Space/Time convergence analysis of a class of conservative schemes for linear wave equations

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Space/Time convergence analysis of a class of conservative schemes for linear wave equations

Juliette Chabassier, Sebastien Imperiale

To cite this version:

Juliette Chabassier, Sebastien Imperiale. Space/Time convergence analysis of a class of conservative

schemes for linear wave equations. Comptes Rendus. Mathématique, Académie des sciences (Paris),

2017, 355 (3), pp.282-289. �10.1016/j.crma.2016.12.009�. �hal-01421882�

Space/Time convergence analysis of a class of conservative schemes for linear wave equations

Juliette Chabassier

S´ ebastien Imperiale

aMagique 3D team – Inria Bordeaux Sud Ouest, 200 avenue de la vieille tour, 33405 Talence Cedex Universit´e de Pau et des Pays de l’Adour, avenue de l’universit´e, 64013 Pau Cedex

bInria and Paris-Saclay University, 1 rue Honor´e d’Estienne d’Orves, 91120 Palaiseau

Received *****; accepted after revision +++++

Abstract

This paper concerns the space/time convergence analysis of conservative two-steps time discretizations for linear wave equations. Explicit and implicit, second and fourth order schemes are considered, while the space discretiza- tion is given and satisfies minimal hypotheses. The convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.

To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

R´esum´e

Ce travail concerne l’analyse de convergence espace/temps de sch´emas en temps `a deux pas pour les ´equations d’onde lin´eaires. Sont consid´er´es des sch´emas explicites et implicites, d’ordre deux et quatre, tandis que la discr´etisation spatiale est donn´ee et satisfait des hypoth`eses minimales. L’analyse de convergence est faite par techniques d’´energie et est valide si le pas de temps est born´e par une quantit´e d´ependant des param`etres de discr´etisation spatiale. En plus de montrer la convergence pour des sch´emas d’ordre quatre r´ecemments intro- duits, la nouveaut´e de ce travail r´eside dans le fait que les estimations ne d´ependent pas de la diff´erence entre le pas de temps et sa plus grande valeur admissible.

Pour citer cet article : A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Email addresses: [email protected](Juliette Chabassier),[email protected]( S´ebastien Imperiale).

Version fran¸caise abr´eg´ee

On consid`ere le probl`eme de propagation d’onde suivant : pour une source donn´ee f ∈ C⁰([0, T], H) trouveru(t)∈V solution, pour tout t∈[0, T],de

∂ttu+Au=f dansV⁰, u(0) = 0 dansV, ∂tu(0) = 0 dansH,

o`u V et H sont deux espaces s´eparables de Hilbert (la norme surH est not´ee | · |) et V<sup>0</sup> le dual de V, l’op´erateurAest autoadjoint deV dansV<sup>0</sup>. Consid´erons donn´ee une discr´etisation en espace s’appuyant sur une famille d’espaces de dimension finie{Vh}h>0avecVh⊂V. Le probl`eme semi-discret s’´ecrit alors : pour un terme sourcefh∈C²([0, T], Vh) trouveruh(t)∈Vh, pour toutt∈[0, T] satisfaisant

∂ttuh+Ahuh=fh, uh(0) = 0, ∂tuh(0) = 0, dansVh,

o`u A<sub>h</sub> est un op´erateur deV<sub>h</sub> dansV<sub>h</sub> autoadjoint born´e (non uniform´ement par rapport `a h) r´esultant de la discr´etisation en espace choisie. Nous supposerons qu’il existe une fonction positiveδ(h) =o(h) telle que pour touth >0 on ait

sup

t∈[0,T]

|u(t)−uh(t)| ≤δ(h).

Nous ´etudions les sch´emas suivants : leθ-sch´ema (not´eTS et dont le sch´ema saute-mouton explicite est un cas particulier), le sch´ema Saute Mouton Stabilis´e (not´eSLF) et le (θ, ϕ)-sch´ema d’ordre quatre (not´e TPS). Soit ∆t >0 le pas de temps et d´efinissonstⁿ:=n∆t. Pour toute suite{v_h^k}k≥0⊂Vh, nous notons

[vⁿ_h]_∆t2:= v_hⁿ⁺¹−2v_hⁿ+v_hⁿ⁻¹

∆t² , {v_hⁿ}θ:=θ vⁿ⁺¹_h + (1−2θ)v_hⁿ+θ v_hⁿ⁻¹, n≥1.

