Strong-stability-preserving additive linear multistep methods

0.41 MB·

by  Yiannis Hadjimichael

Strong-stability-preserving additive linear multistep
methods
∗
Yiannis Hadjimichael David I. Ketcheson
April 7, 2016
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1
0
2 Abstract

r Theanalysisofstrong-stability-preserving(SSP)linearmultistepmethodsisextended
p
A to semi-discretized problems for which different terms on the right-hand side satisfy dif-
ferentforwardEuler(orcircle)conditions. Optimaladditiveandperturbedmonotonicity-

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preservinglinearmultistepmethodsarestudiedinthecontextofsuchproblems. Optimal

perturbed methods attain larger monotonicity-preserving step sizes when the different
]
A forward Euler conditions are taken into account. On the other hand, we show that op-
N timal SSP additive methods achieve a monotonicity-preserving step-size restriction no
. betterthanthatofthecorrespondingnon-additiveSSPlinearmultistepmethods.
h
t
a
m 1 Introduction
[

2 We are interested in numerical solutions of initial value ODEs
v
7 u(cid:48)(t) = F(u(t)), t ≥ t
0
3 (1.1)
6 u(t0) = u0,
3
0 where F : Rm → Rm is a continuous function and u : [t ,∞) → Rm satisfies a monotonicity
0
.
1 property
0
6 (cid:107)u(t+∆t)(cid:107) ≤ (cid:107)u(t)(cid:107), ∀∆t ≥ 0, (1.2)
1
:
v with respect to some norm, semi-norm or convex functional (cid:107)·(cid:107) : Rm → R. In general
i
X F(u(t))mayarisefromthespatialdiscretizationofpartialdifferentialequations;forexample,
r hyperbolicconservationlaws. Asufficientconditionformonotonicityisthatthereexistssome
a
∆t > 0 such that the forward Euler condition
FE
(cid:107)u+∆tF(u)(cid:107) ≤ (cid:107)u(cid:107), 0 ≤ ∆t ≤ ∆t , (1.3)
FE
holds for all u ∈ Rm.
∗Author email addresses: {yiannis.hadjimichael, david.ketcheson}@kaust.edu.sa. This work was sup-
ported by the King Abdullah University of Science and Technology (KAUST), 4700 Thuwal,23955-6900, Saudi
Arabia.
1
In this paper we focus on linear multistep methods (LMMs) for the numerical integration
of (1.1). We denote by u the numerical approximation to u(t ), evaluated sequentially at
n n
times t = t +n∆t, n ≥ 1. At step n, a k-step linear multistep method applied to (1.1) takes
n 0
the form
k−1 k
u = ∑ α u +∆t∑ β F(u ) (1.4)
n j n−k+j j n−k+j
j=0 j=0
and if β = 0, then the method is explicit.
k
We would like to establish a discrete analogue of (1.2) for the numerical solution u in
n
(1.4). Assuming F satisfies the forward Euler condition (1.3) and all α ,β are non-negative,
j j
then convexity of (cid:107) · (cid:107) and the consistency requirement ∑k−1α = 1 imply that (cid:107)u (cid:107) ≤
j=0 j n
max (cid:107)u (cid:107) whenever ∆tβ /α ≤ ∆t for all j. Hence, the monotonicity condition
j n−k+j j j FE
(cid:107)un(cid:107) ≤ max{(cid:107)un−1(cid:107),...,(cid:107)un−k(cid:107)}. (1.5)
is satisfied under a step-size restriction
∆t ≤ C ∆t , (1.6)
LMM FE
where C = min α /β . The ratio α /β is taken to be infinity if β = 0. See [3, Chapter 8]
LMM j j j j j j
and references therein for a review of strong-stability-preserving linear multistep methods
(SSP LMMs).
