Evaluation of Fully Implicit Runge Kutta Schemes for...

J Sci ComputDOI 10.1007/s10915-017-0476-x

Evaluation of Fully Implicit Runge Kutta Schemes forUnsteady Flow Calculations

Antony Jameson1

Received: 14 March 2017 / Revised: 26 May 2017 / Accepted: 31 May 2017© Springer Science+Business Media, LLC 2017

Abstract This paper presents the formulation of a dual time stepping procedure to solve theequations of fully implicit Runge–Kutta schemes. In particular themethod is applied toGaussand Radau 2A schemes with either two or three stages. The schemes are tested for unsteadyflows over a pitching airfoil modeled by both the Euler and the unsteady Reynolds averagedNavier Stokes equations. It is concluded that the Radau 2A schemes are more robust and lesscomputationally expensive because they require a much smaller number of inner iterations.Moreover these schemes seem to be competitive with alternative implicit schemes.

Keywords Implicit Runge–Kutta · Euler equations · Navier–Stokes · URANS · Dualtime-stepping

1 Introduction

The last three decades have seen the emergence of a variety of very fast steady statesolvers for the Euler and Reynolds averaged Navier Stokes (RANS) equations. Whilethese may be based on time marching with explicit Runge–Kutta schemes, implicitalternating direction schemes, or symmetric Gauss Seidel schemes, convergence to asteady state is typically accelerated by the use of techniques such as variable local timestepping or multigrid. Accordingly time accuracy is completely abandoned in these calcula-tions.

While fast and accurate steady state solvers are sufficient for most of the computationalsimulations needed for the preliminary design of a fixed wing aircraft, time accurate sim-ulations are also needed for a variety of important applications, such as flutter analysis, orthe analysis of the flow past a helicopter in forward flight. Applications to real engineer-

Dedicated to Chi-Wang Shu on the occasion of his 60th birthday.

B Antony [email protected]; [email protected]

1 Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305-4035, USA

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ing flows often require meshes with extremely large variations in mesh size, of the orderof 10,000 or more. This is particularly the case for unsteady RANS simulations, for whichthe width of the mesh cells adjacent to the wall needs to be of the order of y+ = 1 indimensionless wall units. In this situation the permissible time step for the numerical stabil-ity of an explicit scheme may be orders of magnitude smaller than that needed to attain thedesired level of accuracy. Consequently explicit schemes become prohibitively expensive.On the other hand, implicit schemes which are A-stable or L-stable allow arbitrarily largetime steps, so one may choose the largest possible time step that will yield the desired accu-racy. The issue is then to find a way to solve the implicit equations which is not excessivelyexpensive.

In 1991 the author proposed the use of a dual time stepping technique in which theequations for each implicit time step are treated as a modified steady state problem whichis solved by marching in a pseudo time variable [1]. The dual time stepping approach hasbeen quite widely adopted, particularly in conjunction with the BDF2 scheme, which isboth A-stable and L-stable. Dahlquist has proved that A-stable linear multi-step schemesare at most second order accurate [2]. In the works of Butcher and other specialists in thenumerical solution of ordinary differential equations it has been shown that it is possibleto design A-stable and L-stable implicit Runge–Kutta schemes which yield higher orderaccuracy [3–5].

Recently there has been considerable interest in whether implicit Runge–Kutta schemescan achieve better accuracy for a given computational cost than the backward differenceformulas. Most of the studies to date have focused on diagonal implicit Runge–Kutta(DIRK) schemes, sometimes called semi-implicit schemes, in which the stages may besolved successively. A third order accurate single diagonal implicit Runge–Kutta (SDIRK)scheme has been successfully used for unsteady flow calculations by Persson [6]. In thesearch for higher order A- and L-stable methods it has been found beneficial to includean initial explicit stage to produce ESDIRK schemes. A scheme that has been widelyinvestigated is a six stage fourth order accurate scheme due to Kennedy and Carpen-ter [7] which was tested for unsteady Navier–Stokes simulations by Jothi Prasad et al.[8]. Recently Boom and Zingg used numerical optimization to identify a six stage fifthorder accurate ESDIRK scheme [9]. Fully implicit Runge–Kutta schemes have not beenwidely used in CFD because of the complexity of solving the coupled equations. Theycan, however, achieve high order accuracy with fewer stages than the SDIRK and ESDIRKschemes.