Nous cherchons une suite {u^k_h}_k≥0 ⊂ Vh telle que uⁿ_h approche uh(tⁿ). Les deux premiers termes sont supoos´es donn´es et pour toutn≥1,

•TS : [uⁿ_h]_∆t2+Ah{uⁿ_h}θ=fh(tⁿ). • SLF : [uⁿ_h]_∆t2+Ahuⁿ_h+∆t²

16 A²_huⁿ_h=fh(tⁿ).

•TPS : [uⁿ_h]_∆t2+Ah{uⁿ_h}θ+ ∆t²

θ− 1 12

A²_h{uⁿ_h}ϕ=fh(tⁿ) + ∆t² ∂tt

12 +

θ− 1 12

Ah

fh(tⁿ).

Pour chaque sch´ema on peut d´efinir une ´energie discr`ete associ´ee, une condition de type CFL assure sa positivit´e en tant que fonctionnelle :

∆t²≤ α ρ(Ah)

o`uα >0 est positif, le plus grand possible, et d´epend du sch´ema. Nous montrons que sous des hypoth`eses portant sur la discrtisation spatiale du probl`eme ainsi que sous condition CFL, le r´esultat de convergence espace/temps suivant : il existeC >0 ind´ependant de ∆t ethtel que

• θ-sch´ema (TS) et sch´ema saute-mouton stabilis´e (SLF) sup

tⁿ∈[0,T]

|u(tⁿ)−uⁿ_h| ≤C

∆t²T²+δ(h) .

• (θ, ϕ)-sch´ema (TPS)

sup

tⁿ∈[0,T]

|u(tⁿ)−uⁿ_h| ≤C

∆t⁴T²+δ(h) .

Notons qu’en particulier ces r´esultats de convergences sont valables lorsque ∆t²=α/ρ(Ah). Pour obtenir ce r´esultat nous nous ramenons `a l’´etude de polynˆomes dont les coefficients sont fonction des param`etres des sch´emas.

2

1. Abstract and semi-discrete wave propagation problem

In this work, we are interested in the simulation of linear wave propagation problems. The most simple example one could think of is given by the following problem: findu(t)∈H₀¹(Ω), for all t ∈[0, T] such that

∂_ttu−∆u=f, u(0) = 0, ∂_tu(0) = 0, in Ω, u= 0 on∂Ω, (1) To study wave propagation in a more abstract framework, following [1], chapter XVIII, we assume given separable Hilbert spacesH andV. The spaceH is equipped with the scalar product (·,·)H. The corre- sponding norms are denoted | · |and k · krespectively. Moreover we assume that V is dense in H,H is identified with its dualH⁰andV is continuously embedded inH. Note that to solve (1) it is standard to chooseH =L²(Ω) andV =H₀¹(Ω). We assume given a continuous hermitian bilinear forma:V×V →R that satisfies

a(v, v)≥0, a(v, v) +Co|v|²≥Cakvk², ∀v∈V, (2) where (Ca, Co) are real positive scalars. From this bilinear form we construct a self-adjoint positive unbounded operatorA:V 7→V⁰. It is defined by

A:u→Au such that hAu, vi=a(u, v), ∀v∈V.

whereV⁰ denotes the dual ofV andh·,·idenotes the duality product. We consider the following abstract wave propagation problems: for a givenf ∈C⁰([0, T], H) findu(t)∈V solution, for allt∈[0, T],of

∂ttu+Au=f inV⁰, u(0) = 0 inV, ∂tu(0) = 0 inH. (3) Existence and uniqueness results for this problem are well known (see again [1], chapter XVIII): there exists a unique functionusolution of (3), it satisfies

u∈C⁰([0, T], V), ∂_tu∈C⁰([0, T], H).

