3.13)

A combination of (2.10), (3.11) and (3.13) implies that for any > 0,

(EN(T))2 (2 + )EN2T + cN2rrT, (U). (3.14)

Furthermore, for any t [0, T],

EN(t)2 = EN(T)2 Tt ddy (EN(y))2dy EN(T)2 + 2ENT ddtENT .Integrating the above with respect to t, yields that

EN2T T

EN(T)2 + 2TENT ddtEN

T

. (3.15)

Moreover, by (2.6), (3.2), (2.10) and (3.12),

ddtENT = ddtENT,N GNT,1T,N + GNT,2T,N GNT,1T + ENT,N GNT,1T + ENT.

This with (3.15) implies that

EN(T)

2 1T

(1 2T T)EN2T 1

GNT,12T. (3.16)

Let be sufficiently small. Substituting (3.16) into (3.14) and using (3.8), we deducethat if (3.12) holds with 4T

< 1, then

EN2T cT

1 4T 2TN2rrT,

2(U). (3.17)

The above with (3.14) leads to

EN(T)

2 c(1 2T T)1 4T 2T N

2rrT, 2

(U). (3.18)

As pointed out before, |U(T) IT,NU(T)| has the same upper-bound as (3.10). Thiswith (3.18) implies that for certain constant c > 0,

|U(T) uN(T)|2 c N2rrT, (U). (3.19)

Similarly, a combination of (3.6) and (3.17) leads to

U uN2T c T N2rrT, (U). (3.20)

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

11/32


Remark 3.1 We see from (3.19) and (3.20) that the errors |U(T) uN(T)| and UuNT,N decay rapidly as Nand rincrease. The convergence rate isO(Nr). Thus, thesmoother the exact solution, the smaller the numerical errors. In other words, thescheme (2.12) possesses the spectral accuracy.

Remark 3.2 In the norms involved in RrT, (U), there exist the weights tr+ (T

t)r+ , = 1, 2. We find from (3.19) and (3.20) that such weights weaken the conditionon the regularity of exact solution. More precisely, for any fixed T, even if d

r+2Udtr+2 =

o

tr+2

2

as t 0 and dr+2U

dtr+2 = o

(T t) r+22 as t T, we still have the convergencerate as

|U(T) uN(T)| = O(Nr), U uNT,N = O(Nr). (3.21)

Accordingly, the scheme (2.12) is convergent for some weakly discontinuous functionU(t). It, in turn, means that f(z, t) could have certain singularity.

Remark 3.3 If dr+2U

dtr+2 L(0, T), then we have

|U(T) uN(T)| c12

Tr+ 3

2 Nr

1

+ T+ T N4

12

dr+2Udtr+2

L(0,T)(3.22)

and

U uNT c12

Tr+2 Nr

1

+ T+ T N4

12dr+2Udtr+2

L(0,T)

. (3.23)

In particular, if d2s+1U

dt2s+1 L(0, T) and T < 1, then we can take r= 2s 1 and N=s 1 in (3.22) and (3.23), to reach that

|U(T) uN(T)| = O T2s+ 1

2s2s+1 , U uNT,N = O T

2s+1s2s+1 . (3.24)Case II In this case, f(z, t) satisfies the condition that for any z1 and z2,

(f(z1, t) f(z2, t))(z1 z2) (z1 z2)2, > 0. (3.25)

As we know, the ordinary differential equations with such one-sided Lipschitzcondition appear frequently in the control theory, which express a strong set-valueddissipativity, cf. [10]. On the other hand, after spectral approximation to some partialdifferential equations, such as parabolic equations, we oftentimes obtain certain

systems with the unknown time-dependent coefficients, which possess the propertylike (3.25).

