AMS subject classifications. 65M06, 65M12, 65M15, 76D05, 35Q30

Transcription

1 CONVERGENCE ANALYSIS OF THE ENERGY AND HELICITY PRESERVING SCHEME FOR AXISYMMETRIC FLOWS JIAN-GUO LIU AND WEI-CHENG WANG Abstact. We give an eo estimate fo the Enegy Helicity Peseving Scheme EHPS in second ode finite diffeence setting on axisymmetic incompessible flows with swiling velocity. This is accomplished by a weighted enegy estimate, along with caeful nonstad local tuncation eo analysis nea the geometic singulaity a fa field decay estimate fo the steam function. A key ingedient in ou a pioi estimate is the pemutation identities associated with the Jacobians, which is also a unique featue that distinguishes EHPS fom stad finite diffeence schemes. Key wods. Incompessible viscous flow, Navie Stokes equation, pole singulaity, consevative scheme, Jacobian, pemutation identity, geometic singulaity AMS subject classifications. 65M06, 65M2, 65M5, 76D05, 35Q30. Intoduction. Axisymmetic flow is an impotant subject in fluid dynamics has become stad textbook mateial e.g. [2] as a stating point of theoetical study fo complicated flow pattens. Although the numbe of independent spatial vaiables is educed by symmety, some of the essential featues complexities of geneic 3D flows emain. Fo example, when the swiling velocity is nonzeo, thee is a voticity stetching tem pesent. This is widely believed to account fo possible singulaity fomation fo Navie-Stokes Eule flows. Fo geneal smooth initial data, it is well known that the solution emains smooth fo shot time in Eule [8] Navie-Stokes flows [9]. A fundamental egulaity esult concening the solution of the Navie-Stokes equation is given in the pioneeing wok of Caffaelli, Kohn Nienbeg [3]: The one dimensional Hausdoff measue of the singula set is zeo. As a consequence, the only possible singulaity fo axisymmetic Navie-Stokes flows would be on the axis of otation. This esult has motivated subsequent eseach activities concening the egulaity of axisymmetic solutions of the Navie-Stokes equation. Some egulaity patial egulaity esults fo axisymmetic Eule Navie- Stokes flows can be found, fo example, in [4] the efeences theein. To date, the egulaity of the Navie-Stokes Eule flows, whethe axisymmetic o not, emains a challenging open poblem. Fo a compehensive eview on the egulaity of the Navie-Stokes equation, see [0] the efeences theein. Due to the subtle egulaity issue, the numeical simulation of axisymmetic flows is also a challenging subject fo computational fluid dynamicists. The ealiest attempt of numeical seach fo potential singulaities of axisymmetic flows dates back to the 90 s [5, 6]. In a ecent wok [], the authos have developed a class of Enegy Helicity Peseving Schemes EHPS fo incompessible Navie-Stokes MHD equations. Thee the authos extended the voticity-steam fomulation of axisymmetic flows given in [5] poposed a genealized voticity-steam fomulation fo 3D Navie-Stokes MHD flows with coodinate symmety. In the case of axisymmetic flows, the majo diffeence between EHPS the fomulation in [5] is the expession numeical discetization of the nonlinea tems. It is shown in [] that all the nonlinea tems in the Navie-Stokes MHD equation, including con- Institute fo Physical Science Technology Depatment of Mathematics, Univesity of Mayl, College Pak, MD jliu@math.umd.edu. Depatment of Mathematics, National Tsing Hua Univesity, HsinChu, Taiwan 300, Taiwan wangwc@math.nthu.edu.tw.

2 2 JIAN-GUO LIU AND WEI-CHANG WANG vection, voticity stetching, geometic souce, Loentz foce electo-motive foce, can be witten as Jacobians. Associated with the Jacobians is a set of pemutation identities which leads natually to the consevation laws fo fist second moments. The pimay featue of the EHPS schemes is the numeical ealization of these consevation laws. In addition to peseving physically elevant quantities, the discete fom of consevation laws povides numeical advantages as well. In paticula, the consevation of enegy automatically enfoces nonlinea stability of EHPS. Fo 2D flows, EHPS is equivalent to the enegy enstophy peseving scheme of Aakawa [], who fist pointed out the impotance of discete consevation laws in long time numeical simulations. Othe than the Jacobian appoach, most of the enegy conseving finite diffeence schemes fo stad flows without geometic singulaity ae based on discetization of the fluid equation in pimitive vaiables. A well known tick that dates back to the 70 s is to take the aveage of consevative nonconsevative discetizations of convection tem Piacsek Williams, [6]. In [4], Moinishi et al futhe exploed compaed vaious combinations among consevative, nonconsevative otation foms of the convection tem. Moe ecently in [8], Vestappen Veldman poposed a discetization fo the convection tem that esulted in a skew-symmetic diffeence opeato theefoe the consevation of enegy could be achieved. A potential difficulty associated with axisymmetic flows is the appeaance of facto which becomes infinite at the axis of otation, theefoe sensitive to inconsistent o low ode numeical teatment nea this pole singulaity. In [], the authos poposed a second ode finite diffeence scheme hled the pole singulaity by shifting the gids half gid length away fom the oigin. Remakably, the pemutation identities theefoe the enegy helicity identities emain valid in this case. Thee ae altenative numeical teatments poposed in liteatues e.g. [6] to hle this coodinate singulaity. Howeve, igoous justifications fo vaious pole conditions ae yet to be established. The pupose of this pape is to give a igoous eo estimate of EHPS fo axisymmetic flows. To focus on the pole singulaity avoid complication caused by physical bounday conditions, we conside hee only the whole space poblem with the swiling components of velocity voticity decaying fast enough at infinity. The eo analysis of numeical methods fo NSE with nonslip physical bounday condition has been well studied. We efe the woks of Hou Wetton [7], Liu Wang [9] to inteested eades. Ou poof is based on a weighted enegy estimate along with a caeful detailed pointwise local tuncation eo analysis. A majo ingedient in ou enegy estimate is the pemutation identities associated with the Jacobians 4.7. These identities ae key to the enegy helicity peseving popety of EHPS fo geneal symmetic flows. Hee the same identities enable us to obtain a pioi estimate even in the pesence of the pole singulaity, see section 5 fo details. To ou knowledge, this is the fist igoous convegence poof fo finite diffeence schemes devised fo axisymmetic flows. In ou pointwise local tuncation eo estimate, a fundamental issue is the identification of smooth flows in the vicinity of the pole. Using a symmety agument, it can be shown [2] that if the swiling component is even in o moe pecisely, is the estiction of an even function on > 0, the vecto field is in fact singula. See Example in section 2 fo details. This is an easily ovelooked mistake that even appeaed in some eseach papes tageted at numeical seach fo potential fomation of finite time singulaities. In addition to the egulaity issue at the axis of symmety,