We introduce the family of finite dimensional spaces{Vh}h>0 withV_h⊂V. As usual, the subscript his devoted to tends to 0 and represents an approximation parameter ofV_h to V. For each hwe define the operatorA_h as A_h:V_h7→V_h and

A_h:u_h→A_hu_h such that (A_hu_h, v_h)_h=a_h(u_h, v_h), ∀vh∈V_h,

where the scalar product onVhdenoted by (·,·)has well as the continuous bilinear formsah(·,·) represent some approximation of (·,·)H anda(·,·) respectively that account for instance for the use of quadrature formulae in the computation of integrals. Similarly we denote by | · |h the norm induced by the scalar product (·,·)h. We assume that there exists a positive constant C_H such that

|vh| ≤CH|vh|h, ∀vh∈Vh

and we assume thatah is an hermitian bilinear form that satisfies the same property as in equation (2) ah(vh, vh)≥0, ah(vh, vh) +Co|vh|²_h≥Cakvhk², ∀vh∈Vh.

Note that this assumption implies that the operator Ah is self-adjoint and non-negative. Its spectrum, denoted Sp(Ah), is a set of finite number of real non-negative eigenvalues (eigenvalues of multiplicity greater than one are counted accordingly). To each eigenvalue λ we associate a real eigenvector w_h^λ defined by

(Ahw_h^λ, vh)h=λ(w_h^λ, vh)h, ∀vh∈Vh, |w^λ_h|h= 1,

such that the family{w^λ_h}_λ∈Sp(Ah)is an orthonormal basis ofVh. The semi-discrete equation we consider reads: for any given source termfh∈C²([0, T], Vh) finduh(t)∈Vh, for allt∈[0, T] such that

∂ttuh+Ahuh=fh, uh(0) = 0, ∂tuh(0) = 0, in Vh. (4)

Existence and uniqueness results for this problem are direct consequences of the theory developed in in- finite dimensional space (chooseH =V =Vh in the paragraph above): there exists a unique solutionuh

of (4), it satisfiesuh∈C¹([0, T], Vh).We introduce the discrepancy erroreh(t) =u(t)−uh(t) it depends on the approximation of the scalar product onH, on the bilinear form ah, on the approximation of the spaceV itself and on the approximation of the source termf. It is expected thatehvanishes whenh→0.

Therefore we make the following assumption

Hypothesis1.1 Convergence of the semi-discrete problem.There exists a positive functionδ(h) = o(h)such that for all h >0 we have

sup

t∈[0,T]

|eh(t)| ≤δ(h).

Such a result is proved for continuous finite element approximation in [2] in the case a(·,·)_h ≡ a(·,·).

In [3] the case of spectral elements is given (see [4] for an introduction to spectral elements). For these convergence results to hold, the continuous solution must be regular enough in time and space. We do not detail the assumptions here. In what follows we specify the extra regularity in time and space we require for the semi-discrete solution uh in order to state the space/time convergence results, such regularity is a consequence of the space/time regularity of the source term.

Hypothesis1.2 Stability of the semi-discrete problem. For a given couple(p, q)with peven, we assume that ∂_t^quh(t) ∈Vh for all t ∈[0, T] and there exists a constantCp,q such that for all h >0 we have

sup

t∈[0,T]

|A^p/2_h ∂_t^quh(t)|h≤Cp,q.

2. Analysis of a family of time discretizations

In the sequel we study the following two-steps conservative time discretizations:θ-Scheme (TS) which is the family of conservative centered Newmark schemes, see [1], such as the classical leap-frog scheme, the Stabilized Leap-Frog scheme (SLF) as introduced in [5] and the higher order (θ, ϕ)-Scheme (TPS) developed in [6]. Let ∆t >0 be the time step, a small parameter devoted to tend to zero and let define tⁿ :=n∆t. To shorten the writing we introduce the following notations, for a series {v_h^k}k≥0⊂V_h,

[vⁿ_h]_∆t2:= v_hⁿ⁺¹−2v_hⁿ+v_hⁿ⁻¹

∆t² , {v_hⁿ}θ:=θ vⁿ⁺¹_h + (1−2θ)v_hⁿ+θ v_hⁿ⁻¹, n≥1.

