A unifying framework for the derivation and analysis of effective classes of one-step methods for ODEs

0.21 MB·English

by  Luigi Brugnano

A UNIFYING FRAMEWORK FOR THE DERIVATION AND ANALYSIS OF
EFFECTIVE CLASSES OF ONE-STEP METHODS FOR ODES∗
LUIGI BRUGNANO†, FELICE IAVERNARO‡, AND DONATO TRIGIANTE§
Abstract. In this paper, we provide a simple framework to derive and analyse several classes of effective one-
step methods. The frameworkconsists in the discretization of a local Fourier expansion of the continuous problem.
Different choices of the basis lead to different classes of methods, even though we shall here consider only the case
1
of an orthonormal polynomial basis, from which a large subclass of Runge-Kutta methods can be derived. The
1
obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian
0
BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see
2 [1] and references therein). A few numerical tests with such methods are also included, in order to confirm the
n effectiveness ofthemethods.
a
J
Key words. Ordinarydifferential equations, Runge-Kutta methods, one-step methods, Hamiltonian problems,

8 HamiltonianBoundaryValueMethods,energypreservingmethods,symplecticmethods,energydrift.

]
A AMS subject classifications. 65L05,65P10.
N
. 1. Introduction.
h
Though I have not always been able to make simple
t
a a difficult thing, I never made difficult a simple one.
m F.G.Tricomi
[
One-stepmethodsarewidelyusedinthenumericalsolutionofinitialvalueproblemsforordinary

3 differential equations which, without loss of generality, we shall assume to be in the form:
v
5 y′(t)=f(y(t)), t∈[t0,t0+T], y(t0)=y0 ∈Rm. (1.1)
6
1 In particular, we consider a very general class of effective one-step methods that can be led back
3 to a local Fourier expansion of the continuous problem over the interval [t0,t0+h], where h is the
. considered stepsize. In general, different choices of the basis result in different classes of methods,
9
0 for which, however, the analysis turns out to be remarkably simple. Though the arguments can be
0 extended to a generalchoiceof the basis,we consider here only the case ofa polynomialbasis,from
1 which one obtains a large subclass of Runge-Kutta methods. Usually, the order properties of such
:
v methods are studied through the classical theory of Butcher on rooted trees (see, e.g., [9, Th.2.13
i onp.153]), almostalwaysresortingto the so calledsimplifying assumptions (see, e.g., [9, Th.7.4on
X
p.208]). Nonetheless,suchanalysisturns outtobe greatlysimplifiedforthe methods derivedinthe
r
a new framework, which is introduced in Section 2. Similar arguments apply to the linear stability
analysis of the methods, here easily discussed through the Lyapunov method. Then, we apply the
same procedure to the case where (1.1) is a canonical Hamiltonian problem, i.e., a problem in the
form
dy 0 I
=J∇H(y), J = m , y(t )=y ∈R2m, (1.2)
dt −Im 0 0 0
(cid:18) (cid:19)
∗Workdeveloped withintheproject“Numericalmethods andsoftwarefordifferentialequations”.
†DipartimentodiMatematica“U.Dini”,Universit`adiFirenze,Italy([email protected]).
‡DipartimentodiMatematica,Universit`adiBari,Italy([email protected]).
§DipartimentodiEnergetica“S.Stecco”, Universit`adiFirenze,Italy([email protected]).
1
where H(y) is a smooth scalar function, thus obtaining, in Section 3, an alternative derivation of
the recently introduced class of energy preserving methods called Hamiltonian BVMs (HBVMs,
see [1, 2, 3] and references therein). A few numerical examples concerning such methods are then
provided in Section 4, in order to make evident their potentialities. Some concluding remarks are
then given in Section 5.
2. LocalFourierexpansionofODEs. Letusconsiderproblem(1.1)restrictedtotheinterval
[t ,t +h]:
0 0
′
y =f(y), t∈[t ,t +h], y(t )=y . (2.1)
0 0 0 0
In order to make the arguments as simple as possible, we shall hereafter assume f to be analytical.