By (3.25), we have ANT,2 2EN2T,N. Inserting this result into (3.11) andtaking = , we obtain

EN(T)2 + EN2T,N cN2rrT,(U). (3.26)

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

12/32


Moreover, |U(T) IT,NU(T)| has the same upper-bound as (3.10). This fact with(3.26) gives that

|U(T) uN(T)|2 cN2rrT,(U) (3.27)

and

U uN2T,N = IT,NU uN2T,N = EN2T,N c

N2rrT,(U). (3.28)

Remark 3.4 The estimates (3.27) and (3.28) show again the spectral accuracy ofscheme (2.12). For fixed T, even if the exact solution has the singularity as describedin Remark 3.2, we still have the convergence rate O(Nr).

Remark 3.5 If

dr+2Udtr+2 L(0, T), then we have from (3.27) and (3.28) that

|U(T) uN(T)| c

1

+ T+ T N4

12

Tr+32 Nr

dr+2Udtr+2

L(0,T)(3.29)

and

U uNT,N c 12

1

+ T+ T N4

12

Tr+32 Nr

dr+2Udtr+2

L(0,T). (3.30)

In particular, if d2s

+1

Udt2s+1 L(0, T) and T < 1, then we may take r= 2s 1 and N=

s 1 in (3.29) and (3.30), to reach that

|U(T) uN(T)| = O

T2s+12s2s+1

, U uNT,N = O

T2s+

12s2s+1

. (3.31)

4 LegendreGauss collocation method with domain decomposition

In the last section, we investigated the LegendreGauss collocation method. The

numerical errors decay very rapidly as N and r increase. However, in actual com-putation, it is not convenient to resolve the discrete system (2.18) with very largemode N. On the other hand, for ensuring the convergence of scheme (2.12), thelength of T is limited sometimes, such as the condition (3.12) with 4T < 1.To remedy these deficiencies, we may use the suggested algorithm with moderatemode N, coupled with domain decomposition. It can be regarded as a special implicitRungeKutta method. This technique simplifies computation, saves work, and stillkeeps the spectral accuracy.

Let M be any positive integer and = TM

. Replacing T by in (2.12) andall formulas in Section 2, we can derive an alternative algorithm, with which weobtain the numerical solution uN1 (t) for 0 t . In particular, uN1 (0) = U0. Next,we evaluate the numerical solutions uNm (t) PN+1(0,) , 2 m M, such that

d

dtuNm

tN,k = fuNm tN,k , m + tN,k , 0 k N, 2 m M,

uNm(0) = uNm1(), 2 m M.(4.1)

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

13/32


Finally, the global numerical solution of (2.11) is given by

uN(t+ m ) = uNm (t), 0 t , 1 m M. (4.2)

The above algorithm is of the BN-stability in the sense of Burrage and Butcher[5], see Appendix of this paper. BN-stability means that the scheme retains thedissipativity of nonlinear problems. In other words, the propagated error arising fromthe starting error can be controlled effectively for long-time calculation.

We now analyze the numerical errors. To do this, let Um(t) = U(t+ m ) for0 t . Then

d

dtUm(t) = f(Um(t), m + t), 0 < t , 1 m M,

Um(0) = Um1(), 2 m M,U1(0) = U0.

(4.3)

Therefore, uNm (t) given by (4.1) is an approximation to Um(t), with the approximate

initial data uNm(0) = uNm1().As before, we shall analyze the numerical error for two different cases.

Case I Let (3.12) hold and 4 < 1. According to (3.19) and (3.20), weobserve that

|U( ) uN( )|2 = |U1( ) uN1 ( )|2 c N2rr, (U1),U uN2 = U1 uN1 2 c N2rr, (U1). (4.4)

Next, let ENm (t) = I, NUm(t) uNm (t). Then

|U(2 ) uN(2 )|2 = |U2( ) uN2 ( )|2 2|U2( ) I, NU2( )|2 + 2

EN2 ( )2

.

(4.5)

Using (3.5), (3.13), (2.10) and (3.8) successively, we deduce that

EN2 ( )

2 (2 + )||EN2 ||2 + EN2 (0)2 + c1 N2rRr,1(U2).Due to (3.16) and (3.8),

||EN2 ||2

1

2

EN2 ( )2 + c

(1

2

)

N2rRr,1(U2).