We seek a series{u^k_h}k≥0⊂V_h such thatuⁿ_h is an approximation ofu_h(tⁿ) solution of equation (4). The two first terms of this series are considered given¹. The following terms of the series are computed using one of the schemes below, forn≥1,

•TS: [uⁿ_h]_∆t2+Ah{uⁿ_h}θ=fh(tⁿ). • SLF: [uⁿ_h]_∆t2+Ahuⁿ_h+∆t²

16 A²_huⁿ_h=fh(tⁿ).

•TPS: [uⁿ_h]_∆t2+A_h{uⁿ_h}_θ+ ∆t²

θ− 1 12

A²_h{uⁿ_h}_ϕ=f_h(tⁿ) + ∆t² ∂tt

12 +

θ− 1 12

A_h

f_h(tⁿ).

By construction (SLF) is an explicit scheme, whereas (TS) is explicit whenθ= 0 and (TPS) is explicit when (θ, ϕ) = (0,0). In this latter case, the scheme corresponds to the modified equation approach

1. In practice, u⁰_h and u¹_h are computed using the initial data and a sufficiently accurate one-step method (such as high-order Runge Kutta methods).

4

presented in [7]. Notice that for the (TPS) the source term has to be modified in order to get adequate accuracy.

All these schemes can be written

P_K(∆t²A_h) [uⁿ_h]_∆t2+P_P(∆t²A_h)A_h{uⁿ_h}1/4=f_hⁿ, (5) withf_hⁿ the source term and with the following definitions of the polynomial functionsP_KandP_P

•TS:PK(x) = 1 +

θ−1 4

x, PP(x) = 1. • SLF:PK(x) = 1−x 4 +x²

64, PP(x) = 1− x 16.

•TPS:PK(x) = 1 +

θ−1 4

x+

ϕ−1

4 θ− 1 12

x², PP(x) = 1 +

θ− 1 12

x.

In the sequel we aim at establishing energy and error estimates for general schemes of the form (5) under some assumptions on P_K and P_P. Let the time discretization error be defined as eⁿ_h :=uh(tⁿ)−uⁿ_h, it satisfies

PK(∆t²Ah) [eⁿ_h]_∆t2+PP(∆t²Ah)Ah{eⁿ_h}_1/4=r_hⁿ. (6) Assuming sufficient regularity of the semi-discrete solution, there existtⁿ⁻¹≤t^n,, t^n,♥, t^n,♣ ≤tⁿ⁺¹ such that for each of the considered schemes, the remainder writes

•TS:rⁿ_h= ∆t² 1

12∂⁴_tuh(t^n,) +θAh∂ttuh(t^n,♥)

. •SLF:r_hⁿ= ∆t² 1

12∂_t⁴uh(t^n,) + 1

16A²_huh(tⁿ)

.

•TPS:rⁿ_h= ∆t⁴ 1

360∂_t⁶u_h(t^n,) + θ

12A_h∂_t⁴u_h(t^n,♥) +ϕ

θ− 1 12

A²_h∂_ttu_h(t^n,♣)

.

This suggests that (TS) and (SLF) are second order accurate and (TPS) is fourth order accurate in time.

For all these schemes, a discrete energy is preserved (in the absence of sources). This energy is computed at the intermediate time stept^n+1/2and is denotedE_h^n+1/2. It is the sum of a kinetic and potential energies, more preciselyE_h^n+1/2=E_K^n+1/2+E_P^n+1/2. When considering equation (6) satisfied by the error, we have

E_K^n+1/2= 1 2

P_K ∆t²Ah

eⁿ⁺¹_h −eⁿ_h

∆t ,eⁿ⁺¹_h −eⁿ_h

∆t

h

(7)

and E_P^n+1/2= 1 2

P_P(∆t²A_h)A_heⁿ⁺¹_h +eⁿ_h

2 ,eⁿ⁺¹_h +eⁿ_h 2

h

. (8)

It can be shown that the total energy satisfies E_h^n+1/2− E_h^n−1/2

∆t =

rⁿ_h,eⁿ⁺¹_h −eⁿ⁻¹_h 2∆t

h

. (9)

A necessary condition for the schemes’ stability is the non-negativity of the kinetic energy (7) and the potential energy (8). Such a condition is achieved if the following hypothesis holds

Hypothesis2.1 There existsα >0 such thatPK(x)andPP(x)are non-negative for all x∈[0, α].