Then, let us fix an orthonormal basis {Pˆ }∞ over the interval [0,1], even though different bases
j j=0
and/orreferenceintervalscouldbe inprincipleconsidered. Inparticular,hereafterweshallconsider
a polynomial basis: i.e., the shifted Legendre polynomials over the interval [0,1], scaled in order to
be orthonormal. Consequently,
1
Pˆ(x)Pˆ (x)dx=δ , degPˆ =j, ∀i,j ≥0,
i j ij j
Z0
where δ is the Kroneckersymbol. We can then rewrite (2.1) by expanding the right-hand side:
ij
∞
1
y′(t +ch)= Pˆ (c)γ (y), c∈[0,1]; γ (y)= Pˆ (τ)f(y(t +τh))dτ. (2.2)
0 j j j j 0
j=0 Z0
X
The basic idea (first sketched in [5]) is now that of truncating the series iafter r terms, which turns
(2.2) into
r−1 1
ω′(t +ch)= Pˆ (c)γ (ω), c∈[0,1]; γ (ω)= Pˆ (τ)f(w(t +τh))dτ. (2.3)
0 j j j j 0
j=0 Z0
X
By imposing the initial condition, one then obtains
r−1 c
ω(t +ch)=y +h γ (ω) Pˆ (x)dx, c∈[0,1]. (2.4)
0 0 j j
j=0 Z0
X
Obviously, ω is a polynomial of degree at most r. The following question then naturally arises:
“how close are y(t +h) and ω(t +h)?” The answer is readily obtained, by using the following
0 0
preliminary result.
Lemma2.1. Letg :[0,h]→Rm beasuitablyregularfunction. Then 1Pˆ (τ)g(τh)dτ =O(hj).
0 j
Proof. Assume, for sake of simplicity, R
∞
g(n)(0)
g(τh)= (τh)n
n!
n=0
X
to be the Taylor expansion of g. Then, for all j ≥0,
∞
1 g(n)(0) 1
Pˆ (τ)g(τh)dτ = hn Pˆ (τ)τndτ =O(hj),
j j
n!
Z0 n=0 Z0
X
2
since Pˆ is orthogonalto polynomials of degree n<j.
j
Asaconsequence,onehasthat(see(2.3))γ (ω)=O(hj). Moreover,foranygivent˜∈[t ,t +h],
j 0 0
we denote by y(s,t˜,y˜) the solution of (2.1)-(2.2) at time s and with initial condition y(t˜) = y˜.
Similarly, we denote by
∂
Φ(s,t˜,y˜)= y(s,t˜,y˜), (2.5)
∂y˜
also recalling the following standard result from the theory of ODEs:
∂
y(s,t˜,y˜)=−Φ(s,t˜,y˜)f(y˜). (2.6)
∂t˜
We can now state the following result, for which we provide a more direct proof, with respect
to that given in [5]. Such proof is essentially based on that of [12, Theorem6.5.1 on pp.165-166].
Theorem 2.2. Let y(t +ch) and ω(t +ch), c ∈ [0,1], be the solutions of (2.2) and (2.3),
0 0
respectively. Then, y(t +h)−ω(t +h)=O(h2r+1).
0 0
Proof. By virtue of Lemma 2.1 and (2.5)-(2.6), one has:
y(t +h)−ω(t +h) = y(t +h,t ,y )−y(t +h,t +h,ω(t +h))
0 0 0 0 0 0 0 0
t0+h d t0+h ∂ ∂
′
= y(t +h,τ,ω(τ))dτ = y(t +h,τ,ω(τ))+ y(t +h,τ,ω(τ))ω (τ) dτ
0 0 0
dτ ∂τ ∂ω
Zt0 Zt0 (cid:18) (cid:19)
1
′
=h Φ(t +h,t +ch,ω(t +ch))(−f(ω(t +ch))+ω (t +ch)) dc
0 0 0 0 0
Z0
1 ∞
=−h Φ(t +h,t +ch,ω(t +ch)) γ (ω)Pˆ (c) dc
0 0 0 j j
 
Z0 j=r
X
∞ 1   ∞
=−h Pˆ (τ)Φ(t +h,t +ch,ω(t +ch))dc γ (ω) = h O(hj)O(hj) = O(h2r+1).
