Seventh International Conferen e onComputational Fluid Dynami s (ICCFD7),Big Island, Hawaii, July 9-13, 2012 ICCFD7-2012-3601Very-high-order onservative dis retization of di�usiveterms with variable vis osityG.A. Gerolymos∗ and I. Vallet∗Corresponding author: georges.gerolymos�upm .fr

∗ Université Pierre-et-Marie-Curie, 4 pla e Jussieu, 75005 Paris, Fran e.Abstra t: We study very-high-order onservative dis retizations for di�usive terms with variablevis osity, whi h are present in the ompressible Navier-Stokes equations, based on the onstru tionof numeri al vis ous �uxes at ell-interfa es. We introdu e a novel onservative approa h for thedis retization of (v(x) u′(x))′ whi h yields O(∆x2s) a ura y on the sten il {i−s, · · · , i, · · · , i+s},thus obtaining the same order-of-a ura y as non onservative methods on the same sten il, andimproving upon previous onservative proposals whi h are O(∆x2⌈ s

2⌉)-a urate on the same sten il.The extension of the s heme to 2-D and 3-D regular Cartesian grids, in luding ross-derivatives, isdes ribed. Several typi al 1-D and 3-D omputational examples substantiate the order-of-a ura yof the method.Keywords: High-Order S hemes, Re onstru tion, Vis ous Terms, Di�usion Equation, Compress-ible Navier-Stokes Equations.1 Introdu tionVery-high-order a ura y is essential in several pra ti al appli ations where the solution ontains widelyvarying spatial and temporal s ales, as in the ase of dire t numeri al simulation (dns) of ompressibleturbulent �ows in physi al spa e [1, 2, 3, 4, 5, 6, 7, 8℄. Sin e su h �ows may ontain sho kwaves [1, 4, 6, 7℄,parti ular are is taken in designing and using very-high-order onservative methods for the onve tivepart of the equations, weno s hemes [9℄ being a widely adopted hoi e [4, 5, 6, 7, 8℄. On the other handthe dis retization of the di�usive (vis ous) terms has re eived less attention. Many authors revert to anon onservative formulation [3℄, the ompa t s heme developed by Lele [10℄ being a popular hoi e [11℄,while others prefer using low-order onservative s hemes for the vis ous terms [8℄, ombined with very-high-order s hemes for the onve tive terms [12℄.However, non onservative approa hes do not warrant global equilibrium of for es in the momentumequation [13℄, whi h is only a hieved approximately, with a ura y depending on the trun ation error of thes heme on the grid used in the parti ular omputation. Therefore, when very oarse grids are used, as eg inthe ase of preliminary ill-resolved dns al ulations, the error in global equilibrium may be not negligible.1It is pre isely su h global equilibrium relations that onservation ensures [13℄.Zingg et al. [14℄ have developed a onservative s heme for the vis ous terms on the sten il si,3,3 := {i −

3, · · · , i+3}, whi h yields an O(∆x4)-a urate approximation of (v(x) u′(x))′. As will be shown in the presentwork the formulation of Zingg et al. [14℄ an be easily extended to the general sten il si,s,s := {i−s, · · · , i+s}to yield O(∆x2⌈ s

2⌉)-a urate numeri al approximation of (v(x) u′(x)

)′. Noti e that the order-of-a ura yobtained for the vis ous terms is lower than the a ura y O(∆x(2s−1)) obtained by the weno(2s − 1) s hemesfor the onve tive terms on the same sten il [15, 16, 8℄. Shen et al. [17℄, working on the same si,3,3 sten ilas Zingg et al. [14℄, developed an alternative O(∆x4)-a urate onservative formulation, whi h they showedto have smaller trun ation error (but the same order-of-a ura y).1eg in fully-developed in ompressible plane hannel �ow [8℄ the global equilibrium relation is 2τw + Ly∂xp = 0, where τw isthe wall-shear-stress ∂xp is the streamwise pressure-gradient, and Ly is the hannel's height1

ICCFD7-3601

Both these onservative approa hes [14, 17℄ onstru t �uxes2 at the ell-interfa es using a fa e-based(and fa e- entered) sten il. Alternatively, Gassner et al. [18℄, extending the generalized Riemann problem[19℄ to di�usive terms, proposed a ell-based re onstru tion, leading to left (l) and right (r) re onstru tingpolynomials on the 2 sides of ea h fa e, whi h are ombined through a di�usion Riemann solver [18℄, to onstru t the interfa e �ux for the di�usive terms. In the present work, we on entrate on the fa e-basedapproa h. Furthermore, we assume that the investigated problems are su� iently smooth, postponing thetreatment of weak solutions (eg through weno re onstru tion [9, 20℄) to a future spe i� study. Throughoutthe paper we work on general sten ils parametrized by the order-parameter s, so that all of the developedrelations and results are s alable to arbitrary order-of-a ura y.2 Fluxes for (v(x)u′(x))′Both Zingg et al. [14℄ and Shen et al. [17℄ developed their s hemes on the sten il si,3,3 := {i − 3, · · · , i + 3}.Shen et al. [17℄ express expli itly the numeri al �uxes [F(vu′),sz ,si,2,3

]i+ 12and [F(vu′),sz ,si−1,2,3

]i− 12, de�nedon the sten ils si,2,3 := {i − 2, · · · , i + 3} and si−1,2,3 = si,3,2 := {i − 3, · · · , i + 2}, respe tively, whi h yieldan O(∆x4)-a urate approximation of (vu′)′i. On the other hand, Zingg et al. [14℄ gave dire tly the �nite-di�eren e expression approximating (vu′)′i to O(∆x4), on the sten il si,3,3 := {i−3, · · · , i+3}. Their method an easily be interpreted in terms of interfa e-�uxes, using re onstru tion on epts [21℄. In the followingwe show how both these s hemes an be extended to arbitrary O(∆x2⌈ s

2⌉) order-of-a ura y on the general entered (around i) sten il si,s,s := {i − s, · · · , i + s}. Then, we present a more ompa t approa h yieldingan asymptoti ally twi e more a urate O(∆x2s) approximation of (vu′)′i on si,s,s := {i − s, · · · , i + s}.The explanation of the s heme of Zingg et al. [14℄ is more intri ate than the original �nite-di�eren inganalysis, in order to bring forward intermediate steps and sten ils (Fig. 1), where a ura y is lost. This moreintri ate analysis, also applied to the s heme of Shen et al. [17℄, highlights the reasons of loss of a ura yand provides guidan e towards improvement.2.1 De�nitionsThe following tools of polynomial re onstru tion [22, 9, 21℄ are used, both to des ribe and extend to arbitraryorder-of-a ura y the methods of Zingg et al. [14℄ and of Shen et al. [17℄, and to develop the present approa h.On a homogeneous 1-D grid

xi = x1 + (i − 1)∆x ∆x = const ∈ R>0 (1a)we de�ne the sten ilsi,M−,M+:= {i − M−, · · · , i + M+} ; M := M− + M+ ≥ 0 (1b)of M + 1 points in the neighbourhood of i, with M− neighbours to the left and M+ neighbours to the right.Assume that the real fun tion f : R −→ R is su� iently smooth, and that there exists a real fun tion

h : R −→ R whose sliding (with x) ell-averages are equal to f(x), ief(x) =

1

∆x

∫ x+ 12∆x

x− 12∆x

h(ζ)dζ ∀x ∈ [a, b][21, (9)℄=⇒ f ′(x) =

h(x + 12∆x) − h(x − 1

2∆x)

∆x∀x ∈ [a, b] (1 )Be ause of (1 ), an approximation to h(x) an serve to de�ne interfa e �uxes at i± 1

2 for the approximationof f ′(x). It simpli�es notation to write, for 2 fun tions f : R → R and h : R → R satisfying (1 )(1 ) ⇐⇒ h = R(1;∆x)(f) (1d)2although the original s heme of Zingg et al. [14℄ gave the expression for the numeri al approximation of „

v(x)u′(x)

«′, it an be easily reinterpreted in terms of interfa e �uxes (�2.2) 2

s = 3

i−

3

i−

2

i−

1

i i+

1

i+

2

i+

3

i + 12

ℓ ∈ {−⌈ s2⌉ + 1, · · · , ⌈ s

2⌉ − 1}s=3= {−1, 0, 1}

Zingg et al. (2000)

ˇ[v]zrnp,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

i − 12

i + 12

i + 32

ˇ[u′]zrnp,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

i − 12

i + 12

i + 32

[F(vu′),zrnp,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

]i+ 12

i + 12

Shen et al. (2009)

ˇ[v]szc,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

i − 12

i + 12

i + 32

ˇ[u′]szc,si+ℓ,2⌈ s

2⌉−2,2⌈ s

2⌉−1,i+ℓ+ 1

2

i − 12

i + 12

i + 32

[F(vu′),szc,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

]i+ 12

i + 12

present

ˇ[v]gv,si,s−1,s

(x)

ˇ[u′]gv,si,s−1,s

(x)