For the schemes we consider we have

•TS (0≤θ <1/4) : α= 4/(1−4θ). • TS (θ≥1/4) : α= +∞. • SLF: α= 16.

•TPS (θ= 0, ϕ= 0) : α= 12. • TPS (θ≥1/4, ϕ≥1/4) : α= +∞.

•TPS ∀(θ, ϕ) : see theorem 5.1 of [6].

Lemma 2.1 Assume Hypothesis 2.1 holds. The energies E_K^n+1/2 and E_P^n+1/2 are non-negative for all n≥0 if∆tsatisfies

∆t²≤ α

ρ(Ah) (CFL condition) (10)

whereρ(A_h)is the largest eigenvalue of A_h.

Note that when α is finite (resp. infinite) the scheme is conditionally (resp. unconditionally) stable.

The non-negativity of the energyE_h^n+1/2 is however not sufficient to ensure the stability of the schemes inH since it is only a semi-norm. In the sequel, the polynomial functionsP_P andP_Kwill be assumed to satisfy the following hypothesis

Hypothesis2.2 We assume that there exists a partitioning of[0, α]into two disjoint subsetsJ_K andJ_P such that there exist two strictly positive constantsC_K,C_P, such that

CK ≤PK(x), ∀x∈ JK, CP ≤x PP(x), ∀x∈ JP. D´efinition 2.3 We define for any vh∈Vh, the projection operators

Π_K(vh) = X

∆t²λ∈JK

λ∈Sp(Ah)

(vh, w^λ_h)hw^λ_h and Π_P(vh) = X

∆t²λ∈JP

λ∈Sp(Ah)

(vh, w^λ_h)hw^λ_h

The operator Π_K (resp. Π_P) corresponds to modal projection operators on subsets of V_h for which the kinetic (resp. potential) part of the energy is a uniform upper bound of the H-norm with respect to h and ∆t. These ideas are specified in the following Lemma.

Lemma 2.2 Assume Hypothesis 2.2 holds. Then, for any h >0 and ∆t >0 satisfying the CFL condi- tion (10), and for anyv_h∈V_h,v_h= Π_K(v_h) + Π_P(v_h)and

|Π_K(v_h)|²_h≤C_K⁻¹ P_K(∆t²A_h)v_h, v_h

h, |Π_P(v_h)|²_h≤C_P⁻¹ ∆t²A_hP_P(∆t²A_h)v_h, v_h

h. (11) Then the following estimates can be written

Lemma 2.3 Let the series {eⁿ_h} satisfy (6) and Hypothesis 2.2 holds. Then for all h > 0 and ∆t >0 satisfying the CFL condition (10), and for all n≥1,

q

E_h^n+1/2≤ q

E_h^1/2+√ 2γ∆t

n

X

`=1

r^`_h

_h (12)

eⁿ⁺¹_{h h}≤√

2 e¹_h

_h+ 2γ tⁿ q

2E_h^1/2+ 4γ²∆t²

n

X

`=1

`

X

k=1

r_h^k

_h, (13) whereγ=C_P^−1/2+C_K^−1/2/2.

It is important to notify here thatγis independent ofhand ∆t, this is the key point to obtain uniform estimates in ∆t. Indeed as pointed out in Remark 10 of [8] the classical stability proof for the (TS) blows up when the time step approaches its greatest admissible value in inequality (10). The constant C_K andC_P (and thereforeγ) are computed for the considered schemes in appendix, we findγ≤√

2 for all schemes.