3 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 3 a efined decay estimate fo the steam function also plays an impotant ole in ou analysis. In geneal, the steam function only decays as Ox at infinity. Accodingly, we have selected an appopiate combination of weight functions that constitute an -homogeneous nom. As a esult, the slow decay of the steam function is compensated by the fast decay of velocity voticity. Oveall, we obtained a second ode eo estimate on axisymmetic flows. The est of this pape is oganized as follows: In section 2, we give a bief eview of the egulaity esults developed in [2], including the chaacteization of pole egulaity fo geneal axisymmetic solenoidal vecto fields solutions of the axisymmetic Navie-Stokes equation 2.2. In section 3, we fomulate a egulaity assumption on solution of Navie-Stokes equation at infinity. We basically assume that the swiling components of velocity voticity decay fast enough at infinity, use it to analyze the decay ate of the steam function. In section 4, we biefly eviewed the enegy helicity peseving popety fo EHPS use it to pove ou main theoem by enegy estimate in section 5. The poof of some technical Lemmas is given in the Appendix. 2. Genealized voticity-steam fomulation fo axisymmetic flows. In this section, we eview the genealized voticity-steam fomulation of axisymmetic Navie-Stokes equation t u + u u + p = ν u u = 0 2. elated egulaity issues. Denote by x-axis the axis of symmety, the axisymmetic Navie-Stokes equation in the cylindical coodinate system x = x, y = cos θ, z = sin θ can be witten as [] u t + 2 J u, ψ = ν 2 2 u, ω t + J ω, ψ = ν 2 2 ω + J u, u, 2.2 ω = 2 2 ψ, whee Ja, b = x a b a x b. In 2.2, ut; x,, ωt; x, ψt; x, epesent the swiling components of velocity, voticity steam function espectively. The quantity ψ is also known as Stokes steam function the fomal coespondence between the solutions of is given by u = ue θ + ψe θ = ψ e x x ψe + ue θ 2.3 whee e x, e e θ ae the unit vectos in the x, θ diections espectively. The voticity-steam fomulation 2.2 has appeaed in [5] with an altenative expession fo the nonlinea tems. In [], the authos have genealized the voticity fomulation to geneal symmetic flows with the nonlinea tems ecast in Jacobians as in 2.2 poposed a class of enegy helicity peseving schemes EHPS based on discetizing 2.2. In section 4 5, we will eview EHPS fo 2.2 give a igoous eo estimate in 2nd ode finite diffeence setting. The eo bound cetainly depends

4 4 JIAN-GUO LIU AND WEI-CHANG WANG on the egulaity of the solution to 2.2. Although 2.2 can be deived fomally fom 2., the equivalence between the two expessions in tems of egulaity of solutions is not quite obvious. An essential peequisite to ou analysis is to chaacteize pope meaning of smoothness of solutions to 2.2. This tuns out to be a subtle issue. Example. Take ux, = 2 e, ω = ψ It is easy to veify that 2.4 is an exact stationay solution of the Eule equation ν = 0 in 2.2. Note that u = O 2 nea the axis 2 ux, Simila functions can be found in liteatues as initial data in numeical seach fo finite time singulaities. Although u C R R +, the following egulaity Lemma fo geneal axisymmetic solenoidal vecto fields shows that u = ue θ is not even in C 2 R 3, R 3 : Lemma 2.. [2] Denote the axisymmetic divegence fee subspace of C k vecto fields by C k s def = {u C k R 3, R 3, θ u x = θ u = θ u θ = 0, u = 0}. 2.5 Then a Fo any u C k s, thee exists a unique u, ψ such that with u = ue θ + ψe θ = ψ e x x ψe + ue θ, > 0, 2.6 ux, C k R R +, 2l ux, 0 + = 0 fo 0 2l k, 2.7 ψx, C k+ R R +, 2l ψx, 0 + = 0 fo 0 2l k b If u, ψ satisfies 2.7, 2.8 u is given by 2.6 fo > 0, then u C k s with a emovable singulaity at = 0. Hee in 2.5 thoughout this pape, the subscipts of u ae used to denote components athe than patial deivatives. The poof of Lemma 2. is based on the obsevation that e θ changes diection acoss the axis of symmety, theefoe u = u θ must admit an odd extension in ode to compensate fo this discontinuity. The details can be found in [2]. Fo simplicity of pesentation, we ecast Lemma 2. as Lemma 2. C k s = {ue θ + ψe θ u C k s R R +, ψ C k+ s R R + } 2.9 whee R R + def = {fx, C k R R +, 2j fx, 0 + = 0, 0 2j k}. 2.0 C k s Fom Lemma 2. Example, it is clea that the pope meaning of smooth solution to 2.2 should be supplemented by the pole conditions 2.7,2.8. In the case of Navie-Stokes equation ν > 0, ou main concen in this pape, 2.2 is an ellipticpaabolic system on a semi-bounded egion > 0. Fom stad PDE theoy, we