Most LMMs have one or more negative coefficients, so the foregoing analysis leads to
C = 0 and thus monotonicity condition (1.5) cannot be guaranteed by positive step
LMM
sizes. However, typical numerical methods for hyperbolic conservation laws U +∇·f(U) =
t
0 involve upwind-biased semi-discretizations of the spatial derivatives. In order to pre-
serve monotonicity using methods with negative coefficients for such semi-discretizations,
downwind-biased spatial approximations may be used. Let F and F(cid:101)be respectively upwind-
and downwind-biased approximations of −∇·f(U). It is natural to assume that F(cid:101)satisfies
(cid:107)u−∆tF(u)(cid:107) ≤ (cid:107)u(cid:107), 0 ≤ ∆t ≤ ∆t , (1.7)
FE
for all u ∈ Rm. A linear multistep method that uses both F and F(cid:101)can be then written as
k−1 k (cid:16) (cid:17)
un = ∑ αjun−k+j+∆t∑ βjF(un−k+j)−β˜jF(cid:101)(un−k+j) . (1.8)
j=0 j=0
If all α are non-negative, then the method is monotonicity preserving under the restriction
j
(1.6) where the SSP coefficient is now C(cid:101) = min α /(β +β˜ ) with β , β˜ non-negative; see
LMM j j j j j j
[3, Chapter 10] and the references therein.
Downwind LMMs were originally introduced in [17, 18], with the idea that F be replaced
by F(cid:101) whenever βj < 0. Optimal explicit linear multistep schemes of order up to six, coupled
withefficientupwindanddownwindWENOdiscretizations,werestudiedin[4].Coefficients
ofoptimalupwind-anddownwind-biasedmethodstogetherwithareformulationofthenon-
linear optimization problem involved as a series of linear programming feasibility problems
2
can be found in [10]. Bounds on the maximum SSP step size for downwind-biased methods
have been analyzed in [11].
Method (1.8) can also be written in the perturbed form
k−1 k (cid:16) (cid:16) (cid:17)(cid:17)
un = ∑ αjun−k+j+∆t∑ β“jF(un−k+j)+β˜j F(un−k+j)−F(cid:101)(un−k+j) , (1.9)
j=0 j=0
where β“ = β −β˜ . We say method (1.9) is a perturbation of the LMM (1.4) with coefficients
j j j
“
βj,andthelatterisreferredtoastheunderlyingmethodfor(1.9). Byreplacing F(cid:101)with F in(1.9)
onerecoverstheunderlyingmethod. Thenotionofaperturbedmethodcanbeusefulbeyond
the realm of downwinding for hyperbolic PDE semi-discretizations. If F satisfies the forward
Eulercondition(1.3)forbothpositiveandnegativestepsizes,thenwecansimplytake F(cid:101)= F.
Insuchcases,theperturbedandunderlyingmethodsarethesame,butanalysisofaperturbed
formofthemethodcanyieldalargerstepsizeformonotonicity,givingmoreaccurateinsight
intothebehaviorofthemethod. See[7]foradiscussionofthisinthecontextofRunge–Kutta
methods, and see Example 2.2 herein for an example using multistep methods. As we will
see in Section 2, the most useful perturbed LMMs (1.9) take a form in which either β or β˜ is
j j
equaltozeroforeachvalueof j. Thus C(cid:101) = min {α /β ,α /β˜ }, andtheclassofperturbed
LMM j j j j j
LMMs (1.9) coincides with the class of downwind LMMs in [17, 18].
In this work, we adopt form (1.8) for perturbed LMMs and consider their application to
the more general class of problems (1.1) for which F and F(cid:101) satisfy forward Euler conditions
under different step-size restrictions:
(cid:107)u+∆tF(u)(cid:107) ≤ (cid:107)u(cid:107), ∀u ∈ Rm, 0 ≤ ∆t ≤ ∆t (1.10a)
FE
(cid:107)u−∆tF(cid:101)(u)(cid:107) ≤ (cid:107)u(cid:107), ∀u ∈ Rm, 0 ≤ ∆t ≤ ∆(cid:102)tFE. (1.10b)
Forafixedorderofaccuracyandnumberofsteps,anoptimalSSPmethodisdefinedtobeany
method that attains the largest possible SSP coefficient. The choice of optimal monotonicity-
preserving method for a given problem will depend on the ratio y = ∆tFE/∆(cid:102)tFE. We analyze
and construct such optimal methods. We illustrate by examples that perturbed LMMs with
larger step sizes for monotonicity can be obtained when the different step sizes in (1.10) are
accounted for.