A combination of the previous two estimates implies

EN2 ( )

2 c EN2 (0)2 + c 1 N2rRr,1(U2) c |U2(0) I, NU2(0)|2 + c |U1( ) uN1 ( )|2 + c 1 N2rRr,1(U2).

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

14/32


With the above inequality, (3.19) and an estimate like (3.10), we have from (4.5) that

|U(2 ) uN(2 )|2 c N2r(r, (U1) + r, (U2)).Repeating the above process, we conclude that for any 1

m

M,

|U(m )uN(m )|2

c N2r1

+

mj=1

(t j + ) r+12 (j t) r+12 dr+2Udtr+22

L2(j,j )

+ 1 N4m

j=1

(tj+ ) r+22 (jt) r+22 dr+2U

dtr+2

2

L2(j,j )

, (4.6)

Similarly,

UuN2L2(0,m )

c N2r1

+ m

j=1(mj+1)

(tj+ ) r+12 (jt) r+12 dr+2Udtr+22

L2(j,j )

+N4

m

j=1 (mj+1)(tj+ )r+2

2 (j

t)

r+22

dr+2U

dtr+

2 2

L2(j,j ) .(4.7)

Remark 4.1 The estimates (4.6) and (4.7) indicate the spectral accuracy of scheme

(4.1). Moreover, for any fixed , even if dr+2U

dtr+2 = o((m t)r+2

2 ) as t m, we stillhave the convergence rate O(Nr).

Remark 4.2 If dr+2U

dtr+2 L(0, m ), then for any 1 m M,

|U(m ) uN(m )| c m 12 r+ 32 Nrdr+2Udtr+2 L(0,m ) . (4.8)U uNL2(0,m ) c mr+2 Nr

dr+2Udtr+2

L(0,m ). (4.9)


dt2s+1 L(0, T) and N= s 1, then for all 1 m M,|U(m ) uN(m )| = O

2ss2s+1 . (4.10)U uNL2(0,T) = TO

2ss2s+1

. (4.11)

As we know, if U C2s+1[0, T], then the numerical error of s-stage implicitLegendreGauss-type RungeKutta method has the accuracy O(2s), as 0.However, for fixed number of stage, the convergence rate is always limited to 2s, no

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

15/32


matter how smooth the exact solution is. Conversely, our method not only possessesthe same convergence rate 2s as 0, but also keeps the spectral accuracy.Moreover, the factor s2s+1 in (4.10) and (4.11) also decreases the numerical error.

Case II Let the condition (3.25) hold. We have from (3.27) that

|U( ) uN( )|2 = |U1( ) uN1 ( )|2 cN2rr, (U).Following the same line as in the derivations of (3.5), (3.10) and (3.27), and using(3.8) with the parameter , we obtain

|U(2 )uN(2 )|2 =|U2( )uN2 ( )|2 2|U2( )I, NU2( )|2+2|I, NU2( )uN2 ( )|2

2|U2( ) I, NU2( )|2 + 2|I, NU2(0) uN2 (0)|2 + c1 N2rRr,1(U2)

2

|U2( )

I, NU2( )

|2

+4

|U2(0)

I, NU2(0)

|2

+4

|U1( )

uN1 ( )

|2

+ c1 N2rRr,1(U2)

cN2rr, (U1) + r, (U2) .Repeating the above process, we conclude that for any 1 m M,

|U(m ) uN(m )|2

cN2r

1

+

m

j=1 (t j + )r+1

2 (j

t)

r+12

dr+2Udtr

+2

2

L2

(j,j )

+1 N4m

j=1

(tj+ ) r+22 (jt) r+22 dr+2Udtr+22

L2(j,j )

. (4.12)Furthermore, let

v, N,m =

m

j=1N

k=0v2

j + tN,k

N,k

12

.