[F(vu′),gv,si,s−1,s]i+ 1

2

i + 12Figure 1: Example for k = 2 =⇒ s⌈ s

2⌉=2 = 3 of the di�erent approximation steps and sten ils used forthe onstru tion of the numeri al �uxes at i + 1

2 for the approximation of [(vu′)′]i, on the sten il si,s,s :=

{i − s, · · · , i + s}s=3= {i − 3, · · · , i + 3}, [F(vu′),zrnp,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

]i+ 12(2f) of Zingg et al. [14℄ (�2.2) and

[F(vu′),sz ,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

]i+ 12(3f) of Shen et al. [17℄ (�2.3), both of whi h use intermediate values at theinterfa es i + ℓ + 1

2 (ℓ ∈ {−⌈ s2⌉+ 1, · · · , ⌈ s

2⌉ − 1}s=3= {−1, 0, +1}), and [F(vu′),gv,si,s−1,s

]i+ 12(6) developed inthe present work (�2.4), whi h uses re onstru tion of interpolating polynomials on the entire sten il (·− ·− ·interfa e where a quantity is approximated, ◦ not used, • point used for the omputation at the interfa e,

� intermediate values at interfa es omputed using ⊙ points).3

and we will all f and h a re onstru tion pair in view of the omputation of the 1-derivative [21℄.We will notepI,M−,M+

(xi + ξ∆x; xi, ∆x; f)[21, (45e)℄

=

M+∑

ℓ=−M−

αI,M−,M+,ℓ(ξ) f(xi + ℓ∆x) (1e)[21, (51b)℄= f(xi + ξ∆x) + O(∆xM+1) (1f)the Lagrange interpolating polynomial on si,M−,M+

:= {i − M−, · · · , i + M+}, of degree M := M− + M+,whi h approximates f(x) to O(∆xM+1), and whose derivative with respe t to x,p′I,M−,M+

(xi + ξ∆x; xi, ∆x; f) := [d

dxpI,M−,M+

](x; xi, ∆x; f)(1e)=

1

∆x

M+∑

ℓ=−M−

α′I,M−,M+,ℓ(ξ) f(xi + ℓ∆x)(1g)(1f)

= f ′(xi + ξ∆x) + O(∆xM ) (1h)approximates f ′(x) to O(∆xM ). The orresponding Lagrange re onstru ting polynomial is de�ned by re-quiring thatpI,M−,M+

(x; xi, ∆x; f) =1

∆x

∫ x+ 12∆x

x− 12∆x

pR1,M−,M+(x; xi, ∆x; f)dζ ∀x ∈ R (1i)This polynomial, whi h is also of degree M [21, Lemma 3.1, p. 277℄, an be represented by

pR1,M−,M+(xi + ξ∆x; xi, ∆x; f)

[21, (45d)℄=

M+∑

ℓ=−M−

αR1,M−,M+,ℓ(ξ) f(xi + ℓ∆x) (1j)[21, (51a)℄= h(xi + ξ∆x) + O(∆xM+1) (1k)and approximates, to O(∆xM+1) [21, Proposition 4.7, p. 292℄, the fun tion h(x), whose sliding (with x) ell-averages are equal to f(x). The Lagrange re onstru ting polynomial (1j) de�nes interfa e �uxes for the omputation of f ′(x) to O(∆xM+1) [21℄. By analogy with (1 ), we have [21, Corollary 4.9, p. 295℄

f ′(xi) + O(∆xM+1)[21, (60)℄

=pR1,M−,M+

(xi + 12∆x; xi, ∆x; f) − pR1,M−,M+

(xi −12∆x; xi−1, ∆x; f)

∆x(1l)

=pR1,M−,M+

(xi + 12∆x; xi, ∆x; f) − pR1,M−+1,M+−1(xi −

12∆x; xi, ∆x; f)

∆x(1m)By (1j), the re onstru ting polynomials at i + 1

2 and i − 12 , al ulated on the sten ils si,M−,M+

:= {i −M−, · · · , i+M+} and si−1,M−,M+

:= {i−1−M−, · · · , i−1+M+} =: si,M−+1,M+−1, respe tively, de�ne theinterfa e �uxes for al ulating f ′i := f ′(xi) to O(∆xM−+M++1) = O(∆xM+1). The fundamental fun tions ofLagrange re onstru tion αR1,M−,M+,ℓ ∈ RM [ξ] are polynomials of degree M whi h an be expli itly al ulatedfrom the analyti al expressions [23℄ of the fundamental fun tions of Lagrange interpolation αI,M−,M+,ℓ ∈

RM [ξ]. Analyti al expressions for αR1,M−,M+,ℓ(ξ) were obtained by Shu [22, (2.19), p. 336℄, and an also beexpressed using the elements of the inverse of the Vandermonde matrix [21, (45g), p. 287℄. These analyti alexpressions [24, (10,11,14), p. 2768℄ are easily implemented using symboli al ulus [25℄ and will be used toexpli itly evaluate and tabulate the oe� ients of the di�erent s hemes.33 In [21, pp. 301�303℄ it was shown that numeri al �uxes for the approximation of the 2-derivative f ′′(x) an be onstru tedby applying twi e the re onstru tion operator (1d). However, the ase of the numeri al approximation of (vu′)′, whi h is studiedhere, is more ompli ated.4

2.2 Zingg et al. [14℄Both the s heme of Zingg et al. [14℄ extended to arbitrary order-of-a ura y here, and the s heme of Shenet al. [17℄ studied below (�2.3), are de�ned on sten ils with an odd number of neighbours on ea h side ofpoint i, si,2k−1,2k−1 := {i − 2k + 1, · · · , i + 2k − 1} (k ∈ N>0), on whi h they are O(∆x2k)-a urate. Forthe purpose of omparison with the present method we des ribe them on the general entered (around i)sten il si,s,s := {i − s, · · · , i + s} (s ∈ N>0), by repla ing k := ⌈ s2⌉, ie on the sten il si,2⌈ s

2⌉−1,2⌈ s

2⌉−1 :=

{i−2⌈ s2⌉+1, · · · , i+2⌈ s

2⌉− 1} (s ∈ N>0), where they are O(∆x2⌈ s2⌉)-a urate. When s = 2k−1 (k ∈ N>0),the entire sten il si,s,s := {i − s, · · · , i + s} (s ∈ N>0) is used, as in the present s heme (�2.4), whereaswhen s = 2k (k ∈ N>0) the s hemes of Zingg et al. [14℄ and of Shen et al. [17℄ do not use the end-points

{i− s, i + s}. Although Zingg et al. [14℄ followed a standard �nite-di�eren ing approa h in developing theirs heme on si,3,3 := {i−3, · · · , i+3}, we des ribe the method in the following in terms of equivalent polynomialinterpolations, both be ause this allows the expli it analyti al al ulation of all s heme- oe� ients in astraightforward manner, but also be ause it lari�es order-of-a ura y relations by (1g, 1l), and the moredetailed results in [21℄.Interpreted in terms of interfa e �uxes[(vu′)′i]zrnp,si,2⌈ s

2⌉−1,2⌈ s

2⌉−1

=[F(vu′),zrnp,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

]i+ 12− [F(vu′),zrnp,si−1,2⌈ s

2⌉−2,2⌈ s

2⌉−1

]i− 12

∆x

=[(vu′)′](xi) + O(∆x2⌈ s2⌉) (2a)the method of Zingg et al. [14℄ omputes the numeri al �ux [F(vu′),zrnp,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

]i+ 12on the sten ilsi,2⌈ s

2⌉−2,2⌈ s

2⌉−1 := {i − 2⌈ s

2⌉ + 2, · · · , i + 2⌈ s2⌉ − 1} (Fig. 1). First, O(∆x2⌈ s

2⌉)-a urate interpolations of

v(x) and u(x) at the ell-interfa es si+ 12,⌈ s

2⌉−1,⌈ s

2⌉−1 := {i − ⌈ s

2⌉ + 32 , · · · , i + ⌈ s

2⌉ −12} are onstru ted.The interpolation at ea h ell-interfa e i + ℓ + 1

2 (ℓ ∈ {−⌈ s2⌉ + 1, · · · , ⌈ s

2⌉ − 1} is obtained on the sten ilsi+ℓ,⌈ s2⌉−1,⌈ s

2⌉ := {i + ℓ − ⌈ s

2⌉ + 1, · · · , i + ℓ + ⌈ s2⌉} whi h is entered around i + ℓ + 1

2 .ˇ[v]zrnp,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

:=pI,⌈ s2⌉−1,⌈ s

2⌉(xi+ℓ + 1

2∆x; xi+ℓ, ∆x; v)(1e)=

s∑

p=−⌈ s2⌉+1

αI,⌈ s2⌉−1,⌈ s

2⌉,p(

12 ) vi+ℓ+p

(1f)= v(xi+ℓ+ 1

2) + O(∆x2⌈ s

2⌉) (2b)