The perturbed methods (1.8) are reminiscent of additive methods, and the latter can be
analyzed in a similar way. Consider the problem
u(cid:48)(t) = F(u(t))+F(cid:98)(u(t))
where F and F(cid:98) may represent different physical processes, such as convection and diffusion
or convection and reaction. Additive methods are expressed as
k−1 k (cid:16) (cid:17)
un = ∑ αjun−k+j+∆t∑ βjF(un−k+j)+βˆjF(cid:98)(un−k+j) ,
j=0 j=0
3
where F and F(cid:98)maysatisfytheforwardEulercondition(1.3)underpossiblydifferentstep-size
restrictions. We prove that optimal SSP explicit or implicit additive methods have coefficients
β = βˆ for all j, hence they lie within the class of ordinary (not additive) LMMs.
j j
The rest of the paper is organized as follows. In Section 2 we analyze the monotonic-
ity properties of perturbed LMMs for which the upwind and downwind operators satisfy
different forward Euler conditions. Optimal methods are derived, and their properties are
discussed. Their effectiveness is illustrated by some examples. Additive linear multistep
methods are presented in Section 3 where we prove that optimal SSP additive LMMs are
equivalent to the corresponding non-additive SSP LMMs. Monotonicity of IMEX linear mul-
tistep methods is discussed, and finally in Section 4 we summarize the main results.
2 Monotonicity-preserving perturbed linear multistep methods
The following example shows that using upwind- and downwind-biased operators allows
the construction of methods that have positive SSP coefficients, even though the underlying
methods are not SSP.
Example 2.1. Let u(cid:48)(t) = F(u(t)) be a semi-discretization of u + f(u) = 0, where F ≈
t x
−f(u) . Consider the two-step, second-order explicit linear multistep method
x
1 1 1 7
un = un−2− ∆tF(un−2)+ un−1+ ∆tF(un−1). (2.1)
2 4 2 4
The method has SSP coefficient equal to zero. Let us introduce a downwind-biased operator
F(cid:101)≈ −f(u)x such that (1.7) is satisfied. Then, a perturbed representation of (2.1) is
1 1 1 1 1
un = un−2+ ∆tF(un−2)− ∆tF(cid:101)(un−2)+ un−1+2∆tF(un−1)− ∆tF(cid:101)(un−1), (2.2)
2 4 2 2 4
in the sense that the underlying method (2.1) is retrieved from (2.2) by replacing F(cid:101) with F.
The perturbed method has SSP coefficient C(cid:101) = 2/9. There are infinitely many perturbed
LMM
representations of (2.1), but an optimal one is obtained by simply replacing F with F(cid:101)in (2.1),
yielding
1 1 1 7
un = un−2− ∆tF(cid:101)(un−2)+ un−1+ ∆tF(un−1), (2.3)
2 4 2 4
with SSP coefficient C(cid:101) = 2/7.
LMM
Remark 2.1. A LMM (1.4) has SSP coefficient C = 0 if any of the following three conditions
hold:
1. α < 0 for some j;
j
2. β < 0 for some j;
j
3. α = 0 for some j for which β (cid:54)= 0.
j j
4
By introducing a downwind operator we can remedy the second condition, but not the first
or the third. Most common methods, including the Adams–Bashforth, Adams–Moulton, and
BDF methods, satisfy condition 1 or 3, so they cannot be made SSP via downwinding.
We consider a generalization of the perturbed LMMs described previously, by assuming
different forward Euler conditions for the operators F and F(cid:101)(see (1.10)).
Definition2.1. AperturbedLMMoftheform(1.8)issaidtobestrong-stability-preserving(SSP)
with SSP coefficients (C,C(cid:101)) if conditions
β ,β˜ ≥ 0, j ∈ {0,...,k},
j j
(2.4)
α −rβ −r˜β˜ ≥ 0, j ∈ {0,...,k−1},
j j j
hold for all 0 ≤ r ≤ C and 0 ≤ r˜ ≤ C(cid:101).
By plugging the exact solution in (1.8), setting F(cid:101)(u(tn)) = F(u(tn)) and taking Taylor expan-
sions around t , it can be shown that a perturbed LMM is order p accurate if
n−k
k−1 k−1 k
∑ α = 1, ∑ jα + ∑(β −β˜ ) = k,
j j j j
j=0 j=0 j=0
(2.5)
k−1 k
∑ α ji+ ∑(β −β˜ )iji−1 = ki, i ∈ {2,...,p}.
j j j
j=0 j=0
The step-size restriction for monotonicity of an SSP perturbed LMM is given by the fol-
lowing theorem.