In the same manner as before, we can prove that

U uN2, N,m

c

N2r

1

+ m

j=1(m j+ 1)

(t j + ) r+12 (j t) r+12 dr+2Udtr+22

L2(j,j )

+1 N4

m

j=1 (mj+1)(tj+ )r+2

2 (j

t)

r+22

dr+2U

dtr

+2

2

L2(j,j ).(4.13)

Remark 4.3 The estimates (4.12) and (4.13) demonstrate again the spectral accuracyof scheme (4.1). For any fixed , even ifU has the singularity as discussed in Remark4.1, we still have the convergence rate O(Nr).

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

16/32


Remark 4.4 If dr+2U

dtr+2 L(0, m ), then for any 0 m M,

|U(m ) uN(m )| c1 +1

12

m12 r+

32 Nr

dr+2Udtr

+2 L(0,m ) . (4.14)

U uN, N,m c

1 + 1

mr+

32 Nr

dr+2Udtr+2

L(0,m ). (4.15)


dt2s+1 L(0, T) and N= s 1, then for all 1 m M,

|U(m )

uN(m )

| =T

12O 2ss2s+1 , U uN, N,m = TO2s

12s2s+1.

(4.16)

5 Systems of differential equations

The proposed algorithms are also applicable to systems of ordinary differentialequations. Let

U(t) = U(1)(t), U(2)(t), , U(n)(t)T ,f (

U(t), t) = f(1)U(t), t, f(2)U(t), t, , f(n)U(t), tT.

We consider the model system

d

dt

U(t) = f U(t), t, 0 < t T,

U(0)

=U0.

(5.1)

The LegendreGauss collocation method for (5.1) is to approximateU(t) byu N(t) (PN+1(0, T))n, such that

d

dtu NtNT,k = f u NtNT,k , tNT,k , 0 k N,

u N(0) = U0.(5.2)

We can derive a matrix form of (5.2), which is similar to (2.18).Let |V|E be the Euclidean norm ofV. If

|f

Z1, t

f

Z2, t

|E |Z1 Z2|E, > 0, 4T < 1, (5.3)

then we have the error estimates similar to (3.19)(3.24).

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

17/32


On the other hand, iff

Z1, t

f

Z2, t

Z1 Z2

Z1 Z2

2

E, > 0, (5.4)

then we can derive the error estimates similar to (3.27)(3.31).The second LegendreGauss collocation method is also available for (5.1). We usethe same notations as before. The corresponding process is to determine u Nm (t) suchthat

d

dt

u Nm

tN,k = f u Nm tN,k , m + tN,k , 0 k N, 1 m M,

u Nm (0) = u Nm1(), 2 m M,u N1 (0) =

U0.

(5.5)

The global numerical solution of(5.1) is given byu N(t+ m ) = u Nm (t), 0 t , 1 m M. (5.6)

Iff

Z, t

fulfills the condition (5.3) or the condition (5.4), then we can derive theerror estimates like (4.6)(4.11) or (4.12)(4.16), respectively. Moreover, the scheme(5.5) is also BN-stable.

For example, we consider the following Hamiltonian system,

d

dt

P(t)

= 4Q(t), 0 < t

T,

d

dtQ(t) = P(t), 0 < t T,

P(0) = P0,Q(0) = Q0.

(5.7)

The corresponding Hamiltonian function is H(P, Q) = 12

P2 + 2Q2. This systempossesses the symplecticity, namely,

P2(t) + 4Q2(t) = P20 + 4Q20. (5.8)

Thus, all points (P(t), Q(t)) always lie on the same elliptic orbit.The collocation method for (5.7) is to seek pN(t) PN+1(0, T) and qN(t)

PN+1(0, T) such that

d

dtpN

tNT,k = 4qNtNT,k , 0 k N,

d

dtqN

tNT,k

= pNtNT,k , 0 k N,pN(0) = P0,qN(0)

=Q0.