ˇ[u′]zrnp,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

:=p′I,⌈ s2⌉−1,⌈ s

2⌉(xi+ℓ + 1

2∆x; xi+ℓ, ∆x; u)(1g)=

1

∆x

⌈ s2⌉

∑

q=−⌈ s2⌉+1

α′I,⌈ s

2⌉−1,⌈ s

2⌉,q(

12 ) ui+ℓ+q

(1h)= u′(xi+ℓ+ 1

2) + O(∆x2⌈ s

2⌉) (2 )By (1f) we know that the interpolation on {i + ℓ− ⌈ s

2⌉+ 1, · · · , i + ℓ + ⌈ s2⌉} is O(∆x2⌈ s

2⌉). The derivative ofthe interpolating polynomial in (2 ) is an O(∆x2⌈ s

2⌉−1)-a urate approximation of the derivative in general,but is O(∆x2⌈ s

2⌉) at i+ ℓ+ 1

2 be ause the sten il si+ℓ,⌈ s2⌉−1,⌈ s

2⌉ := {i+ ℓ−⌈ s

2⌉+1, · · · , i+ ℓ+⌈ s2⌉} is enteredaround i + ℓ + 1

2 (a linear interpolation between 2 points yields an O(∆x2)-a urate approximation of thederivative at the enter of the interval but only O(∆x) elsewhere). Then we use the O(∆x2⌈ s2⌉)-a urateapproximations to vu′(x) at the 2⌈ s

2⌉−1 points i+ℓ+ 12 (ℓ ∈ {−⌈ s

2⌉+1, · · · , ⌈ s2⌉−1} to obtain an O(∆x2⌈ s

2⌉)-a urate re onstru tion of the fun tion [R(1;∆x)(vu′)](x),4 whose sliding with x ell-averages (1 , 1d) areequal to v(x)u′(x). For ease of notation we de�ne as ˇ[vu′]zrnp,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

(x) the Lagrange interpolating4 in (1 ) v(x)u′(x) is f(x) and the unknown fun tion [R(1;∆x)(vu′)](x) is h(x)

5

polynomial de�ned by the 2⌈ s2⌉ − 1 values ˇ[v]zrnp,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

ˇ[u′]zrnp,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

,ˇ[vu′]zrnp,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

(x)(1e):=

⌈ s2⌉−1∑

ℓ=−⌈ s2⌉+1

(

αI,⌈ s2⌉−1,⌈ s

2⌉−1,ℓ

(x − xi+ 12

∆x

)

(

[v]zrnp,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2[u′]zrnp,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

))(1f, 2b, 2 )

= v(x)u′(x) + O(∆x2⌈ s2⌉) (2d)although only the halfpoint-values (2b, 2 ) appear in the �nal s heme.5 The interfa e-�ux is obtained bythe re onstru tion of ˇ[vu′]zrnp,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

(x) (2d) on the sten il si+ 12

,⌈ s2⌉−1,⌈ s

2⌉−1 := {i + 3

2 − ⌈ s2⌉, · · · , i−

12 + ⌈ s

2⌉} whi h is entered around i + ℓ + 12

[F(vu′),zrnp,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

]i+ 12

:=pR1,⌈ s2⌉−1,⌈ s

2⌉−1(xi+ 1

2; xi+ 1

2, ∆x; ˇ[vu′]zrnp,si+ℓ,2⌈ s

2⌉−2,2⌈ s

2⌉−1

)

=

⌈ s2⌉−1∑

ℓ=−⌈ s2⌉+1

(

αR1,⌈ s2⌉−1,⌈ s

2⌉−1,ℓ(0)

(

[v]zrnp,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2[u′]zrnp,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

))

=[R(1;∆x)(vu′)](xi+ 12) + O(∆x2⌈ s

2⌉) (2e)Again, (2e) is O(∆x2⌈ s

2⌉−1)-a urate in general, but is O(∆x2⌈ s

2⌉) at i+ 1

2 be ause the re onstru tion sten ilsi+ 12

,⌈ s2⌉−1,⌈ s

2⌉−1 := {i+ 3

2 −⌈ s2⌉, · · · , i− 1

2 + ⌈ s2⌉}. Repla ing (2b, 2 , 2d) in (2e), gives the �nal expression5

[F(vu′),zrnp,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

]i+ 12

=

1

∆x

⌈ s2⌉−1∑

ℓ=−⌈ s2⌉+1

αR1,⌈ s2⌉−1,⌈ s

2⌉−1,ℓ(0)

( ⌈ s2⌉

∑

p=−⌈ s2⌉+1

αI,⌈ s2⌉−1,⌈ s

2⌉,p(

12 ) vi+ℓ+p

)( ⌈ s2⌉

∑

q=−⌈ s2⌉+1

α′I,⌈ s

2⌉−1,⌈ s

2⌉,q(

12 ) ui+ℓ+q

)

(2f)Using the analyti al expressions for αR1,⌈ s2⌉−1,⌈ s

2⌉−1,m(ξ) [21, (45g), p. 287℄ and for αI,⌈ s

2⌉−1,⌈ s

2⌉,p(ξ) [21,(45h), p. 287℄ it is straightforward to ompute the 2⌈ s

2⌉ − 1 rational onstants αR1,⌈ s2⌉−1,⌈ s

2⌉−1,m(0) (ℓ ∈

{−⌈ s2⌉ + 1, · · · , ⌈ s

2⌉ − 1}), the 2⌈ s2⌉ rational onstants αI,⌈ s

2⌉−1,⌈ s

2⌉,p(

12 ) (p ∈ {−⌈ s

2⌉ + 1, · · · , ⌈ s2⌉}), and the

2⌈ s2⌉ rational onstants α′

I,⌈ s2⌉−1,⌈ s

2⌉,q(

12 ) (q ∈ {−⌈ s

2⌉ + 1, · · · , ⌈ s2⌉}), appearing in the approximation (2f) ofthe numeri al �ux [F(vu′),zrnp,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

]i+ 12.2.3 Shen et al. [17℄The s heme of Shen et al. [17℄ uses the same as Zingg et al. [14℄ si,2⌈ s

2⌉−2,2⌈ s

2⌉−1 := {i − 2⌈ s

2⌉ + 2, · · · , i +

2⌈ s2⌉ − 1} sten il to ompute the numeri al �ux [F(vu′),sz ,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

]i+ 12. The 2 s hemes are quitesimilar, ex ept for the approximation of u′(xi+ℓ+ 1

2) (ℓ ∈ {−⌈ s

2⌉+1, · · · , ⌈ s2⌉−1}), for whi h the entire sten ilsi,2⌈ s

2⌉−2,2⌈ s

2⌉−1 := {i− 2⌈ s

2⌉+ 2, · · · , i + 2⌈ s2⌉− 1} is used by Shen et al. [17℄, ∀ℓ ∈ {−⌈ s

2⌉+ 1, · · · , ⌈ s2⌉− 1}.Interfa e �uxes are de�ned by

[(vu′)′i]sz ,si,2⌈ s2⌉−1,2⌈ s

2⌉−1

=[F(vu′),sz ,si,2⌈ s

2⌉−1,2⌈ s

2⌉−1

]i+ 12− [F(vu′),sz ,si−1,2⌈ s

2⌉−1,2⌈ s

2⌉−1

]i− 12

∆x

= [(vu′)′](xi) + O(∆x2⌈ s2⌉) (3a)5 ˇ[vu′]zrnp,si,2⌈ s

2⌉−2,2⌈ s

2⌉−1

(xi+ℓ+ 1

2

) = pI,⌈ s2⌉−1,⌈ s

2⌉−1(xi+ℓ+ 1

2

;xi+ 1

2

, ∆x; ˇ[vu′]zrnp,si+ℓ,2⌈ s2⌉−2,2⌈ s

2⌉−1

) =

[v]zrnp,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

[u′]zrnp,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

∀ℓ ∈ {−⌈ s2⌉ + 1, · · · , ⌈ s

2⌉ − 1} be ause αI,⌈ s

2⌉−1,⌈ s

2⌉−1,ℓ(m) = δℓm[23℄ 6

Halfpoint values, at the ell-interfa es i + ℓ + 12 (ℓ ∈ {−⌈ s

2⌉ + 1, · · · , ⌈ s2⌉ − 1}) are de�ned, using Lagrangeinterpolation,

ˇ[v]sz ,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

:= ˇ[v]zrnp,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

= (2b) (3b)ˇ[u′]sz ,si+ℓ,2⌈ s

2⌉−2,2⌈ s

2⌉−1,i+ℓ+ 1

2

:=p′I,2⌈ s2⌉−2,2⌈ s

2⌉−1(xi + (ℓ + 1

2 )∆x; xi, ∆x; u)(1g)=

1

∆x

2⌈ s2⌉−1∑

q=−2⌈ s2⌉+2

α′I,2⌈ s

2⌉−2,2⌈ s

2⌉−1,q(ℓ + 1

2 ) ui+q(1h)= u′(xi+ℓ+ 1

2) + O(∆x4⌈ s

2⌉−3)(3 )and used to onstru t the interpolating polynomial of v(x)u′(x), de�ned by