Proof 2.4 (of Lemma 2.3)From Cauchy-Schwartz inequality applied to (9) we get E_h^n+1/2− E_h^n−1/2

∆t ≤ |r_hⁿ|_h

eⁿ⁺¹_h −eⁿ⁻¹_h 2∆t

_h

≤ |rⁿ_h|_h

ΠK

eⁿ⁺¹_h −eⁿ⁻¹_h 2∆t

_h

| {z }

Ξ

+|r_hⁿ|_h

ΠP

eⁿ⁺¹_h −eⁿ⁻¹_h 2∆t

_h

| {z }

Φ

,

where Π_K andΠ_P are the projection operators given by definition 2.3. Now we use the two inequalities of (11) on respectively Ξ and Φ and then recognize the square root of the kinetic and potential parts

6

of the energy. To shorten the notation we introduce the time finite difference of the error: d^n+1/2_h :=

(eⁿ⁺¹_h −eⁿ_h)/∆t. We have 2 Ξ≤

Π_K(d^n+1/2_h ) _h+

Π_K(d^n−1/2_h ) _h

≤C_K^−1/2 r

P_K(∆t²A_h)d^n+1/2_h , d^n+1/2_h

h

+C_K^−1/2 r

P_K(∆t²A_h)d^n−1/2_h , d^n−1/2_h

h

≤C_K^−1/2 q

2E_h^n+1/2+ q

2E_h^n−1/2

,

note that the second inequality is obtained by takingv_h=d^n+1/2_h andv_h=d^n−1/2_h in the first inequality of (11). A similar procedure leads to

Φ≤C_P^−1/2 q

2E_h^n+1/2+ q

2E_h^n−1/2

.

By denotingγ=C_P^−1/2+C_K^−1/2/2 we get from E_h^n+1/2− E_h^n−1/2

∆t ≤γ|rⁿ_h|_h q

2E_h^n+1/2+ q

2E_h^n−1/2

,

which classically leads to inequality (12). The other part of the proof concerns the upper bound on eⁿ⁺¹_h

_h. We write

eⁿ⁺¹_{h h}≤

Π_K(eⁿ⁺¹_h ) _h

| {z }

Ω

+

Π_P(eⁿ⁺¹_h ) _h

| {z }

Υ

Introducing artificially the term eⁿ_h we can write Ω≤ |Π_K(eⁿ_h)|_h+ ∆t

Π_K(d^n+1/2_h )

_h≤ |Π_K(eⁿ_h)|_h+ ∆t C_K^−1/2 r

P_K(∆t²Ah)d^n+1/2_h , d^n+1/2_h

h

≤ |Π_K(eⁿ_h)|_h+ ∆t C_K^−1/2 q

2E_h^n+1/2 The same procedure leads to

Υ≤ |Π_P(eⁿ_h)|_h+ 2 ∆t C_P^−1/2 q

2E_h^n+1/2. We use recursively the former inequalities down ton= 1 to get,

eⁿ⁺¹_{h h}≤

Π_K(e¹_h) _h+

Π_P(e¹_h) _h

| {z }

≤√ 2|^e1h|_h

+2√ 2 ∆t γ

n

X

`=1

q E_h^`+1/2.

We can reuse identity (12)to get (13).

3. Space time convergence result

Letεⁿ_h be the error between the continuous solution and the fully discrete solution. Note that we have u(tⁿ)−uⁿ_h =eh(tⁿ) +eⁿ_h, thanks to this decomposition, assumption 1.1 and lemma 2.3 we can state our final result.

Theorem 3.1 Assume that e⁰_h=e¹_h = 0and assume that Hypothesis 2.2 holds and that Hypothesis 1.2 holds for every sufficiently large (p, q), then, for all h >0 and all ∆t satisfying the CFL condition (2.1) the following uniform convergence results hold

• θ-scheme (TS) sup

tⁿ∈[0,T]

|u(tⁿ)−uⁿ_h| ≤C_H

∆t²γ²T² C0,4

6 + 2θ C_2,2

+δ(h)

.