(5.9)

We can derive its matrix form similar to (2.18). We now multiply the first equationof (5.9) by pN

tNT,k

NT,k and the second one by 4q

N

tNT,k

NT,k, sum the resulting

equations for 0 k N, and put them together. Then we use (2.6) to obtaind

dtpN, pN

T

+ 4

d

dtqN, qN

T

=

d

dtpN, pN

T,N

+ 4

d

dtqN, qN

T,N

= 0.

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

18/32


Therefore pN(T)

2 + 4 qN(T)2 = P20 + 4Q20 (5.10)which simulates the property (5.8) well.

We now analyze the numerical errors. Let IT,N be the same interpolation as inSection 2, and

pGNT(t) = IT,N

d

dtP(t) d

dtIT,NP(t), qG

NT(t) = IT,N

d

dtQ(t) d

dtIT,NQ(t).

Then, we have from (5.7) that

d

dtIT,NP

tNT,k

= 4Q

tNT,k

pGNT

tNT,k

, 0 k N,

ddtIT,NQ tNT,k = PtNT,k qGNT tNT,k , 0 k N. (5.11)

Further, let ENp (t) = pN(t) IT,NP(t) and ENq (t) = qN(t) IT,NQ(t). Subtracting(5.11) from (5.9) yields

d

dtENp

tNT,k

= 4ENq tNT,k+ pGNT tNT,k , 0 k N,d

dtENq

tNT,k

= ENp

tNT,k

+ qGNT

tNT,k

, 0 k N,

EN

p (0) = P0 IT,NP(0),ENq (0) = Q0 IT,NQ(0).

(5.12)

We multiply the first equation and the second equation of (5.12) by 2ENp

tNT,k

NT,kand 8ENq

tNT,k

NT,k, respectively, sum the resulting equations for 0 k N, and put

them together. Then it follows that

2

ENp ,

d

dtENp

T,N

+ 8

ENq ,d

dtENq

T,N

= 2

pGNT, E

Np

T,N

+ 8

qGNT, E

Nq

T,N

.

(5.13)

Moreover, by virtue of (2.6) and (2.10), we have

2

ENz ,

d

dtENz

T,N

=2

ENz ,d

dtENz

T

=ENz (T)2ENz (0)2 , z = p, q,(5.14)

2

zGNT, E

Nz T,N

ENz 2T +1

zGNT2T, z = p, q. (5.15)

Substituting (5.14) and (5.15) into (5.13), we reach thatENp (T)

2+ 4

ENq (T)

2 ENp 2T + 4ENq 2T +

1

pGNT2T

+4qGNT2T +

ENp (0)

2+ 4

ENq (0)

2. (5.16)

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

19/32


We can derive the upper-bounds of zGNTT and

ENz (0)2

, z = p, q, in the samemanner as in the derivations of (3.8) and (3.10). Therefore

ENp (T)

2

+ 4ENq (T)

2

ENp

2

T 4E

Nq

2

T

cN2r

W=P,Q

1

+ TR

rT,1(W) + T1 N4RrT,2(W)

. (5.17)

Furthermore, with the aid of (2.10) and (5.12), we can follow the same line as in thederivation of(3.16) to deduce that

ENp 2T T

ENp (T)2

+ 2TENp T

d

dtENp

T,N

TENp (T)

2

+2T

ENp

T4ENq T,N + pGNTT,N TENp (T)2 + 2TENp T4ENq T + pGNTT

T

ENp (T)2

+ T(1 + )ENp 2T + 16TENq 2T + T pGNT2T.Therefore,

ENp (T)2

1T

(1 T T)ENp 2T 16ENq 2T 1

pGNT2T.

Similarly,

(ENq (T))2 1

T(1 T T)ENq 2T ENp 2T

1

qGNT2T.