ˇ[vu′]sz ,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

(x)(1e):=

⌈ s2⌉−1∑

ℓ=−⌈ s2⌉+1

(

αI,⌈ s2⌉−1,⌈ s

2⌉−1,ℓ

(x − xi+ 12

∆x

)

(

[v]sz ,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2[u′]sz ,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

))(1f, 3b, 3 )

= v(x)u′(x) + O(∆x2⌈ s2⌉) (3d)Then, this polynomial is re onstru ted [21℄ to obtain the interfa e �uxes

[F(vu′),sz ,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

]i+ 12

:=pR1,⌈ s2⌉−1,⌈ s

2⌉−1(xi+ 1

2; xi+ 1

2, ∆x; ˇ[vu′]sz ,si+ℓ,2⌈ s

2⌉−2,2⌈ s

2⌉−1

)

=

⌈ s2⌉−1∑

ℓ=−⌈ s2⌉+1

(αR1,⌈ s

2⌉−1,⌈ s

2⌉−1,ℓ(0)

(

[v]sz ,si+ℓ,⌈ s2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2[u′]sz ,si+ℓ,⌈ s

2⌉−1,⌈ s

2⌉,i+ℓ+ 1

2

))

=[R(1;∆x)(vu′)](x) + O(∆x2⌈ s2⌉) (3e)Repla ing (3b, 3 , 3d) in (3e), gives in analogy with the s heme of Zingg et al. [14℄,5 the �nal expression

[F(vu′),sz ,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

]i+ 12

=1

∆x

⌈ s2⌉−1∑

ℓ=−⌈ s2⌉+1

(

αR1,⌈ s2⌉−1,⌈ s

2⌉−1,ℓ(0)

(s∑

p=−⌈ s2⌉+1

αI,⌈ s2⌉−1,⌈ s

2⌉,p(

12 ) vi+ℓ+p

)( 2⌈ s2⌉−1∑

q=−2⌈ s2⌉+2

α′I,2⌈ s

2⌉−2,2⌈ s

2⌉−1,q(ℓ + 1

2 ) ui+q

)) (3f)The di�eren e ompared to Zingg et al. [14℄ is that Shen et al. [17℄ use a higher-order interpolant forapproximating u′(x), but as they orre tly state in their paper this redu es the magnitude of the trun ationerror on a given grid, without improving the order-of-a ura y.The rational onstants αR1,⌈ s2⌉−1,⌈ s

2⌉−1,m(0) (ℓ ∈ {−⌈ s

2⌉ + 1, · · · , ⌈ s2⌉ − 1}) and αI,⌈ s

2⌉−1,⌈ s

2⌉,p(

12 ) (p ∈

{−⌈ s2⌉+1, · · · , ⌈ s

2⌉}) in the expression (3f) of the numeri al �ux [F(vu′),sz ,si,2⌈ s2⌉−2,2⌈ s

2⌉−1

]i+ 12are the same asthose used in (2f) by Zingg et al. [14℄. The new (2⌈ s

2⌉−2)×(2⌈ s2⌉−1) rational onstants α′

I,2⌈ s2⌉−2,2⌈ s

2⌉−1,q(ℓ+

12 ) (ℓ ∈ {−⌈ s

2⌉+1, · · · , ⌈ s2⌉−1}; q ∈ {−2⌈ s

2⌉+2, · · · , 2⌈ s2⌉−1}) an be easily al ulated using the analyti alexpression for αI,⌈ s

2⌉−1,⌈ s

2⌉,p(ξ) [21, (45h), p. 287℄.2.4 Present s hemeThe s hemes of Zingg et al. [14℄ and of Shen et al. [17℄ yield O(∆x2⌈ s

2⌉)-a urate approximations of (vu′)′ion the sten il si,s,s := {i− s, · · · , i + s}, where entred approximations of u′′

i by standard �nite-di�eren ingyield O(∆x2s)-a ura y [26, (7.6), p. 297℄. Examining the method of Zingg et al. [14℄ (�2.2) it is obvious7

that a ura y is limited both by the a ura y of the substen ils used for approximating vu′ at halfpoints,but also by the �nal re onstru tion from information on 2⌈ s2⌉ − 1 dis rete values ompared to the 2s + 1available dis rete values for v and u on the entire sten il. Shen et al. [17℄ used higher-order approximationsfor u′, but sin e the �nal re onstru tion again only uses 2⌈ s

2⌉−1 dis rete values for (vu′) the resulting s hemeis only O(∆x2⌈ s2⌉)-a urate. The di� ulty with both these approa hes [14, 17℄ lies in the hoi e made of�rst approximating vu′ at halfpoints, and then �nite-di�eren ing [14℄ or re onstru ting [17℄ to obtain theapproximation to (vu′)′ [14℄ or to the appropriate �uxes [17℄. It turns out that we may obtain O(∆x2s)a ura y (roughly twi e more a urate) on the sten il si,s,s := {i− s, · · · , i+ s} by simply interpolating v(x)and u(x) on the entire sten il si,s−1,s := {i − s + 1, · · · , i + s} used for the evaluation of the �ux, using allavailable information at the nodes, and then re onstru ting the approximating polynomial of v(x)u′(x) tode�ne the interfa e �uxes.To avoid loss of information, and hen e order-of-a ura y, we base the present approa h for evaluating the�ux at i+ 1

2 on the O(∆x2s)-a urate interpolating polynomials pI,s−1,s(x; xi, ∆x; v) and pI,s−1,s(x; xi, ∆x; u),so that the �nal approximation[(vu′)′i]gv,si,s,s

=[F(vu′),gv,si,s−1,s

]i+ 12− [F(vu′),gv,si−1,s−1,s

]i− 12

∆x= [(vu′)′](xi) + O(∆x2s) (4a)uses all information ontained in the sten il.2.4.1 Flux at i + 1

2 on the general sten il si,s−1,s := {i − M−, · · · , i + M+}It will prove useful, when working on biased dis retizations for near-boundary points, to develop the expres-sion of the �ux on the general sten il si,s−1,s := {i − M−, · · · , i + M+} (1b). We de�neˇ[v]gv,si,M−,M+

(x) := pI,M−,M+(x; xi, ∆x; v)

(1e)=

M+∑

p=−M−

αI,M−,M+,p

(x − xi

∆x

)

vi+p(1f)= v(x) + O(∆xM+1)(5a)

ˇ[u′]gv,si,M−,M+

(x) :=d

dxpI,M−,M+

(x; xi, ∆x; u)(1g)=

1

∆x

M+∑

q=−M−

α′I,M−,M+,q

(x − xi

∆x

)

ui+q(1h)= u′(x) + O(∆xM )(5b)using Lagrange interpolation on the entire sten il si,M−,M+

:= {i − M−, · · · , i + M+}, and approximatev(x)u′(x) by the produ t of (5a, 5b)

ˇ[vu′]gv,si,M−,M+

(x) :=1

∆x

(s∑

p=−M−

αI,M−,M+,p

(x − xi

∆x

)

vi+p

)(s∑

q=−M−

α′I,M−,M+,q

(x − xi

∆x

)

ui+q

)(5a, 5b)= v(x)u′(x) + O(∆xM ) (5 )The polynomial (5 ) is then re onstru ted on si,M−,M+

, to de�ne the numeri al �ux[F(vu′),gv,si,M−,M+

]i+ 12

:=pR1,M−,M+(x; xi+ 1

2, ∆x; ˇ[vu′]gv,si,M−,M+

)(1g)=

M+∑

ℓ=−M−

αR1,M−,M+,m

(x − xi

∆x

)ˇ[vu′]gv,si,M−,M+

(xi + m∆x)(1h)= [R(1;∆x)(vu′)](x) + O(∆x2s−1) (5d)

8

so that we have �nally[F(vu′),gv,si,M−,M+

]i+ 12

(5 , 5d)=

1

∆x

M+∑

p=−M−

vi+p

M+∑

q=−M−

(

αR1,M−,M+,p(12 ) α′

I,M−,M+,q(p))

︸ ︷︷ ︸

=: a(vu′,gv,M−,M+)pq

ui+q

(5e)where we used the well known fa t that the fundamental fun tions of Lagrange interpolation αI,M−,M+,p(ξ)are = 0 at all integer nodes on the sten il (∀p ∈ {i−M−, · · · , i+M+}\{p}), ex ept at the node ξ = p whereαI,M−,M+,p(p) = 1 [23℄.2.4.2 Flux at internal pointsAt points with su� ient distan e from the boundaries so that the points in the sten il si,s−1,s := {i − s +1, · · · , i + s} be de�ned on the omputational grid, ie for points i ∈ {s, · · · , Ni − s}, the �ux

[F(vu′),gv,si,s−1,s]i+ 1

2

(5e):=

1

∆x

s∑

p=−s+1

(

vi+p

s∑

q=−s+1

(a(vu′,gv,s−1,s)pq

ui+q

)