Theorem2.1. ConsideraninitialvalueproblemforwhichFandF(cid:101)satisfytheforwardEulerconditions
(1.10) for some ∆tFE > 0, ∆(cid:102)tFE > 0. Let a consistent perturbed LMM (1.8) be SSP with SSP
coefficients (C,C(cid:101)). Then the numerical solution satisfies the monotonicity condition (1.5) under a
step-size restriction
∆t ≤ min{C∆tFE,C(cid:101)∆(cid:102)tFE}. (2.6)
Proof. Define αk = Cβk+C(cid:101)β˜k and add αkun to both sides of (1.8) to obtain
k (cid:16) (cid:17)
(1+αk)un = ∑ αjun−k+j−∆tβjF(un−k+j)+∆tβ˜jF(cid:101)(un−k+j) .
j=0
Since the method is SSP with coefficients (C,C(cid:101)) then conditions (2.4) hold for r = C, r˜ = C(cid:101).
Let α = αˆ +α˜ with αˆ = Cβ . Then (2.4) yields α˜ ≥ C(cid:101)β˜ and β ≥ 0,β˜ ≥ 0. Thus, the
j j j j j j j j j
right-hand side can be expressed as a convex combination of forward Euler steps:
k (cid:16) β (cid:17) k (cid:16) β˜ (cid:17)
(1+αk)un = ∑αˆj un−k+j+∆t jF(un−k+j) + ∑α˜j un−k+j−∆t jF(cid:101)(un−k+j) .
αˆ α˜
j=0 j j=0 j
5
Taking norms and using the triangle inequality yields
k (cid:13) β (cid:13) k (cid:13) β˜ (cid:13)
(1+αk)(cid:107)un(cid:107) ≤ ∑αˆj(cid:13)(cid:13)un−k+j+∆t jF(un−k+j)(cid:13)(cid:13)+ ∑α˜j(cid:13)(cid:13)un−k+j−∆t jF(cid:101)(un−k+j)(cid:13)(cid:13).
αˆ α˜
j=0 j j=0 j
Under the step-size restriction ∆t ≤ min{C∆tFE,C(cid:101)∆(cid:102)tFE} we get
β β˜
∆t j ≤ ∆tFE and ∆t j ≤ ∆(cid:102)tFE.
αˆ α˜
j j
Since F and F(cid:101)satisfy (1.10a) and (1.10b) respectively, we have
k k
∑ ∑
(1+α )(cid:107)u (cid:107) ≤ αˆ (cid:107)u (cid:107)+ α˜ (cid:107)u (cid:107),
k n j n−k+j j n−k+j
j=0 j=0
and hence
k−1 k−1
∑ ∑
(cid:107)u (cid:107) ≤ α (cid:107)u (cid:107) ≤ max (cid:107)u (cid:107) α .
n j n−k+j n−k+j j
j=0 0≤j≤k−1 j=0
Consistency requires ∑k−1α = 1 and therefore the monotonicity condition (1.5) follows.
j=0 j
2.1 Optimal SSP perturbed linear multistep methods
We now turn to the problem of finding, among methods with a given number of steps k
and order of accuracy p, the largest SSP coefficients. Since C, C(cid:101) are continuous functions
of the method’s coefficients, we expect that the maximal step size (2.6) is achieved when
C = C(cid:101)∆(cid:102)tFE/∆tFE. It is thus convenient to define y := ∆tFE/∆(cid:102)tFE.
Definition 2.2. For a fixed y ∈ [0,∞) we say that an SSP method (1.8) has SSP coefficient
(cid:8) (cid:9)
C(y) = sup r ≥ 0 : monotonicity conditions (2.4) hold with r˜ = yr
and its corresponding downwind SSP coefficient is C(cid:101)(y) = yC(y). Given a number of steps k
and order of accuracy p an SSP method is called optimal, if it has SSP coefficient
(cid:8) (cid:9)
C (y) = sup C(y) > 0 : C(y) is the SSP coefficient of a k-step method (1.8) of order p .
k,p
α,β,β˜
Next we prove that for a given SSP perturbed LMM with SSP coefficient C(y), we can
construct another SSP method (1.8) with the property that for each j, either β or β˜ is zero.
j j
Example 2.1 is an application of this result.