If5T < 1, then we follow the same line as in the derivations of (3.19) and (3.20)to reach that

|P(T) pN(T)|2 + 4|Q(T) qN(T)|2

c N2r

W=P,Q

(1+T)RrT,1(W) + T1 N4RrT,2(W)

, (5.18)

P pN2T + 4Q qN2 c N2r W=P,Q

T(1 + T)RrT,1(W) + N4RrT,2(W) .(5.19)

The second collocation method for (5.7) is to seek pNm(t) PN+1(0, ) and qNm (t) PN+1(0,) , such that

d

dtpNm

tN,k

= 4qNm

tN,k

, 0 k N, 1 m M,

d

dtqN

mtN,k = pNmtN,k, 0 k N, , 1 m M,pNm (0) = pNm1(), 2 m M,qNm (0) = qNm1(), 2 m M,pN1 (0) = P0,qN1 (0) = Q0.

(5.20)

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

20/32


The global numerical solution of(5.7) is given by

pN(t+ m ) = pNm (t), qN(t+ m ) = qNm (t), 0 t , 1 m M.

It can be checked thatpN(m )

2 + 4 qN(m )2 = P20 + 4Q20, 0 m M. (5.21)Therefore, the points

pN(m ), qN(m )

always lie on the exact orbit. In other words,

the numerical result keeps the exact orbit. Thus the scheme (5.20) is a symplecticRungeKutta scheme (cf. [12]).

Like (4.6) and (4.7), we have that for any 1 m M,

|P(m )

pN(m )

|2

+4|Q(m )

qN(m )

|2

c N2r

W=P,Q

(1 + ) mj=1

(t j + ) r+12 (j t) r+12 dr+2 Wdtr+22

L2(j,j )

+1 N4m

j=1

(t j + ) r+22 (j t) r+22 dr+2 Wdtr+22

L2(j,j )

,(5.22)

P pN2L2(0,m ) + 4Q qN2L2(0,m )

c N2r

W=P,Q

(1+ ) mj=1

(mj+1)(tj+ ) r+12 (jt) r+12 dr+2 Wdtr+2

2L2(j,j )

+N4m

j=1 (mj+1)(tj+ )r+2

2 (jt) r+22 dr+2 Wdtr

+2

2

L2

(j,j )

.

(5.23)

In particular, if d2s+1 P

dt2s+1 ,d2s+1 Qdt2s+1 L(0, T) and N= s 1, then for all 1 m M,

|P(m ) pN(m )| + |Q(m ) qN(m )| = O 2ss2s+1 . (5.24)P pNL2(0,T) + Q qNL2(0,T) = TO

2ss2s+1

. (5.25)

6 Numerical results

In this section, we present some numerical results. The algorithms are implementedby using MATLAB, and all calculations are carried out with a computer of CPUP4 3.0G, Mother Board I865PE/FSB 800/Dual Channel DDR400.

7/29/2019 art%3A10.1007%2Fs10444-008-9067-6

21/32


6.1 LegendreGauss collocation methods

Case I We first use scheme (2.18) to solve problem (2.11) with the condition (3.12).We take the test function

U(t) = (t+ 1) 32 + 5sin2t

which oscillates and grows to infinity as t . The corresponding right term at(2.11) is

f(U(t), t)=exp

1

5sin (U(t))

+ 3

2(t+1) 12 +10cos2texp

1

5sin((t+1) 32 +5sin2t)

.

Clearly, it fulfills the condition (3.12) with =1

5 e

1

5 . Therefore, as predicted by (3.19)and (3.20), for any T < 1

4= 5

4e

15 , the point-wise absolute errors |U(T) uN(T)|

and the global absolute errors U uNT decay exponentially as N .For description of numerical errors, we introduce the notations

ENT,pa = |U(T) uN(T)|, ENT,aa = U uNT,N.

In Fig. 1, we plot the point-wise absolute errors log10 of EN1,pa and the global

absolute errors log10 ofEN1,aa with various values ofN. They indicate that for T

art%3a10.1007%2fs10444-008-9067-6

Documents