) (6a)a(vu′,gv,s−1,s)pq

(5e):=αR1,s−1,s,p(

12 ) α′

I,s−1,s,q(p) ∈ Q

{p ∈ {−s + 1, · · · , s}q ∈ {−s + 1, · · · , s}

(6b)is onstru ted at the interfa e i + 12 . Using the analyti al expressions for αR1,s−1,s−1,m(ξ) [21, (45g), p.287℄ and for αI,s−1,s,p(ξ) [21, (45h), p. 287℄ it is straightforward to ompute by (6b) the (2s)2 rational onstants a(vu′,gv,s−1,s)pq

(p, q ∈ {−s + 1, · · · , s}) appearing in the de�nition (6a) of the numeri al �ux[F(vu′),gv,si,s−1,s

]i+ 12, whi h were tabulated for s ∈ {1, · · · , 6} (Tab. 1).2.5 RemarkIt is easy to verify that all of the 3 methods (�2.2, �2.3, and �2.4) yield the same basi O(∆x2) approximationon the sten il si,1,1 := {i − 1, i, i + 1} (s = 1 =⇒ ⌈ s

2⌉ = ⌈ 12⌉ = 1 = s)(2f, 3f, 6a) s=1

=⇒ [F(vu′),gv,si,0,1]i+ 1

2= [F(vu′),zrnp,si,0,1

]i+ 12

= [F(vu′),sz ,si,0,1]i+ 1

2=

vi+1 + vi

2

ui+1 − ui

∆x(7)whi h is widely used in many solvers [8℄.2.6 Comparison on si,3,3A �rst omputational veri� ation of the 3 s hemes was performed (Fig. 2), on the sten il si,3,3 := {i −

3, · · · , i + 3}, for 2 sets of fun tions, v(x) and u(x), studied in Shen et al. [17℄, sz 1 (�2.8.1) and sz 2(�2.8.2). Dis retization was performed on a uniform grid of Ni points (Nc = Ni−1 intervals), using Nph = 3phantom nodes to ompute (vu′)′ at the boundary points, without reverting to biased sten ils. The L∞norm of the error for these test- ases is de�ned by the uns aled errorEL∞(Nc) := max

i∈{1,··· ,Nc+1}|(vu′)′num − (vu′)′exact| (8a)and the asso iated rate-of- onvergen e, between 2 onse utive levels of grid re�nement, Nc and Ncc

, byr nvrgL∞

(Nc) := −log10 EL∞(Nc) − log10 EL∞(Ncc

)

log10

(Nc

Ncc

) (8b)All of the 3 s hemes a hieve their theoreti al order-of-a ura y (Fig. 2), O(∆x4) for Zingg et al. [14℄and Shen et al. [17℄, and O(∆x6) for the present method, already at the oarsest grid of Ni = 21 points9

Table 1: Rational onstants a(vu′,gv,s−1,s)pq:= αR1,s−1,s,p(

12 ) α′

I,s−1,s,q(p) ∈ Q (6b) appearing in the expres-sion of the numeri al �ux [F(vu′),gv,si,s−1,s]i+ 1

2= ∆x−1

∑sp=−s+1 vi+p

∑sq=−s+1 a(vu′,gv,s−1,s)pq

ui+q (6a) ofthe present s heme for the omputation of (vu′)′i (4a) (s ∈ {1, · · · , 6}).q = −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 +6

s p

6 −5 83711153679680

−1504

51008

−5504

5336

−160

172

−5588

51344

−54536

15040

−160984

−4 −67304920

−32568769854400

675544

−673696

672772

−672640

673300

−675544

6712936

−6744352

67249480

−673049200

−3 −107762300

10734650

35834317463600

−1072310

1072310

−1072475

1073300

−1075775

10713860

−10748510

107277200

−1073430350

−2 −4433430350

443207900

−44320790

−3291495821200

4433465

−4434950

4437425

−44313860

44334650

−443124740

443727650

−4439147600

−1 −565336590400

56532494800

−5653332640

565355440

6048715821200

−565319800

565339600

−565383160

5653221760

−5653831600

56534989600

−565364033200

0 −1810764033200

181074656960

−18107698544

18107155232

−1810738808

−18107166320

1810727720

−1810777616

18107232848

−18107931392

181075821200

−1810776839840

+1 1810776839840

−181075821200

18107931392

−18107232848

1810777616

−1810727720

18107166320

1810738808

−18107155232

18107698544

−181074656960

1810764033200

+2 565364033200

−56534989600

5653831600

−5653221760

565383160

−565339600

565319800

−6048715821200

−565355440

5653332640

−56532494800

565336590400

+3 4439147600

−443727650

443124740

−44334650

44313860

−4437425

4434950

−4433465

3291495821200

44320790

−443207900

4433430350

+4 1073430350

−107277200

10748510

−10713860

1075775

−1073300

1072475

−1072310

1072310

−35834317463600

−10734650

107762300

+5 673049200

−67249480

6744352

−6712936

675544

−673300

672640

−672772

673696

−675544

32568769854400

67304920

+6 160984

−15040

54536

−51344

5588

−172

160

−5336

5504

−51008

1504

−83711153679680

5 −4 −71293175200

1140

−170

145

−140

150

−190

1245

−11120

111340

−3 2322680

11063705600

−23630

23540

−23540

23720

−231350

233780

−2317640

23181440

−2 127181440

−12710080

−215939200

1271080

−1271440

1272160

−1274320

12712600

−12760480

127635040

−1 473635040

−47347040

4735880

17501151200

−4731680

4733360

−4737560

47323520

−473117600

4731270080

0 16271270080

−1627105840

162717640

−16273780

−162712600

16272520

−16277560

162726460

−1627141120

16271587600

+1 −16271587600

1627141120

−162726460

16277560

−16272520

162712600

16273780

−162717640

1627105840

−16271270080

+2 −4731270080

473117600

−47323520

4737560

−4733360

4731680

−17501151200

−4735880

47347040

−473635040

+3 −127635040

12760480

−12712600

1274320

−1272160

1271440

−1271080

215939200

12710080

−127181440

+4 −23181440

2317640

−233780

231350

−23720

23540

−23540

23630

−11063705600

−2322680

+5 −111340

11120

−1245

190

−150

140

−145

170

−1140

71293175200

4 −3 36339200

−140

380

−124

132

−3200

1240

−11960

−2 −295880

−84116800

29280

−29336

29504

−291120

294200

−2935280

−1 −13935280

1392520

653350400

−139504

1391008

−1392520

13910080

−13988200

0 −53388200

5338400

−5331400

−5333360

533840

−5332800

53312600

−533117600

+1 533117600

−53312600

5332800

−533840

5333360

5331400

−5338400

53388200

+2 13988200

−13910080

1392520

−1391008

139504

−653350400

−1392520

13935280

+3 2935280

−294200

291120

−29504

29336

−29280

84116800

295880

+4 11960

−1240

3200

−132

124

−380

140

−36339200

3 −2 −1373600

112

−112

118

−148

1300

−1 275

1390

−415

215

−245

1150

0 371200

−37120

−37180

3760

−37240

371800

+1 −371800

37240

−3760

37180

37120

−371200

+2 −1150

245

−215

415

−1390

−275

+3 −1300

148

−118

112

−112

1373600

2 −1 1172

−14

18

−136

0 −736

−724

712

−772

+1 772

−712

724

736

+2 136

−18

14

−1172

2 0 −12

+12

+1 −12

+12(Fig. 2). More interestingly, the present s heme systemati ally has a lower error, even on the oarsest grid.Improvement by the present method is higher for the sz 2 (�2.8.2) test- ase (Fig. 2), for whi h u(x) is awavy fun tion.To obtain an O(∆x2s) a ura y, the sten il width in reases linearly with s, and the omputational10

omplexity in reases quadrati ally with s (Tab. 2). If the 3 methods are ompared, for the same order-of-a ura y (Tab. 2), it is seen that, for large s, the present method obtains the same a ura y as the previouslypublished approa hes [14, 17℄, on an asymptoti ally twi e more ompa t sten il, and with roughly half the omputational omplexity.Table 2: Number of numeri al onstants ( ooe� ients) appearing in the s hemes of Zingg et al. [14℄ (2f), ofShen et al. [17℄ (3f), and in the present method (6), omputational omplexity (number of both additionsand multipli ations), and dis retization sten il-width, at interior points.s heme order number of oe� ients omplexity sten il-widthznrp [14℄ O(∆x2s) 6s − 1 16s2 − 1 4s − 2sz [17℄ O(∆x2s) 4s2 − 2s + 1 24s2 − 18s + 3 4s − 2present O(∆x2s) 4s2 8s2 + 2s− 1 2s2.7 Near-boundary pointsWhen i ≤ s − 1 or i ≥ Ni−s there are not enough points in the omputational domain to ompute the �ux[F(vu′),gv,si,s−1,s

]i+ 12(6) and one must use biased sten ils. The near-boundary interfa es, in the neighbour-hood of the boundary i = 1, requiring biased sten ils, are

{0 + 12

︸ ︷︷ ︸

1− 12

, 1 + 12 , · · · , (s − 1) + 1

2} = {s − ℓb + 12 ; ℓb = s, · · · , 1} (9a)For these interfa es (9a), we use the sten ils