j 0 0 0 j
j=r(cid:18)Z0 (cid:19) j=r
X X
The previous result reveals the extent to which the polynomial ω(t), solution of (2.3), approxi-
mates the solution y(t) of the originalproblem (2.1) on the time interval [t ,t +h]. Obviously, the
0 0
valueω(t +h)mayserveastheinitialconditionforanewIVPintheform(2.3)approximatingy(t)
0
onthe time interval[t +h,t +2h]. Ingeneral,setting t =t +ih, i=0,1,..., andassuming that
0 0 i 0
an approximation ω(t) is available on the interval [ti−2,ti−1], one can extend the approximation to
the interval [ti−1,ti] by solving the IVP
r−1 1
ω′(ti−1+ch)= Pˆj(c) Pˆj(τ)f(ω(ti−1+ch))dτ, c∈[0,1], (2.7)
j=0 Z0
X
the initial value ω(ti−1) having been computed at the preceding step. The approximationto y(t) is
thus extended on an arbitraryinterval [t ,t +Nh], and the function ω(t) is a continuous piecewise
0 0
polynomial. As a direct consequence of Theorem 2.2, we obtain the following result.
Corollary 1. Let T = Nh, where h > 0 and N is an integer. Then, the approximation to
the solution of problem (1.1) by means of (2.7) at the grid-points ti = ti−1+h, i = 1,...,N, with
ω(t )=y , is O(h2r) accurate.
0 0
3
Wenowwanttocomparetheasymptoticbehaviorofω(t)andy(t)ontheinfinitelengthinterval
[t ,+∞) in the case where f is linear or defines a canonical Hamiltonian problem. To this end we
0
introduce the the infinite sequence {ω }≡{ω(t )}.
i i
Remark 1. Though in general, the sequence {ω } cannot be formally regarded as the outcome
i
of a numerical method, under special situations, this can be the case. For example, when f is a
polynomial, the integrals in (2.3) may be explicitly determined and the IVP in (2.3) is evidently
equivalent to a nonlinear system having as unknowns the coefficients of the polynomial ω expanded
along a given basis (for example, the polynomyal ω may be computed by means of the method of
undetermined coefficients). This issue, as well as details about how to manage the integrals in
the event that the integrands do not admit an analytical primitive function in closed form, will be
thoroughly faced in Section 3.
2.1. Linear stability analysis. For the linear stability analysis, problem (2.1) becomes the
celebrated test equation
′
y =λy, ℜ(λ)≤0. (2.8)
By setting
α −β
λ=α+iβ, y =x +ix , x=(x ,x )T, A= ,
1 2 1 2 β α
(cid:18) (cid:19)
with i the imaginary unit, problem (2.8) can be rewritten as
′
x =Ax, t∈[t ,t +h], x(t ) given. (2.9)
0 0 0
Consequently, the corresponding truncated problem (2.3) becomes
r−1 1
ω′(t +ch)=A Pˆ (c) Pˆ (τ)∇V(ω(t +τh))dτ, c∈[0,1], (2.10)
0 j j 0
j=0 Z0
X
where
1
V(x)= xTx (2.11)
2
is a Lyapunov function for (2.9). From (2.10)-(2.11) one readily obtains
1
∆V(ω(t ))=V(ω(t +h))−V(ω(t )) = h ∇V(ω(t +τh))Tω′(t +τh)dτ
0 0 0 0 0
Z0
r−1 1 T 1
=h Pˆ (τ)∇V(ω(t +τh))dτ A Pˆ (τ)∇V(ω(t +τh))dτ
j 0 j 0
j=0(cid:20)Z0 (cid:21) (cid:20)Z0 (cid:21)
X
r−1 1 2
=αh Pˆ (τ)ω(t +τh)dτ .
j 0
j=0(cid:13)Z0 (cid:13)2
X(cid:13) (cid:13)
(cid:13) (cid:13)
Last equality follows by taki(cid:13)ng the symmetric part o(cid:13)f A. We observe that
r−1 1 2
ω 6=0 =⇒ Pˆ (τ)ω(t +τh)dτ >0,
j 0
j=0(cid:13)Z0 (cid:13)2
X(cid:13) (cid:13)
(cid:13) 4 (cid:13)
(cid:13) (cid:13)
since, conversely, this would imply ω(t +ch) = ρ·Pˆ (c) for a suitable ρ 6= 0 and, therefore (from
0 r
(2.10)), Pˆ′ ≡0 which is clearly not true. Thus for a generic y 6=0,
r 0
∆V(ω(t ))<0⇐⇒ℜ(λ)<0 and ∆V(ω(t ))=0⇐⇒ℜ(λ)=0.