5 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 5 need to assign one only one bounday condition fo each of the vaiables ψ, u ω. An obvious choice is the zeoth ode pat of the pole conditions 2.7,2.8: ψx, 0 = ux, 0 = ωx, 0 = It is theefoe a natual question to ask whethe a smooth solution of 2.2, 2. in the class ψt; x, C 0, T ; C k+ R R + ut; x, C 0, T ; C k R R ωt; x, C 0, T ; C k R R + will give ise to a smooth solution of 2.2. In othe wods, is the pole condition 2.7,2.8 automatically satisfied if only the zeoth ode pat 2. is imposed? The answe to this question is affimative: Theoem 2.2. [2] a If u, p is an axisymmetic solution to 2. with u C 0, T ; Cs k, p C 0 0, T ; C k R 3 k 3. Then thee is a solution ψ, u, ω to 2.2 in the class ψt; x, C 0, T ; Cs k+ R R + ut; x, C 0, T ; Cs k R R ωt; x, C 0, T ; Cs k R R + u = ue θ + ψe θ. b If ψ, u, ω is a solution to 2.2,2. in the class 2.2 with k 3. Then ψ, u, ω is in the class 2.3, u def = ue θ + ψe θ C 0, T ; Cs k thee is an axisymmetic scala function p C 0 0, T ; C k R 3 such that u, p is a solution to 2.. The poof of Theoem 2.2 can be found in [2]. We emak hee that Theoem 2.2 not only establishes the equivalence between fo classical solutions, the fact that smooth solutions to 2.2 automatically satisfies the pole condition 2.3 is also cucial to ou local tuncation eo analysis. See the Appendix fo details. 3. Regulaity assumption on solutions of Navie Stokes equation at infinity. The focus of this pape is the convegence ate of EHPS in the pesence of the pole singulaity. To sepaate difficulties avoid complications intoduced by physical boundaies, we only conside the whole space poblems with solutions decaying apidly at infinity. To be moe specific, we estict ou attention to the case whee the suppots of the initial data ux, 0 ωx, 0 ae essentially compact. Since 2.2 is a tanspot diffusion equation fo u ω with initially finite speed of popagation, we expect u ω to be essentially compactly suppoted, at least fo shot time. In the case of linea tanspot diffusion equations, the solution togethe with its deivatives will then decay faste than polynomials at infinity fo t > 0. Some igoous esults concening the spatial decay ate fo the solutions of axisymmetic flows can be found in [4] the efeence theein. In paticula, it is shown in [4] that both u ω decay algebaically at infinity as long as this is the case initially. Hee we make a stonge, yet plausible assumption along this diection. The pecise fom of ou assumption is

6 6 JIAN-GUO LIU AND WEI-CHANG WANG fomulated in tems of weighted noms is less stingent than the analogy we daw fom linea tanspot diffusion equations, see Assumption below. To quantify ou assumption, we fist intoduce a family of -homogeneous composite noms coesponding function spaces which tun out to be natual fo ou pointwise enegy estimate: Definition 3.. a l,α,β = l +l 2=l + α + x β l x l2 a L R R + 3. a k,α,β = 0 l k a k l,α l,β 3.2 Note that the noms 3., 3.2 ae well defined fo functions in Cs k decay popely at infinity. We denote by C k,α,β s = {ax, C k s R R + that R R +, a k,α,β < }. 3.3 In section 5, we will show that EHPS is second ode accuate povided the solution satisfies { ψ, ω C 0, T ; C 4,α+ 7 2,β s Cs 4,2α+2,2β u C, α > 0, T ; Cs 4,2α+2,2β Cs,2,0 2, β > In view of 3.4, we fomulate ou egulaity assumption as Assumption. ψ, ω C 0, T ; C 4,γ,δ s, u C 0, T ; Cs 4,5,δ, γ > 4, δ > Although we expect u, ω thei deivatives to decay faste than any polynomial at infinity, the same expectation is not ealizable fo ψ. As we will see, geneically ψ only decays like Ox at infinity. Nevetheless, we will show that Assumption is still ealizable if ω decays fast enough. To analyze the decay ate of ψ, we stat with the integal expession fo ψ. Fom the voticity-steam elation the identification ψ = ω ψx, = ψ z x, y, z y=,z=0, ωx, = ω z x, y, z y=,z=0, one can deive the following integal fomula fo ψ [7]: whee ψx, = 0 ωx, Kx x,, dx d 3.6 Kx x,, = 2π cos θ 4π 0 dθ x x 2 + cos θ 2 + sin θ 2 = 2 2 π π 2 0 cos 2 θ ρ dθ 3.7 +ρ ρ ++ρ

7 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 7 ρ 2 ± = x x 2 + ± cos θ 2 + sin θ 2 As a consequence, we have the following fa field estimate fo K: Lemma 3.2. l x m Kx x,, C l,m x, x l m as x Poof. We will deive a fa field estimate fo the integ in 3.7. We fist conside a typical tem with lim x l x m ρ whee x 0, 0 c 0 ae some constants. With the change of vaiables we can ewite the x deivatives by ρ 2 = x x c = σ cos λ x x 0 = σ sin λ ρ = σ2 + c 2 0 = σ σ σ2 + c λ λ σ2 + c 2 0 = σ ρ cos λ x ρ = x σ2 + c 2 0 = xσ σ σ2 + c xλ λ σ2 + c 2 0 = σ ρ sin λ Theefoe by induction, l x m ρ = P l,m cos λ, sin λq l,m σ, ρ whee P l,m cos λ, sin λ is a polynomial of degee l+m in its aguments Q l,m σ, ρ a ational function of σ ρ of degee l m. By degee of a ational function we mean the degee of the numeato subtacting the degee of the denominato. Since σ = O x ρ = O x 2 + 2, we conclude that l x m l x m ρ = O x l m. We can now apply the agument above Leibniz s ule to get J l,m ρ + ρ ρ + + ρ = j whee J l,m is a finite intege, σ ± ρ ± ae defined by P l,m l,m j cos λ +, sin λ +, cos λ, sin λ Q j σ +, ρ +, σ, ρ, ± cos θ = σ ± cos λ ± x x 0 = σ ± sin λ ±