The purpose of the present study is to investigate the feasibility of using dual timestepping to solve the equations of fully implicit Runge–Kutta schemes, in which thestage equations are fully coupled. The investigation focuses in particular on the two andthree stage Gauss schemes [5] and the two and three stage Radau 2A schemes [5,10].The flow past a pitching airfoil is taken as a representative example of an engineer-ing problem for which an explicit scheme would be prohibitively expensive. Preliminaryresults were presented at the 22nd AIAA Computational Fluid Dynamics Conference inJune 2015 [11]. Section 2 reviews the concept of dual time stepping, while Section 3summarizes the implicit Runge Kutta schemes to be evaluated. Section 4 outlines the for-mulation of dual time stepping for implicit Runge Kutta schemes, and Section 5 describesa fast solution technique for Euler and RANS equations. Section 6 presents the results ofnumerical experiments for the pitching airfoil. Finally Section 7 presents the conclusionsof the investigation, namely that the Radau 2A schemes are superior to the alterna-tives.

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2 Review of Dual Time Stepping

The Euler equations for gas dynamics can be expressed in conservation law form as

∂w∂t

+ ∂

∂xifi (w) = 0.

Here the state vector is

w =

⎡⎢⎢⎢⎢⎣

ρ

ρu1ρu2ρu3ρE

⎤⎥⎥⎥⎥⎦

,

where ρ is the density, ui are the velocity components and E is the total energy. The fluxvectors are

fi = uiw + p

⎡⎢⎢⎢⎢⎣

0δi1δi2δi3ui

⎤⎥⎥⎥⎥⎦

where the pressure isp = (γ − 1)ρ(E − uiui ).

A semi-discrete finite volume scheme is obtained directly approximating the integral formon each computational cell

d

dt

∫cell

w dV +∫cell boundary

ni fi dS = 0,

where ni are the components of the unit normal to the cell boundary. This leads to a semi-discrete equation with the general form

Vdwdt

+ R(w) = 0, (1)

where w now denotes the average value of the state in the cell. V is the cell volume, orin the two dimensional case, the cell area. R(w) is the residual resulting from the spacediscretization. Applications to external aerodynamics typically use grids in which the cellarea or volume varies by many orders of magnitude between the body and the far field, andthis is a principal reason for using implicit schemes for time accurate simulations. Introducingsuperscripts n to denote the time level, the second order backward difference formula (BDF2)for time integration is

3V

2�twn+1 − 2V

�twn + V

2�twn−1 + R

(wn+1) = 0. (2)

In the dual time stepping scheme this equation is solved by marching the equation

dwdτ

+ R∗(w) = 0 (3)

to a steady state, where the modified residual is

R∗(w) = R(w) + 3V

2�tw − 2V

�twn − V

2�twn−1. (4)

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In solving equation 4 one is free to use every available acceleration technique for fast steadystate solutions without regard for time accuracy. In the author’s work it was shown thatmultigrid techniques can be very effective for this purpose.

3 Review of Implicit Runge Kutta Schemes

Applied to equation 1, the two stage Gauss scheme takes the form

ξξξ1 = wn − �t

V(a11R(ξξξ1) + a12R(ξξξ2)) ,


Vt (a21R(ξξξ1) + a22R(ξξξ2)) ,

wn+1 = wn − �t

2V(R(ξξξ1) + R(ξξξ2)) ,

(5)

where the matrix A of coefficients is

A =[

14

14 −

√36

14 +

√36

14

](6)

and the stage values correspond to Gauss integration points inside the time step with the

values(12 −

√36

)�t and

(12 +

√36

)�t .

The three stage Gauss scheme takes the form


V(a11R(ξξξ1) + a12R(ξξξ2) + a13R(ξξξ3)) ,


V(a21R(ξξξ1) + a22R(ξξξ2) + a23R(ξξξ3) ,


V(a31R(ξξξ1) + a32R(ξξξ2) + a33R(ξξξ3) ,

wn+1 = wn − �t

18V(5R(ξξξ1) + 8R(ξξξ2) + 5R(ξξξ3) ,

(7)


A =⎡⎢⎣

536

29 −

√1515

536 −

√1530

536 +

√1530

29

536 −

√1524

536 +

√1530

29 +

√1515

536

⎤⎥⎦ (8)

and the stage values correspond to the intermediate times(12 −

√1510

)�t , 1

2�t , and(12 +

√1510

)�t within the time step.

The two stage Gauss scheme is fourth order accurate, while the three stage Gauss schemeis sixth order accurate. Both schemes are A-stable but not L-stable. The Radau 2A schemesinclude the end of the time interval as one of the integration points, corresponding to Radauintegration. Consequently they have an order of accuracy 2s − 1 for s stages. They have theadvantages, on the other hand, that the last stage value is the final value, eliminating the needfor an extra step to evaluate wn+1, and that they are L-stable.