0 + 12 = 1 − 1

2 : { 1︸︷︷︸

=0−(−1)

, · · · , 2s + 1︸ ︷︷ ︸

=0+(2s+1)

} =s0,−1,2s+1 (9b)1 + 1

2 : { 1︸︷︷︸

=1−0

, · · · , 2s + 1︸ ︷︷ ︸

=1+(2s)

} =s1,0,2s (9 )2 + 1

2 : { 1︸︷︷︸

=2−1

, · · · , 2s + 1︸ ︷︷ ︸

=2+(2s−1)

} =s2,1,2s−1 (9d)...s − 1 + 1

2 : { 1︸︷︷︸

=(s−1)−(s−2)

, · · · , 2s + 1︸ ︷︷ ︸

=(s−1)+(s+2)

}=ss−1,s−2,s+2 (9e)ie the same set of points, {1, · · · , 2s+1}, to ompute the �ux. The sten il {1, · · · , 2s+1} in ludes one more ell in the sten il with ompared to internal points (�2.4.2). Symmetri ally, the set {Ni−2s, · · · , Ni} is usedfor the interfa es{Ni + 1

2 , Ni − 1 +1

2, Ni − 2 + 1

2 , · · · , Ni − (s − 1) + 12} = {Ni − (s − ℓb) + 1

2 ; ℓb = s, · · · , 1} (9f)near the boundary i = Ni. For both boundaries (9a, 9f), ℓb = s orresponds to ompletely extrapolatedinterfa es (outside of the omputational domain).2.8 Computational examplesTo demonstrate that the proposed s heme a hieves its theoreti al order-of-a ura y, we �rst examine the nu-meri al omputation of (v(x)u′(x))′ for several test-fun tions (�2.8.1��2.8.2). Then, we apply the proposeds heme to the omputation of laminar Couette �ow of air (�2.8.3).11

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(

v(x)u′(x))′

discretization-error

Shen et al. (2009) test cases

s = 3 ; Nph = s

szc1

v(x) := 1100e−2x

u(x) :=1 − e−20x

1 − e−20

u✛

v ✲

(vu′)′✛

u;(v

u′ )′

✻

v

✻

x ✲

szc2

v(x) := 110e2x

u(x) := sin(10x)

u✛(vu′)′ ✲

v

✻

u;v

✻

(vu′ )′

✻

x ✲

log10 eL∞

∆x−6

∆x−4

Zingg et al.; s = 3 Shen et al.; s = 3 present; s = 3

Nc= ∆x

−1 ✲

log10 eL∞

∆x−6

∆x−4

Zingg et al.; s = 3 Shen et al.; s = 3 present; s = 3

Nc= ∆x

−1 ✲Figure 2: L∞-norm error eL∞ (8a), as a fun tion of the number of grid- ells Nc = Nj − 1, of thenumeri al approximation of (vu′)′ on the sten il si,3,3 := {i − 3, · · · , i + 3} by the present method(4a, 6), for the test- ases (10, 11) of Shen et al. [17℄, using progressively re�ned omputational grids(Ni = 21, 41, 81, 161, 321, 641, 1281, 2561, 5121, 10241, 20481, 40961, 81921 points) with Nph = 3 phantomnodes, and omparison with the previous approa hes of Zingg et al. [14℄ (2a, 2f) and of Shen et al. [17℄ (3a,3f).12

2.8.1 v(x) := 1100e−2x, u(x) := (1 − e−20)−1(1 − e−20x); x ∈ [0, 1]This test- ase studied by Shen et al. [17℄

u(x) := sin 10x

v(x) := 110e2x

}

=⇒ (v(x)u′(x))′ =−22e−22x

5(1 − e−20)(10)The proposed s heme either Nph = s phantom nodes to the mesh to avaoid the use of biased sten ils or

Nph = 1 phantom node, a hieves the theoreti al order-of-a ura y for s ∈ {1, · · · , 9} (Fig. 3).2.8.2 v(x) := 110e2x, u(x) := sin 10x; x ∈ [0, 1]This test- ase studied by Shen et al. [17℄

u(x) := sin 10x

v(x) := 110e2x

}

=⇒ (v(x)u′(x))′ = −2e2x(5 sin 10x − cos 10x) (11)Again the s heme a hieves its theoreti al order-of-a ura y for s ∈ {1, · · · , 9} (Fig. 4).2.8.3 Laminar ompressible Couette �owFor laminar ompressible fully developed Couette �ow [27, pp. 190�192℄ between a �xed adiabati wall aty = 0 and a moving isothermal wall at y = δ, the ompressible Navier-Stokes equations [27, pp. 190�192℄simplify (under the assumption of fully developed unidire tional �ow parallel to the 2 plates, ~V = u(y)~ex,whi h satis�es automati ally the steady ontinuity equation) to

p =const (12a)0 =

d

dy

(

µ(T )du

∂y

) (12b)0 =µ(T )

(du

dy

)2

+d

dy

(

λ(T )dT

dy

) (12 )where y is the oordinate normal to the plates and to the �ow, u is the x-wise velo ity- omponent, T is thetemperature, µ(T ) is the dynami oe� ient of vis osity depending on T [28℄, λ(T ) is the oe� ient of heat ondu tivity depending on T [28℄. We onsider �ow of air, withµ(T )

[28℄= µ0

[T

Tµ0

] 32 Sµ + Tµ0

Sµ + T; µ0 = µ(Tµ0

) = 17.11 × 10−6 Pa s ; Tµ0= 273.15 K ; Sµ = 110.4 K(12d)

λ(T )[28℄= λ0

µ(T )

µ0[1 + Aλ(T − Tµ0

)] ; λ0 = λ(Tµ0) = 0.0242 W m−1 K−1 ; Aλ = 0.00023 K−1; (12e)Under the assumption that µ (12d) and λ (12e) are fun tions of T only, (12) is a system of 2 odes (12b,12 ) for the 2 variables u and T , with boundary- onditions

u(0) =0 (13a)dT

dy(0) =0 (13b)

u(δ) =ue > 0 (13 )T (δ) =Te > 0 (13d)The derivatives dy(µ(T )dyu) in (12b) and dy(λ(T )dyT ) in (12 ) are dis retized using the present methodon the sten il si,s,s := {i − s, · · · , i + s} for interior points (�2.4), whi h provides O(∆x2s) a ura y, andappropriately biased sten ils at near-boundary points (�2.7), redu ing the theoreti al order-of-a ura y to13

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rcnvrgL∞

s = 9

s = 6

s = 3

s = 1

rcnvrgL∞

s = 7

s = 4

rcnvrgL∞

s = 8

s = 5

s = 2

log10 eL∞

s = 1

s = 3

s = 6s = 9

log10 eL∞

s = 7s = 4

log10 eL∞

s = 8

s = 5

s = 2

(

v(x)u′(x))′

discretization-error: szc1

s = 9 (2s = 18)

s = 8 (2s = 16)

s = 7 (2s = 14)

s = 6 (2s = 12)

s = 5 (2s = 10)

s = 4 (2s = 8)

s = 3 (2s = 6)

s = 2 (2s = 4)

s = 1 (2s = 2)

Nph = s

Nph = 1 x ∈ [0, 1]

u(x) :=1 − e−20x

1 − e−20

0 0.5 1

v(x) := 1100e−2x

0 0.5 1

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

x ✲

0 1

x ✲

0 1

Figure 3: L∞-norm error eL∞ (8a) and rate-of- onvergen e r nvrgL∞(8b), as a fun tion of the number ofgrid- ells Nc = Nj − 1, of the numeri al approximation of (vu′)′ by the present method (6, 9) with s ∈

{1, · · · , 9}, for the sz 1 test- ase (10) of Shen et al. [17℄, using Nph ∈ {1, s} phantom nodes on progressivelyre�ned omputational grids (Ni = 21, 41, 81, 161, 321, 641, 1281, 2561, 5121, 10241, 20481, 40961, 81921 pointsdepending on the value of the order-parameter s).14

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rcnvrgL∞

s = 9

s = 6

s = 3

s = 1

rcnvrgL∞

s = 7

s = 4

rcnvrgL∞

s = 8

s = 5

s = 2

log10 eL∞

s = 1

s = 3

s = 6

s = 9

log10 eL∞

s = 7

s = 4

log10 eL∞

s = 8

s = 5

s = 2

(

v(x)u′(x))′

discretization-error: szc2

s = 9 (2s = 18)

s = 8 (2s = 16)

s = 7 (2s = 14)

s = 6 (2s = 12)

s = 5 (2s = 10)

s = 4 (2s = 8)

s = 3 (2s = 6)

s = 2 (2s = 4)

s = 1 (2s = 2)

Nph = s

Nph = 1 x ∈ [0, 1]

u(x) := sin(10x)

0 0.5 1

v(x) := 110e2x

0 0.5 1

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

Nc = ∆x−1 ✲

x ✲

0 1

x ✲

0 1

Figure 4: L∞-norm error eL∞ (8a) and rate-of- onvergen e r nvrgL∞(8b), as a fun tion of the number ofgrid- ells Nc = Nj − 1, of the numeri al approximation of (vu′)′ by the present method (6, 9) with s ∈