0 0
Again,theabovecomputationcanbeextendedtoanyinterval[ti−1,ti]and,fromthediscreteversion
ofthe Lyapunovtheorem(see,e.g.,[12, Th.4.8.3onp.108]),we havethatthe sequenceω tends to
i
zero if and only if ℜ(λ)<0, while it remains bounded whenever ℜ(λ)=0, whatever is the stepsize
h>0 used. The following result is thus proved.
Theorem 2.3. The continuous solution y(t) of (2.8) and its discrete approximation ω have
i
the same stability properies, for any choice of the stepsize h>0.
2.2. The Hamiltonian case. In the case where problem (1.1) is Hamiltonian, i.e., (1.2), the
approximationprovidedbythe polynomialω in(2.3)-(2.4)inheritsaveryimportantpropertyofthe
continuous problem, i.e., energy conservation. Indeed, it is very well known that for the continuous
solution one has, by virtue of (1.2),
d
H(y(t))=∇H(y(t))Ty′(t)=∇H(y(t))TJ∇H(y(t))=0,
dt
due to the fact that matrix J is skew-symmetric. Consequently, H(y(t))= H(y ) for all t. For the
0
truncated Fourier problem, the following result holds true.
Theorem 2.4. H(ω(t +h))=H(ω(t ))≡H(y ).
0 0 0
Proof. From (2.3), considering that f(ω)=J∇H(ω) and JTJ =I, one obtains:
H(ω(t +h))−H(y )=
0 0
1 1 r−1
=h ∇H(ω(t +τh))Tω′(t +τh)dτ = h ∇H(ω(t +τh))T Pˆ (τ)γ (ω)dτ
0 0 0 j j
Z0 Z0 j=0
X
r−1 1 T r−1
=h ∇H(ω(t +τh))Pˆ (τ)dτ γ (ω) = h γ (ω)TJγ (ω) = 0,
0 j j j j
j=0(cid:18)Z0 (cid:19) j=0
X X
since J is skew-symmetric.
3. Discretization. Clearly, the integrals in (2.3), if not directly computable, need to be nu-
merically approximated. This can be done by introducing a quadrature formula based at k ≥ r
abscissae {c }, thus obtaining an approximation to (2.3):
i
r−1 k
u′(t +ch)= Pˆ (c) b Pˆ (c )f(u(t +c h)), c∈[0,1]. (3.1)
0 j ℓ j ℓ 0 ℓ
j=0 ℓ=1
X X
where the {b } are the quadrature weights, and u is the resulting polynomial, of degree at most r,
ℓ
approximating ω. It can be obtained by solving a discrete problem in the form:
r−1 k
u′(t +c h)= Pˆ (c ) b Pˆ (c )f(u(t +c h)), i=1,...,k. (3.2)
0 i j i ℓ j ℓ 0 ℓ
j=0 ℓ=1
X X
5
Let q be the order of the formula, i.e., let it be exact for polynomials of degree less than q (we
observethatq ≥k ≥r). Clearly,sinceweassumef tobe analytical,choosingk largeenough,along
with a suitable choice of the nodes {c }, allows to approximate the given integral to any degree
i
of accuracy, even though, when using finite precision arithmetic, it suffices to approximate it to
machine precision. We observe that, since the quadrature is exact for polynomials of degree q−1,
then its remainderdepends onthe q-th derivative of the integrandwith respect to τ. Consequently,
considering that Pˆ(i)(c)≡0, for i>j, one has
j
1 k
∆ (h) ≡ Pˆ (τ)f(u(t +τh))dτ − b Pˆ (c )f(u(t +c h)) = O(hq−j), (3.3)
j j 0 ℓ j ℓ 0 ℓ
Z0 ℓ=1
X
j =0,...,r−1. Thus, (3.1) is equivalent to the ODE,
r−1 1
u′(t +ch)= Pˆ (c)(γ (u)−∆ (h)), c∈[0,1], γ (u)= Pˆ (τ)f(u(t +τh))dτ, (3.4)
0 j j j j j 0
j=0 Z0
X
with u(t )=y , in place of (2.3). The following result then holds true.