Lemma 2.1. Consider a k-step perturbed LMM (1.8) of order p with SSP coefficient C(y) for a given
y. Then, we can construct a k-step SSP method (1.8) of order p with SSP coefficient at least C(y) that
satisfies β β˜ = 0 for each j. Moreover, both perturbed methods correspond to the same underlying
j j
method.
6
Proof. Suppose there exists an k-step SSP method (1.8) of order p with SSP coefficient C(y)
for some y ∈ [0,∞), such that β ≥ β˜ > 0 for j ∈ J ⊆ {0,1,...,k} and β˜ > β > 0 for
j j 1 j j
j ∈ J ⊆ {0,1,...,k}. Clearly J ∩ J = ∅. Define
2 1 2
(cid:40) (cid:40)
β −β˜ , if j ∈ J , 0, if j ∈/ J ,
β∗ = j j 1 β˜∗ = 2
j 0, if j ∈/ J , j β˜ −β , if j ∈ J .
1 j j 2
Observe that conditions (2.4) with r = C(y), r˜ = C(cid:101)(y) and the order conditions (2.5) are
satisfiedwhen β ,β˜ arereplacedby β∗,β˜∗. Therefore,themethodwithcoefficients(α,β∗,β˜∗)
j j j j
has SSP coefficient at least C(y) and satisfies β∗β˜∗ = 0 for each j. Finally, the definition of
j j
β∗j and β˜∗j leaves βj −β˜j invariant, thus substituting F(cid:101) = F in method (1.8) with coefficients
(α,β,β˜) or (α,β∗,β˜∗) yields the same underlying method.
The next Corollary is an immediate consequence of Lemma 2.1.
Corollary 2.1. Let k, p and y be given such that C (y) > 0. Then there exists an optimal SSP
k,p
perturbed LMM (1.8) with SSP coefficient C (y) that satisfies β β˜ = 0 for each j.
k,p j j
Based on Lemma 2.1 we have the following upper bound for the SSP coefficient of any
perturbed LMM (1.8). This extends Theorem 2.2 in [11].
Theorem2.2. Giveny ∈ [0,∞),anyperturbedLMM(1.8)ofordergreaterthanonesatisfiesC(y) ≤
2.
Proof. Consider a second-order optimal SSP perturbed LMM with SSP coefficient C = C(y)
and C(cid:101)= yC(y) for some y ∈ [0,∞). Then, from Lemma 2.1 there exists an optimal method
with the at least SSP coefficient C and coefficients (α,β,β˜) such that β β˜ = 0 for each j.
j j
Suppose y > 0 and define δ = β +yβ˜ and
j j j
(cid:40)
1 if β˜ = 0
σ = j
j
−1/y if β = 0.
j
Sinceeither β or β˜ iszero,then β −β˜ = σδ forall j. Letγ = α −Cδ for j ∈ {0,...,k−1}.
j j j j j j j j j
Taking p = 2, r = C, and r˜ = C(cid:101)in (2.5), the second order conditions can be written as
k−1
∑
γ +Cδ = 1, (2.7)
j j
j=0
k−1
∑
jγ +(jC +σ)δ = k−σ δ , (2.8)
j j j k k
j=0
k−1
∑ j2γ +(j2C +2jσ)δ = k(k−2σ δ ). (2.9)
j j j k k
j=0
7
Multiplying (2.7), (2.8) and (2.9) by −k2, 2k and −1, respectively and adding all three expres-
sions gives
k−1
∑ −(k−j)2γ +(cid:0)−C(k−j)2+2σ(k−j)(cid:1)δ = 0. (2.10)
j j j
j=0
Since the method satisfies conditions (2.4) for r = C and r˜ = C(cid:101), then all coefficients γ and δ
j j
are non-negative. Therefore, there must be at least one index j such that the coefficient of δ
0 j0
in (2.10) is non-negative. Note that if β = 0, then σ < 0; hence it can only be that β˜ = 0
j0 j0 j0
and β (cid:54)= 0. Thus,
j0
−C(k−j )2+2(k−j ) ≥ 0,
0 0
which implies
2
C ≤ ≤ 2 (2.11)
k−j
0
since k−j ≥ 1. If now y = 0, define δ = β +β˜ and σ = sign(β −β˜ ). Using γ = α −Cβ
0 j j j j j j j j j
and performing the same algebraic manipulations as before we get
k−1
∑ −(k−j)2(γ +Cβ )+2σ(k−j)δ = 0. (2.12)
j j j j
j=0
Again, there must be at least one index j in (2.12) for which the coefficient of δ is non-
0 j0
negative, thus δ = β (cid:54)= 0 and this yields the inequality (2.11).
j0 j0
Remark 2.2. For given values k,p,y, it may be that there exists no method with positive
SSP coefficients. However, from (2.4) and Theorem 2.2 if a method exists with bounded
SSP coefficient, then the existence of an optimal method follows since the feasible region is
compact.