8 8 JIAN-GUO LIU AND WEI-CHANG WANG P l,m j, Ql,m j ae polynomials ational functions of degees l + m, 2 l m in thei aguments espectively. The Lemma follows by integating θ ove 0, π 2 in 3.7. We close this section by noting that ψ exhibit slow decay ate at infinity as a consequence of 3.6 Lemma 3.2. Moe pecisely, ψx, Ox in geneal. This may seem to aise the question whethe Assumption is ealizable at all. Indeed, using a simila calculation as in the poof of Lemma 3.2, one can deive the following Poposition 3.3. If γ + δ < k + 2 ω C k,γ,δ s fo sufficiently lage γ s. As a consequence, we see that the ange of γ δ in 3.5 is δ, then ψ C k,γ,δ not void povided ω decays fast enough at infinity. This justifies Assumption. 4. Enegy Helicity Peseving Scheme. In this section, we outline the deivation of the discete enegy helicity identities fo EHPS. A key ingedient in the deivation is the efomulation of nonlinea tems into Jacobians. The details can be found in []. We intoduce the stad notations: x x φx + 2, φx 2 D x φx, =,, D φx, = x φx, + 2 φx, 2, D x φx, = φx + x, φx x,, D φx, = 2 x h = D x, D, h = D, D x. φx, + φx,. 2 The finite diffeence appoximation of 2 the Jacobians ae given by 2 hψ = D x D x ψ + D D ψ J h f, g = { 3 h f h g + h f h g + h g } h f 4. Altogethe, the second ode finite diffeence vesion of EHPS is: t u h + 2 J h u h, ψ h = ν 2 h 2 u h t ω h + J h ωh, ψ h = ν 2 h 2 ω h + J h uh, u h 4.2 ω h = 2 h + 2 ψ h To deive the discete enegy helicity identity, we fist intoduce the following discete analogue of weighted inne poducts a, b h = i= j= ab i,j x 4.3

9 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 9 [a, b] h = i= j= the coesponding noms D x ad x b i 2,j + i= j= D ad b i,j x + a, b h a 2 0,h = a, a h, a 2,h = [a, a] h 4.5 whee the gids have been shifted [3] to avoid placing the gid points on the axis of otation: x i = i x, i = 0, ±, ±2,, j = j, j =, 2, j= f j 2 = 2 f 2 + j=2 f j The evaluation of D 2 h tems in 4.2 at j = involves the dependent vaiables u h, ψ h, ω h the stetching facto h 3 = θ = at the ghost points j = 0. In view of Lemma 2., we impose the following eflection bounday condition acoss the axis of otation: u h i, 0 = u h i,, ψ h i, 0 = ψ h i,, ω h i, 0 = ω h i,. 4.8 Futhemoe, we take even extension fo the coodinate stetching facto h 3 = θ = which appeas in the evaluation of the Jacobians at j = : h 3 i, 0 = h 3 i,. 4.9 We will show in the emaining sections that the extensions indeed give ise to a discete vesion of enegy helicity identity optimal local tuncation eo. As a consequence, second ode accuacy of EHPS is justified fo axisymmetic flows. Remak. At fist glance, the extension 4.9 may seem to contadict 4.6 on the ghost points j = 0. A less ambiguous estatement of 4.9 is to incopoate it into 4.2 as t u h + J 2 h u h, ψ h = ν 2 h u 2 h t ω h + J h ωh, ψ h ω h = 2 h + 2 ψ h = ν 2 h 2 ω h + J h uh, u h on x i, j, j 4.0 The following identities ae essential to the discete enegy helicity identity the eo estimate: Lemma 4.. Suppose a, b, c satisfies the eflection bounday condition ai, 0 = ai,, bi, 0 = bi,, ci, 0 = ci,

10 0 JIAN-GUO LIU AND WEI-CHANG WANG define T h a, b, c := 3 i= j= c h a h b + a h b h c + b h c h a x. 4. i,j Then c i,j J h a, b i,j x = T h a, b, c, 4.2 i= j= a, 2 h + 2 b h = [a, b] h. 4.3 Poof. : We fist deive 4.2. In view of 4. 4., it suffices to show that c h a h b = a h c h b 4.4 j i i,j c h b h a = i j i,j b h c h a 4.5 o, since thee is no bounday tems in the x diection, simply i= j= f D g i,j = i= j= g D f i,j 4.6 with f = c g = b D x a a D x b. Using the summation-by-pats identity see fo example, [5] o [], it is staight fowad to veify that i= j= f D g i,j = i= j= g D f i,j f i,0 g i, + g i,0 f i,. i= In the deivation of the discete enegy helicity identities see below, a typical tiplet a, b, c is given by, say, a = ψ h, b = u h c = u h. Fom the eflection bounday condition , we see that f i,0 = f i,, g i,0 = g i,. This gives 4.6, theefoe 4.4, Next we deive 4.3. Fom the identity f j g j+ g 2 j = f 2 j f j g j 2 2 f + f 0 g 2 j= = 0, it is easy to show that 2 i= j= j= a i,j D D b i,j = i= j= D a i,j 2 j 2 D b i,j 2.

11 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME Theefoe 4.3 follows. Fom 4., we can easily deive the pemutation identities T h a, b, c = T h b, c, a = T h c, a, b, T h a, b, c = T h b, a, c. 4.7 Moeove, fom 4.2, 4.3, it follows that υ, t u h h + T h u h, ψ h, υ = ν υ, 2 h 2 u h h [ϕ, t ψ h ] h + T h ω h, ψ h, ϕ = ν ϕ, 2 h 2 ω h h + T h u h, u h, ϕ 4.8 ξ, ω h h = [ξ, ψ h ] h fo all υ, ϕ ξ satisfying υi, 0 = υi,, ϕi, 0 = ϕi,, ξi, 0 = ξi,. As a diect consequence of the pemutation identity 4.7, we take υ, ϕ = u h, ψ h in 4.8 ecove the discete enegy identity d dt 2 u h, u h h + [ψ h, ψ h ] h + ν[u h, u h ] h + ω h, ω h h = Similaly, the discete helicity identity d dt u h, ω h h + ν[u h, ω h ] h ω h, 2 h 2 u h h = follows by taking υ, ϕ = ω h, u h in 4.8. Remak 2. In the pesence of physical boundaies, the no-slip bounday condition gives u n = τ ψ = 0, u τ = n ψ = 0, u e θ = u = whee τ = n e θ e θ is the unit vecto in θ diection. When the coss section Ω is simply connected, 4.2 eads: u = 0, ψ = 0, n ψ = 0 on Ω It can be shown that the enegy helicity identities 4.9, 4.20 emain valid the pesence of physical bounday conditions []. The numeical ealization of the no-slip condition 4.22 intoduced in [] is second ode accuate seems to be new even fo usual 2D flows. The convegence poof fo this new bounday condition will be epoted elsewhee. 5. Enegy estimate the main theoem. In this section, we poceed with the main Theoem of eo estimate. We denote by ψ h, u h, ω h the numeical solution satisfying t u h + 2 J h u h, ψ h = ν 2 h 2 u h t ω h + J h ωh, ψ h = ν 2 h 2 ω h + J h uh, u h 5. ω h = 2 h + 2 ψ h