0 0
Theorem 3.1. In the above hypotheses: y(t +h)−u(t +h)=O(hp+1), with p=min(q,2r).
0 0
Proof. TheproofisquitesimilartothatofTheorem2.2: byvirtueofLemma2.1and(3.3)-(3.4),
one obtains
y(t +h)−u(t +h) = y(t +h,t ,y )−y(t +h,t +h,u(t +h))
0 0 0 0 0 0 0 0
t0+h d t0+h ∂ ∂
′
= y(t +h,τ,u(τ))dτ = y(t +h,τ,u(τ))+ y(t +h,τ,u(τ))u(τ) dτ
0 0 0
dτ ∂τ ∂u
Zt0 Zt0 (cid:18) (cid:19)
1
′
=h Φ(t +h,t +ch,u(t +ch))(−f(u(t +ch))+u(t +ch)) dc
0 0 0 0 0
Z0
1 r−1 ∞
=h Φ(t +h,t +ch,u(t +ch)) Pˆ (c)∆ (h)− γ (u)Pˆ (c) dc
0 0 0 j j j j
 
Z0 j=0 j=r
X X
r−1 1  
=h Pˆ (τ)Φ(t +h,t +ch,u(t +ch))dc ∆ (u)
j 0 0 0 j
j=0(cid:18)Z0 (cid:19)
X
∞ 1
−h Pˆ (τ)Φ(t +h,t +ch,u(t +ch))dc γ (u)
j 0 0 0 j
j=r(cid:18)Z0 (cid:19)
X
r−1 ∞
=h O(hj)O(hq−j) − h O(hj)O(hj) = O(hq+1)+O(h2r+1).
j=0 j=r
X X
As an immediate consequence, one has the following result.
Corollary 2. Letq betheorder ofthequadratureformula definedbytheabscissae {c }. Then,
i
the order of the method (3.2) for approximating (1.1), with y =u(t +h), is p=min(q,2r).
1 0
Concerningthelinearstabilityanalysis,byconsideringthataquadratureformulaoforderq ≥2r
6
isexactwhentheintegrandisapolynomialofdegreeatmost2r−1,thefollowingresultimmediately
derives from Theorem 2.3.
Corollary 3. Let q be the order of the quadrature formula defined by the abscissae {c }. If
i
q ≥2r, then the method (3.2) is perfectly A-stable.1
In the case r =1, the above results apply to the methods in [10] (see also [11]).
3.1. Runge-Kutta formulation. By setting, as usual, u = u(t +c h), f = f(u ), i =
i 0 i i i
1,...,k, (3.2) can be rewritten as
r−1 ci k
u =y +h Pˆ (τ)dτ b Pˆ (c )f , i=1,...,k. (3.5)
i 0 j ℓ j ℓ ℓ
j=0Z0 ℓ=1
X X
Moreover, since q ≥ r ≥ degu, one has y = u(t +h) ≡ y +h k b f . Consequently, the
1 0 0 ℓ=1 ℓ ℓ
methods which Corollary 2 refers to are the subclass of Runge-Kutta methods with the following
P
tableau:
c
1
.. A=(a )≡ b r−1Pˆ (c ) ciPˆ (τ)dτ
. ij j ℓ=0 ℓ j 0 ℓ . (3.6)
ck (cid:16) P R (cid:17)
b ... b
1 k
In particular, in [4] it has been proved that when the nodes {c } coincide with the k Gauss points
i
in the interval [0,1], then
A=APPTΩ,
whereA∈Rk×k isthematrixintheButchertableauofthek-stagesGaussmethod,P =(Pˆj−1(ci))∈
Rk×r, and Ω=diag(b ,...,b ). In sucha way,when k =r, one obtains the classicalr-stages Gauss
1 k
collocation method. Consequently, (3.6) can be regardedas a generalizationof the classical Runge-
Kutta collocation methods, (3.2) being interpreted as extended collocation conditions.