• Stabilized leap-frog scheme (SLF) sup

tⁿ∈[0,T]

|u(tⁿ)−uⁿ_h| ≤C_H

∆t²γ²T² C_0,4

6 +C_4,0 8

+δ(h)

.

• (θ, ϕ)-scheme (TPS) sup

tⁿ∈[0,T]

|u(tⁿ)−uⁿ_h| ≤CH

∆t⁴γ²T² C_0,6

180 +θC_2,4

6 + 2ϕ θ− 1 12

C4,2

+δ(h)

. Note that, in particular these convergence results hold if ∆t²=α/ρ(Ah).

Appendix: Expression of γ for the considered numerical schemes

• TS (0 ≤ θ < 1/4). We are looking for a partitioning J_K∪ J_P of the interval [0,_(1−4θ)⁴ ] and two positive constantCK andCP such that

CK≤PK(x) = 1 +

θ−1 4

x, ∀x∈ JK, CP ≤x PP(x) =x, ∀x∈ JP

We see that the first inequality cannot be fulfilled forx= _(1−4θ)⁴ ∈ J_Kand the second one forx= 0∈ J_P. The proposed partitioning is the following

JK= 0, a²

, CK= 1 +

θ−1 4

a², JP =

a², 1 (1/4−θ)

, CP =a²

wherea can be chosen according toθ in order to minimizeγ =C_P^−1/2+C_K^−1/2/2. The peculiar optimal value is given in Lemma 2.3 of [9], it reads

a(θ) =

s 4

(1−4θ)^2/3+ (1−4θ); γ(θ) = 1

a(θ)+ 1

p4−(1−4θ)a(θ)² Especially ifθ= 0 we getγ=√

2.

• TS (θ ≥ 1/4). In this case the polynomial functions P_K(x) and P_P(x) are positive on the whole interval [0,+∞] and moreoverP_K(x)≥1∀x≥0. Therefore one can chooseJ_K= [0,+∞] andJ_P ={∅}.

As a consequence one can setC_K= 1 and sinceC_P has no influence and can be chosen formally equal to +∞. We getγ= 1/2.

•SLF. We are looking for a partitioningJ_K∪ J_P of the interval [0,16] and two constant C_K and C_P such that

CK≤ 1−x

8 ²

= 1−x 4 +x²

64, ∀x∈ JK, CP ≤x 1− x

16

, ∀x∈ JP

8

Out of symmetry with respect to x = 8, we look for J_K = [0, 8(1−a)[∪]8(1 +a), 16] and J_P = [8(1−a), 8(1 +a)]. ThenC_K=a² andC_P = 4(1−a²). Therefore

γ=C_P^−1/2+C_K^−1/2

2 = 1

2√

1−a² + 1 2a. The choice ofathat minimizes the value ofγisa= ^√1

2 and leads toγ=√ 2.

• TPS (θ = 0, ϕ = 0). The approach is similar to the one presented above. We are looking for a partitioningJ_K∪ J_P of the interval [0,12] and two constantC_K andC_P such that

C_K≤1−x 4 +x²

48, ∀x∈ J_K, C_P ≤x 1− x

12

, ∀x∈ J_P

We look forJK= [0, 6(1−a)[∪]6(1 +a), 12] andJP = [6(1−a), 6(1 +a)], this choice gives some values forC_KandC_P that can be optimized to minimizeγ. The optimal choice isa= ^√1

3 andγ=√ 2.

•TPS (θ≥1/4, ϕ≥1/4).In the case of the (θ, ϕ)-scheme withθ≥1/4 andϕ≥1/4, the polynomial functionsP_K(x) andP_P(x) read

P_K(x) = 1 +

θ−1 4

x+

ϕ−1

4 θ− 1 12

x², P_P(x) = 1 +

θ− 1 12

x.

The polynomial P_K(x) and P_P(x) are positive on the whole interval [0,+∞] and P_K(x) ≥ 1 ∀x ≥ 0.

Therefore one can chooseJ_K= [0,+∞] andJ_P ={∅}. We getγ= 1/2.

References

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