12 2 JIAN-GUO LIU AND WEI-CHANG WANG ψ, u, ω the exact solution to 2.2, t u + 2 J h u, ψ = ν 2 h 2 u + E t ω + J h ω, ψ = ν 2 h 2 ω + J h u, u + E ω = 2 h + 2 ψ + E 3 whee the local tuncation eos E j can be deived by subtacting 2.2 fom 5.2: E = 2 J h Ju, ψ ν 2 h 2 u E 2 = J h J ω, ψ ν 2 h 2 ω J h J u, u 5.3 E 3 = 2 h 2 ψ Fom , we see that t u u h + 2 J hu, ψ J h u h, ψ h = ν 2 h 2 u u h + E 5.4 t ω ω h + J h ω, ψ J h ωh, ψ h = ν 2 h 2 ω ω h + J h u, u J h uh, u h + E2 5.5 ω ω h = 2 h + 2 ψ ψ h + E Lemma 5. Lemma 5.2 below ae key to ou eo estimate. The pemutation identities 4.7 associated with EHPS esults in exact cancellation among the nonlinea tems leads to an exact identity 5.7. The estimates fo the tilinea fom in 5.3, 5.4 then funishes necessay inequalities fo ou a pioi eo estimate. The poof fo Lemma 5.2 the local tuncation eo analysis, Lemma 5.4, is given in the Appendix. Lemma t u u h 2 0,h + ψ ψ h 2,h + ν u u h 2,h + ω ω h 2 0,h = u u h, E h + ψ ψ h, E 2 t E 3 h + ν ω ω h, E 3 h T u uh h, u u h, ψ T h ψ ψ h, ω ω h, ψ + T h ψ ψh, u, u u h 5.7 Poof. We take the weighted inne poduct of u u h with 5.4 to get 2 t u u h 2 0,h + u u h, J 2 h u, ψ J h u h, ψ h h = ν u u h, 2 h 2 u u h h + u u h, E h. 5.8

13 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 3 The second tem on the left h side of 5.8 can be ewitten as u u h, 2 J h u, ψ J h u h, ψ h h = T u uh h, u, ψ T u uh h, u h, ψ h = T h u uh, u u h, ψ ψ h + T u uh h, u u h, ψ + T u uh h, u, ψ ψ h. 5.9 In addition, fom 4.3 we have Thus ν u u h, 2 h 2 u u h h = ν[u u h, u u h ] h = ν u u h 2,h. 2 t u u h 2 0,h T u uh h, u u h, ψ ψ h + ν u u h 2,h = u u h, E h T u uh h, u u h, ψ T u uh h, u, ψ ψ h. 5.0 Similaly, we take the weighted inne poduct of ψ ψ h with 5.5 poceed as to get 2 t ψ ψ h 2,h + T h ψ ψ h, ω ω h, ψ = T h ψ ψ h, u u h, u u h + T h ψ ψh, u, u u h +T h ψ ψ h, u u h, u Next, we apply 4.3 twice to get + ψ ψ h, E 2 t E 3 h + ν ψ ψ h, 2 h 2 ω ω h h. 5. ν ψ ψ h, 2 h 2 ω ω h h = ν 2 h 2 ψ ψ h, ω ω h h = ν ω ω h 2 0,h+ν ω ω h, E 3 h 5.7 follows. This completes the poof of this Lemma. We now poceed with the estimate fo the tilinea fom T h : Lemma 5.2. Fo a, b c Cs 2 R R +, we have T h a, b, c 5.2 C a,h b,h c,2,0 5.3 T h a, b, c C a 0,h b,h c 2,2, Poof. : See section 7.. Fom Lemma 5. Lemma 5.2, we can theefoe deive 2 t u u h 2 0,h + ψ ψ h 2,h + ν u u h 2,h + ω ω h 2 0,h u u h, E h + ψ ψ h, E 2 t E 3 h + ν ω ω h, E 3 h +C u u h 0,h u u h,h ψ 2,2,0 + C ω ω h 0,h ψ ψ h,h ψ 2,2,0 +C ψ ψ h,h u u h,h u,2,0. 5.5

14 4 JIAN-GUO LIU AND WEI-CHANG WANG Since a 0,h a,h, we can futhe estimate the fist few tems on the ight h side of 5.5 by u u h, E h = u u h, E h ν 4 u u h 2,h + ν E 2 0,h ψ ψ h, E 2 t E 3 h ψ ψ h 2,h + E 2 t E 3 2 0,h ω ω h, E 3 h 2 ω ω h 2 0,h + 2 E 3 2 0,h. Applying Hölde s inequality on the emaining tems of 5.5, we have deived the following Poposition t u u h 2 0,h + ψ ψ h 2,h + ν 4 u u h 2,h + ω ω h 2 0,h ψ ψ h 2,h + C ν E 2 0,h + E 2 2 0,h + te 3 2 0,h + ν E 3 2 0,h + C ν u u h 2 0,h ψ 2 2,2,0 + C ν ψ ψ h 2,h ψ 2 2,2,0 + C ν ψ ψ h 2,h u 2,2,0 5.6 With Poposition 5.3, it emains to estimate E 0,h, E 2 0,h, t E 3 0,h E 3 0,h. We summaize the esults in the following Lemma. Lemma 5.4. Let ψ, u, ω C 0, T ; C 4 s be a solution of the axisymmetic Navie Stokes equation 2.2 E, E 2, E 3 be defined by 5.2. Then we have the following pointwise local tuncation eo estimate fo α, β R: x E C + 2α + x 2β x E 2 C + 2α + x 2β ψ 4,α+ 7 2,β u 4,α+ 7 2,β + u 4,2α+2,2β 5.7 ψ 4,α+ 7 2,β ω 4,α+ 7 2,β + u 2 4,α+ 7 2,β + ω 4,2α+2,2β 5.8 x t E 3 C + 2α + x 2β t ψ 4,2α+2,2β 5.9 x E 3 C + 2α + x 2β ψ 4,2α+2,2β 5.20 Poof. See section 7.2. Fom Lemma , ou main esult follows:

15 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 5 Theoem 5.5. Let ψ, u, ω be a solution of the axisymmetic Navie Stokes equation 2.2 satisfying Then sup [0,T ] ψ, ω C 0, T ; C 4,γ,δ s, u C 0, T ; Cs 4,5,δ, γ > 4, δ > T u uh 2 0,h + ψ ψ h 2,h + u u h 2,h+ ω ω h 2 0,hdt C x log whee C = Cψ, u, ν, T. Poof. Fom Lemma 5.4, we have i= j= E 2 0,h + E 2 2 0,h + te 3 2 0,h C x j x + j 4α + x i 4β ψ, u, ω 4 4,α+ 7 2,β + u, ω, t ψ 2 4,2α+2,2β Similaly, E 3 2 0,h C x i= j= x j + j 4α + x i 4β ψ 2 4,2α+2,2β.. Since i= j= j x + j 4α + x i 4β C, fo α > 2, β > 4 i= j= x j + j 4α + x i 4β C log, fo α > 0, β > 4, it follows that E 2 0,h + E 2 2 0,h + t E 3 2 0,h C x ψ, u, ω 4 4,γ,δ + u, ω, t ψ 2 4,γ,δ 5.23 E 3 2 0,h C x log ψ 2 4,γ,δ 5.24 povided γ > 4, δ > 2. Unde assumption 5.2, we have, in paticula, ψ Cs 2,2,0, u Cs,2,0. It follows fom Poposition , 5.24 that 2 t u u h 2 0,h + ψ ψ h 2,h + ν 4 u u h 2,h + ω ω h 2 0,h C u u h 2 0,h + C ψ ψ h 2,h + C x4 + 4 log The eo estimate 5.22 then follows fom Gonwall s inequality.

16 6 JIAN-GUO LIU AND WEI-CHANG WANG 6. Conclusion. The impotance subtlety of the pole singulaity has been a majo difficulty in theoetical analysis algoithm design fo axisymmetic flows. The numeical analysis nea the pole singulaity is much moe complicated than that of stad smooth flows. The pincipal ingedients of ou eo analysis ae a The fact that smooth solutions to 2.2 automatically satisfy the pole condition thus belong to the class 2.3. This symmety popety plays an essential ole in the local tuncation eo analysis. b Pope fomulation discetization of the nonlinea tems. Hee the Jacobian fomulation along with the distinctive discetization 4. esult in exact cancellation among the nonlinea tems in the enegy estimate theefoe leads to consevation identities in discete setting. These ingedients may also seve as a guideline of algoithm design fo axisymmetic flows. In addition, the slow decay of the steam function at infinity poses exta technical difficulties in analyzing the whole space poblem. This difficulty is caefully esolved by choosing a popely weighted -homogeneous nom 3.2. On the one h, 3.2 takes into account the local behavio of the swiling components nea the pole singulaity. On the othe h, it incopoates fee paametes so that the slow decay of the steam function can be popely compensated by tuning the paametes though caeful analysis. 7. Appendix: poof of technical lemmas. 7.. Estimate fo the tilinea fom T h poof of Lemma 5.2. We stat with the following basic identities: Poposition 7.. Define Ãxf i,j = 2 f i+,j + f i,j, Ãf i,j = 2 f i,j+ + f i,j. Then the following estimates hold fo j : D a C Ãa + C D a 7. D a Ãa C 2 + C D a 7.2 Ãa Cà a 7.3 D a à a, x D x a Ãx a 7.4 Remak 3. As in Remak, the stetching facto in the aguments of the left h side of satisfy the even extension 4.9. A moe pecise statement fo, say, 7. is given by D a i,j C Ãa i,j + C j D a i,j, j. Fo simplicity of pesentation, we will adopt the expession as in though est of the pape.

17 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 7 Poof of Poposition 7.: It is easy to veify that D fg = Ãf D g + Ãg D f, Dx fg = Ãxf D x g + Ãxg D x f A staight fowad calculation shows that à j C j, D j C à j C j, D j C 2 j fo j. The estimates then follows. The poof fo 7.4 is also staight fowad. Poof of Lemma 5.2: We begin with the poof of 5.3. We exp the left h side as T h a, b, c = 3 c, 2 h a h b h + a, h b h c h + b, h c h a h = 3 I + I 2 + I 3 estimate the I j s tem by tem. Fistly, we have I = c 2, h a h b h = c, D a D x b + D x a D b h, theefoe the estimate I C c, Ãa + D a D x b + Ãb + D b D x a h C a,h b,h c 0,,0 follows fom 7., Hölde inequality the inequality c = c c 0,,0. Secondly, we have I 2 C a, Ãb + D b D x c h + C a, D c D x b h + C a, A c D x b h = C a, Ãb + D b D x c h + C a, D c D x b h + C a, A c D x b h C a,h b,h c 0,,0 + c,2,0 C a,h b,h c,2,0 The estimate fo I 3 is simila 5.3 follows. Next we poceed with 5.4. Since T h a, b, c = a, 2 J hb, c h a 0,h 2 J hb, c 0,h,