3.2. Hamiltonian Boundary Value Methods (HBVMs). When considering a canonical
Hamiltonian problem (1.2), the discretization of the integrals appearing in (2.3) by means of a
Gaussian formula at k nodes results in the HBVM(k,r) methods introduced in [3].2 For such
methods we derive, in a novel way with respect to [1, 2, 3], the following result.
Corollary 4. For all k ≥r, HBVM(k,r) is perfectly A-stable and has order 2r. The method
is energy conserving for all polynomial Hamiltonians of degree not larger than 2k/r.
1I.e.,itsabsolutestabilityregioncoincideswiththeleft-halfcomplexplane,C−,[6].
2Adifferentdiscretization,basedatk+1Lobatto abscissae,waspreviouslyconsideredin[2].
7
Proof. The result on the order and linear stability easily follow from Corollaries 2 and 3,
respectively. Concerning energy conservation, one has
1
H(u(t +h))−H(y ) = h ∇H(u(t +τh))Tu′(t +τh)dτ
0 0 0 0
Z0
1 r−1 k
=h ∇H(u(t +τh))T Pˆ (τ) b Pˆ (c )J∇H(u(t +c h))dτ
0 j ℓ j ℓ 0 ℓ
Z0 j=0 ℓ=1
X X
r−1 1 T k
=h Pˆ (τ)∇H(u(t +τh))dτ J b Pˆ (c )∇H(u(t +c h)) = 0,
j 0 ℓ j ℓ 0 ℓ
j=0(cid:20)Z0 (cid:21) "ℓ=1 #
X X
provided that
1 k
Pˆ (τ)∇H(u(t +τh))dτ = b Pˆ (c )∇H(u(t +c h)). (3.7)
j 0 ℓ j ℓ 0 ℓ
Z0 ℓ=1
X
In the case where H is a polynomial of degree ν, this is true provided that the integrand is a
polynomial of degree at most 2k−1. Consequently, νr−1≤2k−1, i.e., ν ≤2k/r.
In the case of general Hamiltonian problems, if we consider the limit as k → ∞ we recover
formulae(2.3), whichhavebeen calledHBVM(∞,r) (or,moreingeneral, ∞-HBVMs)[1,3]: by the
way, (2.4) is nothing but the Master Functional Equation in [1, 3]. In the particular case r =1, we
derive the averaged vector field in [14] (for a related approach see [7]).
Remark 2. We observe that, in the case of polynomial Hamiltonian systems, if (3.7) holds true
for k=k∗, then
∗ ∗
HBVM(k,r)≡HBVM(k ,r)≡HBVM(∞,r), ∀k ≥k .
That is, (3.1) coincides with (2.3), for all k ≥ k∗. In the non polynomial case, the previous con-
clusions continue “practically” to hold, provided that the integrals are approximated within machine
precision.
Remark 3. As is easily deduced from the arguments in Section 3.1, the HBVM(r,r) method is
nothing but the r-stages Gauss method of order 2r (see also [3]).
4. Numerical Tests. We here provide a few numerical tests, showing the effectiveness of
HBVMs, namely of the methods obtained in the new framework, when the problem (1.1) is in the
form(1.2). In particular,we consider the Keplerproblem, whoseHamiltonian is (see, e.g.,[8, p.9]):
−1
H [q , q , p , p ]T = 1 p2+p2 − q2+q2 2 .