{1, · · · , 9}, for the sz 2 test- ase (11) of Shen et al. [17℄, using Nph ∈ {1, s} phantom nodes on progressivelyre�ned omputational grids (Ni = 21, 41, 81, 161, 321, 641, 1281, 2561, 5121, 10241, 20481, 40961, 81921 pointsdepending on the value of s).15

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rcnvrgL∞

s = 9

s = 8

s = 7

s = 6

s = 5

s = 4

s = 3

s = 2

s = 1

eL∞

(s = 9) ∆x−18

∆x−16 (s = 8)

∆x−14 (s = 7)

∆x−12 (s = 6)

∆x−10 (s = 5)

∆x−8 (s = 4)

∆x−6 (s = 3)

∆x−4 (s = 2)

∆x−2 (s = 1)

Couette-flow numerical solution error

s = 9 (2s = 18)

s = 8 (2s = 16)

s = 7 (2s = 14)

s = 6 (2s = 12)

s = 5 (2s = 10)

s = 4 (2s = 8)

s = 3 (2s = 6)

s = 2 (2s = 4)

s = 1 (2s = 2) 0

0.2

0.4

0.6

0.8

1

0 200 400 600 800

0 500 1000 1500

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0.2

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0.02 0.03 0.04 0.05 0.06

Nc = ∆x−1 ✲ Nc = ∆x−1 ✲

µ (Pa s−1) ✲

λ (W m−1 K−1) ✲

yδ

✻

λ✛

✻

µ ✲

❄

T (K) ✲

u (m s−1) ✲

yδ

✻

Te = 200 K

❅❅❘

ue = 1220 m s−1✻

T ′(0) = 0

�✠

u ✲

✻

T ✲

❄

Figure 5: L∞-norm error eL∞ (15) and rate-of- onvergen e r nvrgL∞(8b), as a fun tion of the numberof grid- ells Nc = Nj − 1, for the di�eren e from the analyti al solution of numeri al omputations of ompressible laminar Couette �ow, using the present method (14) for s = 1, · · · , 9, on progressively re�ned omputational grids (Nx = 21, 41, 81, 161, 321, 641, 1281, 2561, 5121 points).

O(∆x2s−1). The derivative dyu appearing in the sour e-term µ(T )(dyu)2, representing heating due to vis ousfri tion in (12 ), is dis retized using standard entered O(∆y2s) �nite-di�eren es [29℄ on the sten il si,s,s :={i−s, · · · , i+s} for interior points, and using O(∆y2s) biased �nite-di�eren es at the near-boundary-points.The adiabati -wall boundary- ondition (13a) is also dis retized using biased O(∆y2s) �nite-di�eren es. Theglobal a ura y is therefore O(∆y2s−1) for s ≥ 2, and O(∆y2) for s = 1 for whi h all points, ex ept at the16

walls where b s are applied instead, are internal points.A homogeneous grid of Nj points (j = 1 is the lower �xed adiabati wall, j = Nj is the upper isothermalwall, (Nj − 1)∆y = δ). The �ow is initialized byu0,j =

yj

δue (14a)

T0,j =Te (14b)and the nonlinear algebrai system of equations[F(µdyu),gv,si,s−1,s

]i+ 12− [F(µdyu),gv,si,s−1,s

]i− 12

∆y=0 (14 )

µ(Tn,j)

[du

dy

]2

j

+[F(λdyT ),gv,si,s−1,s

]i+ 12− [F(λdyT ),gv,si,s−1,s

]i− 12

∆y=0 (14d)are solved using quasi-Newton iteration based on an O(∆x2) approximate Ja obian, orresponding to an

O(∆x2) dis retization on the restri ted sten il si,1,1 [30℄.Numeri al results were obtained for s ∈ {1, · · · , 9}, on progressively �ner grids (Fig. 5), and their a ura ywas assessed by omparison with the semi-analyti al solution of (12, 13), whi h was determined with apre ision better than 30 signi� ant digits. Sin e the problem (12, 13) involves 2 variables (u and T ), we usedthe nondimensional error-normEL∞(Nc) := max

j∈{1,··· ,Nc+1}

(|unum − uexact|

ue,|Tnum − Texact|

Te

) (15)fun tion of the number of grid- ells Nc = Nj − 1. This norm (15) was then used in (8b) to ompute therate-of- onvergen e (Fig. 5).For all of the studied values of the order-parameter s ∈ {1, · · · , 9}, the s heme rea hes its theoreti al order-of-a ura y (Fig. 5), the s = 9 s heme rea hing numeri al noise after Nc = 2560 ells, where r nvrgL∞(Nc =

2560, s = 9) ≅ 17. It appears indeed that, for this parti ular problem, and for the range of grids used, themethod is super onvergent, rea hing and ex eeding O(∆x2s) order-of-a ura y (Fig. 5), whereas O(∆x2s−1)order-of-a ura y is expe ted, be ause of the biased sten ils used at the near-boundary-points.3 Multidimensional extension3.1 Vis ous stresses in the Navier-Stokes equationsFor a standard Newtonian onstitutive relation, the vis ous stress tensor τ is of the formτ = 2µS +

(µb − 2

3µ)tr(S)I3 (16a)where S := 1

2

(

grad~V + (grad~V )t) is the rate-of-strain tensor, µ = µ(ρ, T ) is the dynami vis osity, µb =

µb(ρ, T ) is the bulk vis osity, I3 is the identity tensor in the Eu lidian 3-D spa e E3, ρ is the �uid density andT the stati temperature. As a onsequen e, the vis ous for e per unit volume divτ reads, by straightforward omputation,

divτ =

[

∂

∂x

((µb + 4

3µ) ∂u

∂x

)

+∂

∂y

(

µ∂u

∂y

)

+∂

∂z

(

µ∂u

∂z

)

+∂

∂x

((µb − 2

3µ)(

∂v

∂y+

∂w

∂z

))

+∂

∂y

(

µ∂v

∂x

)

+∂

∂z

(

µ∂w

∂x

)

︸ ︷︷ ︸ ross-derivatives ]

~ex

17

+

[

∂

∂x

(

µ∂v

∂x

)

+∂

∂y

((µb + 4

3µ) ∂v

∂y

)

+∂

∂z

(

µ∂v

∂z

)

+∂

∂x

(

µ∂u

∂y

)

+∂

∂y

((µb − 2

3µ)(

∂w

∂z+

∂u

∂x

))

+∂

∂z

(

µ∂w

∂y

)

︸ ︷︷ ︸ ross-derivatives ]

~ey

+

[

∂

∂x

(

µ∂w

∂x

)

+∂

∂y

(

µ∂w

∂y

)

+∂

∂z

((µb + 4

3µ) ∂w

∂z

)

+∂

∂x

(

µ∂u

∂z

)

+∂

∂y

(

µ∂v

∂z

)

+∂

∂z

((µb − 2

3µ)(

∂u

∂x+

∂v

∂y

))

︸ ︷︷ ︸ ross-derivatives ]

~ez (16b)and ontains both terms whi h an be dis retized using the previously developed s heme for (v(x)u(x))′and ross-derivatives, eg ∂x(µ∂yu), whi h require a di�erent method. It turns out that the dis retizationof ross-derivatives is simpler than the method (� 2.4) required to dis retize the linewise derivatives x, eg∂x(µ∂xu).For a �uid with a Newtonian onstitutive relation (16a) and following a linear Fourier heat-�ux law

~q = −λ(ρ, T )gradT (16 )where ~q is the heat-�ux ve tor, the power per unit volume due to fri tion and heat ondu tion reads, bystraightforward di�erentiationdiv(

~V .τ − ~q)

=∂

∂x

[(µb + 4

3µ)u

∂u

∂x+ µv

∂v

∂x+ µw

∂w

∂x− λ

∂T

∂x

]

+∂

∂x

[(µb − 2

3µ)u

(∂v

∂y+

∂w

∂z

)

+ µv∂u

∂y+ µw

∂u

∂z

]

︸ ︷︷ ︸ ross-derivatives+

∂

∂y

[

µu∂u

∂y+(µb + 4

3µ)v∂v

∂y+ µw

∂w

∂y− λ

∂T

∂y

]

+∂

∂y

[

µu∂v

∂x+(µb − 2

3µ)v

(∂u

∂x+

∂w

∂z

)

+ µw∂v

∂z

]

︸ ︷︷ ︸ ross-derivatives+

∂

∂z

[

µu∂u

∂z+ µv

∂v

∂z+(µb + 4

3µ)w

∂w

∂z− λ

∂T

∂z

]

+∂

∂z

[

µu∂w

∂x+ µv

∂w

∂y+(µb − 2

3µ)w

(∂u

∂x+

∂v

∂y

)]

︸ ︷︷ ︸ ross-derivatives (16d)3.2 Numeri al �uxes for ross-derivatives (∂x(v∂yu))The approa h followed for the dis retization of ross-derivatives is a generalization to higher-order of a u-rar y of the method used by Zingg et al. [14℄ and Shen et al. [17℄. First, the derivative ∂yu is approximatedat the points of the omputational grid6 using standard entered O(∆x2s) �nite-di�eren ing (equivalentlydi�erentiation of the Lagrange interpolating polynomial on the sten il[

[ ˇ∂yu]s i,0,0j,s,sk,0,0

!