By combining conditions (2.4) and (2.5), and setting
γ = α −rβ −r˜β˜ for j ∈ {0,...,k−1}, (2.13)
j j j j
the problem of finding optimal SSP perturbed LMMs (1.8) can be formulated as a linear
programming feasibility problem:
LP 1. For fixed k ≥ 1, p ≥ 1 and a given y ∈ [0,∞), determine whether there exist non-
negative coefficients γ , j ∈ {0,...,k−1} and β ,β˜ , j ∈ {0,...,k} such that
j j j
k−1 k−1 k
∑ γ +rβ +r˜β˜ = 1, ∑ j(γ +rβ +r˜β˜ )+ ∑(β −β˜ ) = k,
j j j j j j j j
j=0 j=0 j=0
(2.14)
k−1 k
∑(γ +rβ +r˜β˜ )ji+ ∑(β −β˜ )iji−1 = ki, i ∈ {2,...,p},
j j j j j
j=0 j=0
for some value r ≥ 0 and r˜ = yr.
8
Expressing (2.14) in a compact form facilitates the analysis of the feasible problem LP 1.
Let the vector
a := (1,j,j2,...,jp)(cid:124) ∈ Rp+1, (2.15)
j
and denote by a(cid:48) the derivative of a with respect to j, namely a(cid:48) = (0,1,2j,...,pjp−1)(cid:124).
j j j
Define
(cid:40)
±xa(cid:48) if j = k,
b±(x) := k (2.16)
j a ±xa(cid:48) otherwise.
j j
The conditions (2.14) can be expressed in terms of vectors a ,b±(·):
j j
k−1 k k
∑ γ a +r∑ β b+(r−1)+r˜∑ β˜ b−(r˜−1) = a . (2.17)
j j j j j j k
j=0 j=0 j=0
The number of non-zero coefficients of an optimal SSP perturbed LMM is given by Theo-
rem 2.3. The following lemma is a consequence of Carathéodory’s theorem, which states that
if a vector x belongs to the convex hull of a set S ⊆ Rn, then it can be expressed as a convex
combination of n+1 vectors in S. The proof appears in Appendix A.
Lemma 2.2. Consider a set S = {x ,...,x } of distinct vectors x ∈ Rn, j ∈ {1,...,m}. Let
1 m j
C = conv(S) be the convex hull of S. Then the following statements hold:
(a) Any non-zero vector in C can be expressed as a non-negative linear combination of at most n
linearly independent vectors in S.
(b) Suppose the vectors in S lie in the hyperplane {(1,v) : v ∈ Rn−1} of Rn. Then any non-zero
vector in C can be expressed as a convex combination of at most n linearly independent vectors
in S.
Theorem 2.3. Let k,p be positive integers such that 0 < C (y) < ∞ for a given y ∈ [0,∞). Then
k,p
there exists an optimal perturbed LMM (1.8) with SSP coefficient C = C (y) that has at most p
k,p
non-zero coefficients γ, i ∈ {0,...,k−1} and β ,β˜ , j ∈ {0,...,k}.
i j j
Proof. Consider an optimal LMM (1.8) with coefficients (α,β,β˜) and SSP coefficient C (y) >
k,p
0, for a given y ∈ [0,∞). From Lemma 2.1 an optimal method can be chosen such that
β β˜ = 0foreach j. Using(2.13)wecanperformachangeofvariablesandconsiderthevector
j j
of coefficients x(r) = (cid:0)γ(r),β(r),β˜(r)(cid:1)(cid:124) ∈ R3k+2, x(r) ≥ 0. We will show that x has at most p
non-zero coefficients. Suppose on the contrary that x has at least p+1 non-zero coefficients
γ ,...,γ ,β ,...,β ,β˜ ,...,β˜ ,
i1 im j1 jn l1 ls
where 0 ≤ i < ··· < i ≤ k−1, 0 ≤ j < ··· < j ≤ k and 0 ≤ l < ··· < l ≤ k.