18 8 JIAN-GUO LIU AND WEI-CHANG WANG it suffices to give a pointwise estimate fo the integ J h b, c: 3J h b, c = D b D x c D x b D c + D b D x c D x b D c + D x c D b D c D x b = D bi + Ã D x c D x bi + Ãx D c + Ã Ãxb D Dx c +Ãx Ãc D x D b + D x cãx D b D cã D x b = D bi + Ã D x c D x bi + Ãx D c + Ã Ãxb D Dx c + 2 x2 D Dx bd 2 xc 2 2 D Dx bd 2 c + D x cãx D b D cã D x b Hee I is the identity opeato we have used the identities 7.5 Ã x = 2 x2 D 2 x + I, Ã = 2 2 D 2 + I in the second equality of 7.5. Fom 7., the fist two tems on the ight h side of 7.5 can be estimated by D bi + Ã D x c C 2 D b + Ãb x c L 7.6 D x bi +Ãx D c C 2 D x b c+ c L C2 D x b c 0,0,0 + c,,0 7.7 Fom , we can similaly estimate the emaining tems in 7.5: Ã Ãxb D Dx c C 2 Ã + Ãx b x c L C 2 Ã + Ãx b c,,0 + c 2,2,0, x2 D Dx bdxc 2 C x2 Ã D x bdxc 2 C 2 Ã D x b c 2,2,0, D Dx bd 2 c C Ã D x b 2 c L C 2 Ã D x b c 2,2,0, 7.0 Ãx D b D x c C 2 Ãx D b x c L C 2 Ã x D b + Ã b c,,0 7. Ã D x b D c C 2 Ã D x b c,, Fom , we can estimate the weighted L 2 nom of 2 J h b, c by 2 J hb, c 0,h C D x b + D b + b 0,h c 2,2,0 C b,h c 2,2,0 5.4 follows.

19 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME Local tuncation eo analysis poof of Lemma 5.4. In this subsection, we poceed with the local tuncation eo estimate. All the assetions in Lemmas ae pointwise estimates on the gid points x i, j, j. Fo bevity, we omit the indices i, j wheneve it is obvious. We stat with the estimates of the diffusion tems in 5.3. Lemma 7.2. If a C 4 s R R + α 0, β 0 R, we have 2 h 2 a C x α0 + x β0 a 4,α0+2,β h 2 a C x α0 + x a 4,α0+2,β β Poof. Since a Cs 4 R R +, the odd extension of a given by { ax,, if 0 ãx, = ax,, if < 0 is in C 4 R 2. It follows that 2 ha = D 2 x + D 2 + D a = 2 a+ 2 x2 xa 4 ξ, a x,η a x,η2 7.5 is valid fo all j with ξ x x, x + x η, η 2, +. Thus 2 h 2 a C x x a ξ, + 4 a x,η + 3 a x,η 2 C x a 4,α0 +2,β 0 + α ξ + a 4,α 0 +2,β 0 + a 3,α0 +,β 0 β 0 +η + a 3,α 0 +,β 0 + a 2,α0,β 0 α x β 0 +η 2 α 0 + x β 0 C x α 0 + x β 0 a 4,α0+2,β 0 This gives 7.3, togethe with 7.4 as a diect consequence. Next we poceed with the estimates fo the Jacobians, stating with thei typical factos: Lemma 7.3. Fo a C 4 s R R +, α, β R, we have D x a = x a + O x2 + α + x β a 3,α,β 7.6 D x a = x a + O 2 x 2 + α + x β a 3,α,β 7.7 D a = a + O α + x β a 3,α+3,β 7.8

20 20 JIAN-GUO LIU AND WEI-CHANG WANG D a = a + O 2 Poof. We begin with Since + α + x β a 3,α+3,β 7.9 D x x f = x2 6 3 xf ξ,, ξ x x, x + x, it follows that D x x a x 2 = 3 6 x a ξ, C x 2 + α + x β a 3,α,β D x x a = x2 6 x 3 2 a ξ, C 2 x 2 + α + x β a 3,α,β Fo , the estimate is moe complicated due to ou eflection bounday condition We estimate fo j > j = sepaately. When j >, we have D f = f x,η, η, +. Theefoe D a 2 = 3 a 6 x,η C α + x β a 3,α+3,β D a C a x,η C 2 When j =, we have a j= = C α + x β a 3,α+3,β + a 2,α+2,β + a,α+,β a j= C 2 3 In addition, since = 2, we apply 4.9 to get D a j= = a a 2 = C follows. 7.9 can be poved similaly, D a j= = a C 2 a a2 + a C α + x β a,α+,β 3 2 a a = a a, C 2 + α + x β a 0,α,β, + α + x β a 0,α,β

21 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 2 Theefoe a 2 C 2 a 2 2 C 2 + α + x β a 0,α,β. D a C 2 j= + α + x β a 0,α,β In addition, a j= 2 a a + 2 j= C 2 a,α+,β + a 0,α,β + α + x β, 7.9 follows. We now continue with the pointwise estimate fo the Jacobi tems J ha, b Ja, b J a h, b J a, b. Since 3 J 2 h a, b = D x a D b D a D x b + D a D x b b D a + 2 D 2 b D x a 2 a D x b 3J a h, b = D x a D b D a D x b + D a D x b b D a + D b D x a a D x b,, it suffices to estimate the tems in individually. We summaize them as the following Lemma 7.4. If a, b C 4 s R R + α, α 2, β, β 2 R, then D a D x b a xb + D b D x a b x a C x α +α 2 + x a β +β 2 3,α+ 5 b ,β 3,α2+ 5 2,β2 D x a D b x a b + D x a D b xa b C x α +α 2 + x a β +β 2 3,α+ 5 b ,β 4,α2+ 7 2,β2 D a D x b a x b + D 2 a D x b 2 a x b C x α +α 2 + x a β +β 2 3,α+ 5 b ,β 4,α2+ 7 2,β2 Poof. Since ae valid fo any α, β R, we have D x a = x a + O x2 + α+λ + x β a 3,α+λ,β 7.25 D x a = x a + O 2 x 2 + α+λ + x β a 3,α+λ,β

22 22 JIAN-GUO LIU AND WEI-CHANG WANG D a = a + O α+λ + x β a 3,α+λ+3,β 7.27 D a = a + O 2 + α+λ + x β a 3,α+λ+3,β 7.28 D x b = x b + O x2 + α2+µ + x β2 b 3,α2+µ,β D x b = x b + O 2 x 2 + α2+µ + x β2 b 3,α2+µ,β D b = b + O α2+µ + x β2 b 3,α2+µ+3,β D b = b + O 2 + α2+µ + x β2 b 3,α2+µ+3,β fo any λ, µ R. We apply 7.27, 7.30 with λ = 2, µ = 5 2 to get D a D x b a xb = D a D x b a D x b + a D x b a xb = O D a x b 2 3,α + 5,β 2 + O 2 + α 2 a + x β 3 x 2 b 3,α ,β 2 + α x β 2 Moeove, since it follows that 3 a + α x β a,α+ 2,β D x b = x bξ, 2 + α x β2 b,α2+ 2,β2, D a D x b a xb C x a 3,α + 5 2,β b,α 2 + 2,β 2 + a,α + 2,β b 3,α ,β 2 + α +α 2 + x β +β C x a 3,α + 5 2,β b 3,α ,β 2 + α +α 2 + x β +β 2