]

ijk

(1g):=

1

∆y

+s∑

ℓ=−s

α′I,s,s,ℓ(0)ui,j+ℓ,k

(1h)=

∂u

∂y

∣∣∣∣ijk

+ O(∆y2s) (17a)6the dire tion i, j, k of the homogeneous Cartesian grid used, are aligned with the x, y, z dire tions of the Cartesian systemof oordinates. 18

on the sten il s(i,0,0j,s,sk,0,0

)

:={(i, j − s, k), · · · , (i, j + s, k)

} (17b)where, as usual, the O(∆y2s−1) a ura y of the derivative of the Lagrange interpolating polynomial be omesO(∆y2s) at the point ijk, be ause the sten il is entered with respe t to this point [26, (7.6), p. 297℄. Then,the Lagrange interpolating polynomial

[ˇv∂yu]s i,s−1,s

j,s ,sk,0 ,0

!

,j,k

(xi + ξ∆x)(1e):=

+s∑

ℓ=−s+1

αI,s−1,s,ℓ(ξ) vi+ℓ,j,k

[

[ ˇ∂yu]s i+ℓ,0,0j ,s,sk ,0,0

!

]

i+ℓ,j,k(1f, 16a)= [v∂yu]jk (xi + ξ∆x) + O(∆x2s−1, ∆y2s) (17 )is re onstru ted to obtain the required �ux

F

(v∂yu),s i,s−1,sj,s ,sk,0 ,0

!

i+ 12

,j,k

:=pR1,s−1,s

xi + 1

2∆x; xi, ∆x;[

ˇv∂yu]s i,s−1,s

j,s ,sk,0 ,0

!

,j,k

(1i, ??)

=+s∑

ℓ=−s+1

αR1,s−1,s,ℓ(12 ) vi+ℓ,j,k

[

[ ˇ∂yu]s i+ℓ,0,0j ,s,sk ,0,0

!

]

i+ℓ,j,k(1f, 16a)=

[R(1;∆x)(v∂yu)jk

](xi+1/2) + O(∆x2s−1, ∆y2s) (17d)where by [24, (8b), p. 2767℄ O(∆x2s) a ura y is re overed, in (17d) for the parti ular value ξ = 1

2 . Finally

F

(v∂yu),s i,s−1,sj,s ,sk,0 ,0

!

i+ 12,j,k

−

F

(v∂yu),s i−1,s−1,sj ,s ,sk ,0 ,0

!

i− 12

,j,k

∆x=

∂

∂x

(

v∂u

∂y

)∣∣∣∣ijk

+ O(∆x2s, ∆y2s) (17e)Similar relations apply for the di�erent ross-derivatives appearing in (16b, 16d). At near-boundary points,we use biased sten ils, in analogy with (�2.7).3.3 A ura y testTo he k that the appli ation of the proposed s heme for the line-derivatives (�2.4), oupled with the usualapproa h for the ross-derivatives (�3.2), returns the theoriti al order-of-a ura y numeri al results for divτ(16b) and div(

~V .τ − ~q) (16d) orresponding to the hypotheti al �eld

u(x, y, z) := sin 4πx sin 2πy sin 3πz

v(x, y, z) := sin 5πx sin 4πy sin 3πz

w(x, y, z) := sin 5πx sin 3πy sin 7πz

T (x, y, z) := sin 6πx sin 8πy sin 9πz + 2

µ(x, y, z) :=exyz

λ(x, y, z) :=exyz

µb :=0 (18)are ompared (Fig. 6) with the analyti al solution obtained by straightforward di�eren iation in (16b, 16d).19

-10

-5

0

5

10

10 100 1000 10000

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

10 100 1000 10000

-10

-5

0

5

10

10 100 1000 10000

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

10 100 1000 10000

-10

-5

0

5

10

10 100 1000 10000

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

10 100 1000 10000

rcnvrgL∞

s = 9

s = 6

s = 3

s = 1

rcnvrgL∞

s = 7

s = 4

rcnvrgL∞

s = 8

s = 5

s = 2

log10 eL∞

s = 1

s = 3

s = 6s = 9

log10 eL∞

s = 7s = 4

log10 eL∞

s = 8

s = 5

s = 2

discretization-error for divτ and div(

~V .τ − ~q) s = 9 (2s = 18)

s = 8 (2s = 16)

s = 7 (2s = 14)

s = 6 (2s = 12)

s = 5 (2s = 10)

s = 4 (2s = 8)

s = 3 (2s = 6)

s = 2 (2s = 4)

s = 1 (2s = 2)

Nph = s

Nph = 1

Nph = 0

12Nc = ∆x−1 ✲

12Nc = ∆x−1 ✲

12Nc = ∆x−1 ✲

12Nc = ∆x−1 ✲

12Nc = ∆x−1 ✲

12Nc = ∆x−1 ✲Figure 6: L∞-norm error eL∞ (19) and rate-of- onvergen e r nvrgL∞

(8b), as a fun tion of the numberof grid- ells Nc = Ni − 1 = Nj − 1 = Nk − 1, of the numeri al approximation of divτ and divτ by thepresent method (6, 9) with s ∈ {1, · · · , 9}, for the test- ase (18), using Nph ∈ {0, 1, s} phantom nodes onprogressively re�ned omputational grids (Ni = 21, 41, 81, 161, 321 points depending on the value of theorder-parameter s). 20

The �elds (18) and the oordinates (x, y, z) are assumed to be nondimensional quantities, so that thenorm (19) is used.EL∞(Nc) := max

i,j,k

{∣∣∣[(divτ )x]num − [(divτ )x]exact

∣∣∣,

∣∣∣[(divτ )y]num − [(divτ )y]exact

∣∣∣,

∣∣∣[(divτ )z]num − [(divτ )z ]exact

∣∣∣,

∣∣∣[div(~V · τ ) − ~q]num − [div(~V · τ − ~q)]exact

∣∣∣

}

; i, j, k ∈ {1, · · · , Nc + 1} (19)4 Con lusionsThe present work de�nes numeri al �uxes for very-high-order �nite-volume onservative dis retization of(µu′), appli able to the vis ous terms of the Navier-Stokes equations. Future work in ludes the developmentof weno dis retizations of these terms for �ows with dis ontinuities.The present s heme is almost twi e more a urate on a given sten il si,s,s than previous approa hes [14, 17℄based on �nite di�eren ing (equivalently yields the same a ura y as previous approa hes on almost twi emore ompa t, for large s, sten ils). It is also on eptually simpler, in that it simply uses polynomialinterpolation and re onstru tion on the entire sten il in lieu of 2 levels of �nite di�eren ing on substen ils.The extension of the method to unstru tured �nite-volume meshes is straightforward, by using edge-basedre onstru tion.A knowledgmentsComputer resour es were made available by idris� nrs (www.idris.fr). The authors are listed alphabeti- ally.All the omputer programs developed and used in the present work are open sour e and an be foundat http://aerodynami s.sour eforge.net. The pa kage in ludes all the re onstru tion routines (in f90language), and their appli ation to the various test- ases.Referen es[1℄ N. A. Adams and K. Shari�. A high-resolution hybrid ompa t-eno s heme for sho k/turbulen eintera tion problems. J. Comp. Phys., 127:27�51, 1996.[2℄ J. Sesterhenn. A hara teristi -type formulation of the Navier-Stokes equations for high-order upwinds hemes. Comp. Fluids, 30:37�67, 2001.[3℄ N. D. Sandham, Q. Li, and H. C. Yee. Entropy splitting for high-order numeri al simulation of om-pressible turbulen e. J. Comp. Phys., 178:307�322, 2002.[4℄ S. Pirozzoli. Conservative hybrid ompa t-weno s hemes for sho k-turbulen e intera tion. J. Comp.Phys., 178:81�117, 2002.[5℄ D. Ponziani, S. Pirozzoli, and F. Grasso. Development of optimized weno s hemes for multis ale ompressible �ows. Int. J. Num. Meth. Fluids, 42:953�977, 2003.[6℄ M. P. Martín, E. M. Taylor, M. Wu, and V. G. Weirs. A bandwidth-optimized weno s heme for e�e tivedire t numeri al simulation of ompressible turbulen e. J. Comp. Phys., 220:270�289, 2006.[7℄ E. M. Taylor, M. Wu, and M. P. Martín. Optimization of nonlinear error for weighted essentiallynon-os illatory methods in dire t numeri al simulations of ompressible turbulen e. J. Comp. Phys.,223:384�397, 2007.[8℄ G. A. Gerolymos, D. Séné hal, and I. Vallet. Performan e of very-high-order upwind s hemes for dnsof ompressible wall-turbulen e. Int. J. Num. Meth. Fluids, 63:769�810, July 2010.21

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