1 m 1 n 1 s
Assume that the set
(cid:110) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:111)
S = a ,...,a ,b+ 1 ,...,b+ 1 ,b− 1 ,...,b− 1
i1 im j1 r jn r l1 yr ls yr
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spans Rp+1. Let r˜ = yr; then the system of equations (2.17) can be written as A(r)x(r) = a ,
k
where
 
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
A(r) =  ai1 ... aim rb+j1 1r ... rb+jn 1r yrb−l1 y1r ... yrb−ls y1r .
Let x = (x ,x )(cid:124) be a permutation of x such that x(cid:124) ∈ Rp+1 is a strictly positive vector
and xp(cid:124) ∈ R3Bk−pN+1 is non-negative. The columns of A(Br) can be permuted in the same way,
N
yielding A (r) = [B(r) | N(r)], where B ∈ R(p+1)×(p+1) and N ∈ R(p+1)×(3k−p+1). Hence, the
p
columns of B and N are associated with x and x , respectively. From our assumption there
B N
must be a subset of S that forms a basis for Rp+1, hence A(r) can be permuted in such a way
so that B(r) has full rank. Therefore, A (r)x (r) = a gives x(cid:124)(r) = B−1(r)(cid:0)a − N(r)x(cid:124)(cid:1).
p p k B k N
Since x (r) > 0, there exists (cid:101) > 0 such that x∗ = x (r+(cid:101)) > 0. Note that we can choose
B B B
to perturb only x and keep x invariant. Let x∗ = (x∗,x )(cid:124), then A (r+(cid:101))x∗ = a . But
B N p B N p p k
thiscontradictstotheoptimalityofthemethodsincewecanconstructa k-stepSSPperturbed
LMM of order p and coefficients given by x∗ and SSP coefficient C (y)+(cid:101).
k,p
Now, assume that the set S does not span Rp+1. Then the vectors in set S lie in the
hyperplane {(1,v) : v ∈ Rp} ⊂ Rp+1 and they are linearly dependent. If the method is
explicit then β = β˜ = 0 and a lies in the convex hull of S. Therefore, from part (b) of
k k k
Lemma 2.2 the vector a can be expressed as a convex combination of p vectors in S. In the
k
case the method is implicit, assume without loss of generality that β > 0 and divide (2.17)
k
by (1+rβ ). The vector (1+rβ )−1a belongs to the convex hull of S and thus from part (a)
k k k
of Lemma 2.2 it can be written as a non-negative linear combination of p vectors in S.
Furthermore, uniqueness of optimal perturbed LMMs can be established under certain
±
conditionsonthevectors a ,b . Thefollowinglemmaisageneralizationof[12,Lemma3.5].
j j
Lemma 2.3. Consider an optimal perturbed LMM (1.8) with SSP coefficient C = C (y) > 0 and
k,p
C(cid:101)= yC (y) for a given y ∈ [0,∞). Let the indices
k,p
0 ≤ i < ··· < i ≤ k−1, 0 ≤ j < ··· < j ≤ k, 0 ≤ l < ··· < l ≤ k,
1 m 1 n 1 s
where m+n+s ≤ p be such that γ ,...,γ ,β ...,β ,β˜ ,...,β˜ are the positive coefficients
i1 im j1 jn l1 ls
in (1.8). Let us also denote the sets I = {0,...,k}, I = {i ,...,i }, J = {j ,...,j }, J =
1 1 m 1 1 n 2
{l ,...,l }. Assume that the function
1 s
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
F(v) = det v,a ,...,a ,b+ 1 ,...,b+ 1 ,b− 1 ,...,b− 1
i1 im j1 C jn C l1 C(cid:101) ls C(cid:101)
is either strictly positive or strictly negative, simultaneously for all v = a , i ∈ I \(I ∪{k}),
i 1
v = b+j (1/C), j ∈ I\ J1 and v = b−l (cid:0)1/C(cid:101)(cid:1), l ∈ I\ J2. Then (1.8) is the unique optimal k-step SSP
perturbed LMM of order p.
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