23 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 23 Similaly, fom , we have D x a D b x a b = D x a D b D x a b + D x a b x a b = O D x a b 3,α ,β 2 + O + α 2 2 b x 2 a 3,α + 2 5,β + x β 2 + α x β 7.34 C 2 a,α + 2,β b 3,α ,β 2 + C x 2 a 3,α + 5 b,β 2,α2 +,β α +α 2 + x β +β 2 + α +α 2 + x β +β 2 C x a 3,α + 5 2,β b 3,α ,β 2 + α +α 2 + x β +β 2 The estimate 7.22 then follows fom Fo 7.23, we have D x f D g x f g = D x f D g + D x x f g = x f D g ξ, + 6 x2 3 xf g ξ2,η = x f D g ξ, + f D x g ξ, + 6 x2 xf 3 g ξ2,η We poceed with individual tems in 7.35, taking f = a g = b. Fom 7.30 with µ = 2, we have a x D b C x a 2 b 3,α2+ 5 2,β2 Similaly, fom 7.32 a D x b + α x β 2 C 2 + α +α 2 + x β +β 2 a,α+ 2,β b 3,α2+ 5 2,β2 C 2 a + α x b + x β 2 3,α2+ 7 2,β2 C 2 + α +α 2 + x β +β 2 a 0,α 2,β b 4,α ,β2 x 2 3 x a b x,η C x 2 3 a x b + a 3 x b C x 2 a 3,α + 5 2,β b,α 2 + 2,β 2 + a 0,α 2,β b 3,α ,β 2 + α +α 2 + x β +β 2 C x 2 a 3,α + 5 2,β b 4,α ,β 2 + α +α 2 + x β +β 2

24 24 JIAN-GUO LIU AND WEI-CHANG WANG Theefoe, D x a D b x a b C x a 3,α+ 5 2,β b 4,α2+ 7 2,β2 + α+α2 + x β+β2. Using the same agument as above, one can deive D x a D b xa b C x2 + 2 a 3,α+ 5 2,β b 4,α2+ 7 2,β2 + α+α2 + x β+β2 theefoe 7.23 is poved. We continue with Fo the fist tem, we can wite D a D x b a x b = D a D x x b + D a x b. Since a, b C 4 s R R +, by extending a, b to odd functions acoss = 0, we see that the extended a D x b is in C 4 R 2, thus theefoe D a D x x b = a D x x b x,η = a D x x b + a D x x b x,η = x2 6 a x,η 3 xb ξ,η + a x,η 3 x b ξ2,η D a D x x b C x a 2,α+ b 2,β 4,α2+ 7 2,β α+α2 β+β2 + x Similaly, the extended a x b is in C 3 R 2, we have D a x b = a x b x,η C a 2 3,α+ 5 b 2,β 4,α2+ 7 2,β α+α2 + x β+β2 Fom , we have the following estimate fo the fist tem of 7.24: D a D x b a x b C x a 3,α+ 5 2,β b 4,α2+ 7 2,β2 + α+α2 + x β+β The second tem in 7.24 can be teated similaly, D 2 a D x b 2 a x b = D 2 a D x x b + D 2 a x b 7.39 Again, the extensions of 2 a D x x b 2 a x b ae both in C 3 R 2, we can diectly estimate these two tems by D 2 a D x x b = 2 a D x x b x,η = 2 a D x x b x,η + 2 a D x x b x,η = C x 2 a + 2a 3 xb ξ,η + a 3 x b ξ 2,η 7.40

25 CONVERGENCE ANALYSIS OF ENERGY AND HELICITY PRESERVING SCHEME 25 D 2 a x b = a x b x,η = a x b x,η 7.4 Fom , we have D 2 a D x x b C x 2 a,α + 2,β b 3,α ,β 2 + a 0,α 2,β b 4,α ,β 2 + α +α 2 + x β +β C x 2 a,α + 2,β b 4,α ,β 2 + α +α 2 + x β +β 2 D 2 a x b C a 2 3,α+ 5 b 2,β 4,α2+ 7 2,β α+α2 + x β+β2 Fom 7.39, , we conclude that D 2 a D x b 2 a x b C x2 + 2 a 3,α+ 5 b 2,β 4,α2+ 7 2,β α+α2 + x β+β2 The estimates imply Thus the poof of Lemma 7.4 is completed. As a diect consequence of Lemma 7.4, we have the pointwise estimate fo the Jacobians Lemma 7.5. If a, b C 4 s R R +, then J ha, b Ja, b C x a 4,α+ 7 b 2,β 4,α2+ 7 2,β2 + α+α2 + x β+β2 a a J h, b J, b C x a 4,α+ 7 b 2,β 4,α2+ 7 2,β2 + α+α2 + x β+β2 fo any α, α 2, β, β 2 R. Fom 5.3, Lemma 7.2 Lemma 7.5, we can easily deive This completes the poof of Lemma 5.4. Acknowledgments. The authos would like to thank anonymous efeees D. YinLiang Huang fo thei valuable suggestions that help to impove this pape. J.-G. Liu is sponsoed in pat by NSF gant DMS W.C. Wang is sponsoed in pat by NSC of Taiwan unde gant numbe M REFERENCES [] A. Aakawa, Computational design fo long-tem numeical integation of the equations of fluid motion: two dimensional incompessible flow. Pat I, J. Compute. Phys [2] G. K. Batchelo, An intoduction to fluid dynamics, Cambidge Univesity Pess, Cambidge, 999. [3] L. Caffaelli, R. Kohn L. Nienbeg, Patial egulaity of suitable weak solutions of the Navie-Stokes equations, Comm. Pue Appl. Math

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