1 2 1 2 2 1 2 1 2
When started at
(cid:0) (cid:1) (cid:0) (cid:1) T(cid:0) (cid:1)
1−e, 0, 0, (1+e)/(1−e) , e∈[0,1),
it has an elliptic perio(cid:16)dic orbit of periodp2π and eccentricit(cid:17)y e. When e is close to 0, the problem
is efficiently solved by using a constant stepsize. However, it becomes more and more difficult as
e→1, so that a variable-stepintegrationwould be more appropriatein this case. We now compare
the following 6-th order methods for solving such problem over a 1000 periods interval:
• HBVM(3,3), i.e., the GAUSS6 method, which is a symmetric and symplectic method;
8
10−5 100
HBVM(4,3)
10−6
10−1
10−7
GAUSS6
NIAN ERROR1100−−98 ON ERROR1100−−32 GAUSS6 HBVM(15,3)
MILTO10−10 HBVM(4,3) OLUTI
HA S10−4
10−11
10−5
10−12
HBVM(15,3)
10−13 10−6
10−2 10−1 100 10−2 10−1 100
STEPSIZE STEPSIZE
Fig. 4.1. Kepler problem, e = 0.6, Hamiltonian (left plot) and solution (right plot) errors over 1000 periods
with a constant stepsize.
10−4 102
101
10−6 GAUSS6
100
R GAUSS6
N ERRO10−8 HBVM(4,3) ERROR10−1 HBVM(4,3)
NIA ON 10−2
MILTO10−10 OLUTI10−3
A S
H
HBVM(15,3)
10−4
10−12
10−5
HBVM(15,3)
10−14 10−6
100 101 102 103 100 101 102 103
PERIODS PERIODS
Fig.4.2. Keplerproblem,e=0.99,Hamiltonian(leftplot)andsolution(rightplot)errorsover1000periodswith
a variable stepsize, Tol=10−10. Note that HBVM(4,3) isnot energy preserving, on the contrary of HBVM(15,3).
• HBVM(4,3), which is symmetric [2] but not symplectic nor energy preserving, since the
Gauss quadrature formula of order 8 is not enough accurate, for this problem;
• HBVM(15,3), which is practically energy preserving,since the Gauss formula of order 30 is
accurate to machine precision, for this problem.
In the two plots in Figure 4.1 we see the obtained results when e = 0.6 and a constant stepsize is
used: as one can see,the Hamiltonian is approximatelyconservedfor the GAUSS6 and HBVM(4,3)
methods, and (practically) exactly conserved for the HBVM(15,3) method. Moreover, all methods
exhibitthesameorder(i.e.,6),withtheerrorconstantoftheHBVM(4,3)andHBVM(15,3)methods
much smaller than that of the symplectic GAUSS6 method. On the other hand, when e=0.99, we
9
consider a variable stepsize implementation with the following standard mesh-selection strategy:
1/(p+1)
Tol
h =0.7·h ,
new n
err
(cid:18) n(cid:19)
where p = 6 is the order of the method, Tol is the used tolerance, h is the current stepsize, and
n
err is an estimate of the local error. According to what stated in the literature, see, e.g., [15,
n
pp.125–127], [13, p.235], [8, pp.303–305], this is not an advisable choice for symplectic methods,
for which a drift in the Hamiltonian appears, and a quadratic error growth is experienced, as is
confirmed by the plots in Figure 4.2. The same happens for the method HBVM(4,3), which is not
energypreserving. Conversely,forthe(practically)energypreservingmethodHBVM(15,3),no drift
in the Hamiltonian occurs and a linear error growth is observed.
Remark 4. Weobservethat morerefined(thoughmoreinvolved)meshselection strategiesexist
for symplecticmethods aimedtoavoid thenumericaldrift intheHamiltonian andthequadratic error
growth(see,e.g.,[8,ChapterVIII]).However,wewanttoemphasizethattheyarenomorenecessary,
when using energy preserving methods, since obviously no drift can occur, in such a case.
5. Conclusions. In this paper, we have presented a general framework for the derivation and
analysis of effective one-step methods, which is based on a local Fourier expansion of the problem
at hand. In particular, when the chosen basis is a polynomial one, we obtain a large subclass of
Runge-Kutta methods, which can be regardedas a generalization of Gauss collocation methods.
When dealing with canonical Hamiltonian problems, the methods coincide with the recently
introduced class ofenergy preservingmethods namedHBVMs. A few numerical tests seem to show
that such methods are less sensitive to a wider class of perturbations, with respect to symplectic,
or symmetric but non energy conserving, methods. As matter of fact, on the contrary of the latter
methods, they can be conveniently associated with standard mesh selection strategies.
REFERENCES
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