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For the solution of equation 1, the two stage Radau 2A scheme takes the form

ξ1 = wn − �t

V(a11R(ξ1) + a12R(ξ2)) ,

ξ2 = wn − �t

V(a21R(ξ1) + a22R(ξ2)) ,

wn+1 = ξ2

(9)


A =[ 512 − 1

12

34

14

](10)

and the stage values correspond to Radau integration points at 13�t and �t .

The three stage Radau 2A scheme takes the form

ξ1 = wn − �t

V(a11R(ξ1) + a12R(ξ2) + a13R(ξ3)) ,

ξ2 = wn − �t

V(a21R(ξ1) + a22R(ξ2) + a23R(ξ3)) ,

ξ3 = wn − �t

V(a31R(ξ1) + a32R(ξ2) + a33R(ξ3)) ,

wn+1 = ξ3,

(11)


A =⎡⎢⎣

88−7√6

360296−169

√6

1800−2+3

√6

225296+169

√6

180088+7

√6

360−2−3

√6

22516−√

636

16+√6

3619

⎤⎥⎦ (12)

and the stage values correspond to the Radau integration points 4−√6

10 �t , 4+√6

10 �t and �t .The following two single diagonal implicit Runge Kutta (SDIRK) schemes are also

included in this study for comparison. The first is the two stage second order accurate scheme(SDIRK2)

ξ1 = wn − �t

Va11R(ξ1),

ξ2 = wn − �t

V(a21R(ξ1) + a22R(ξ2)) ,

wn+1 = ξ2,

(13)

with the coefficient array

A =[1 − 1

2

√2 0

12

√2 1 − 1

2

√2

](14)

and stage values at intermediate times (1 − 12

√2)�t and �t .

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The second is the three stage third order accurate scheme (SDIRK3)

ξ1 = wn − �t

Va11R(ξ1),

ξ2 = wn − �t

V(a21R(ξ1) + a22R(ξ2)) ,

ξ3 = wn − �t

V(a31R(ξ1) + a32R(ξ2) + a33R(ξ3)) ,

wn+1 = ξ3,

(15)

with the coefficient array

A =⎡⎣

λ 0 012 (1 − λ) λ 0

14

(−6λ2 + 16λ − 1) 1

4

(6λ2 − 20λ + 5

)λ

⎤⎦ (16)

where λ = 0.4358665215 and stage values at the intermediate times λ�t , 12 (1 + λ)�t and

�t . Both these schemes are L-stable and the stage values can be solved sequentially. It willbe shown in the next section, however, that dual time stepping can be used to solve all ofthese schemes with an inexpensive preconditioner.

4 Formulation of the Dual Time Stepping Method for Fully ImplicitRunge Kutta Schemes

In order to clarify the issues it is useful to consider first the application of the two stage Gaussscheme to the scalar equation

du

dt= au (17)

where a is a complex coefficient lying in the left half plane. A naive application of dual timestepping would simply add derivatives in pseudo time to produce the scheme

dξ1

dτ= a(a11ξ1 + a12ξ2) + un − ξ1

�tdξ2

dτ= a(a21ξ1 + a22ξ2) + un − ξ2

�t

(18)

which may be written in vector form as

dξ

dτ= Bξ + c (19)

where

B =[a11a − 1

�t a12aa21a a22a − 1

�t

], c = 1

�t

[un

un

]

For equation (19) to converge to a steady state the eigenvalues of B should lie in the left halfplane. These are the roots of

det(λI − B) = 0

or

λ2 − λ

((a11 + a22)a − 2

�t

)+ a11a22a

2 − (a11 + a22)a

�t+ 1

�t2− a12a21a

2 = 0

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Substituting the coefficient values for the Gauss scheme given in equation (6), we find that

λ = 1

4a − 1

�t± ia

√1

48

Then if a = p + iq

λ = 1

4p ± q

√1

48− 1

�t+ i

(1

4q ± p

√1

48

)

and for small �t one root could have a positive real part even when a lies in the left plane.In order to prevent this we canmodify equation (18) bymultiplying the right hand side by a

preconditioning matrix. It is proposed here to take the inverse of the Runge–Kutta coefficientarray A as the preconditioning matrix. Here

A−1 = 1

D

[a22 −a12a21 a11

]

where the determinant of A is

D = a11a22 − a12a21

Setting

r1 = a(a11ξ1 + a12ξ2) + un − ξ1

�t

r2 = a(a21ξ1 + a22ξ2) + un − ξ2

�t

the preconditioned dual time stepping scheme now takes the form

dξ1

dτ= (a22r1 − a12r2)D

= aξ1 + a22D�t

(un − ξ1

) − a12D�t

(un − ξ2

)

dξ2

dτ= (a11r2 − a21r1)/D

= aξ2 + a11D�t

(un − ξ2

) − a21D�t

(un − ξ1

)

which may be written in the vector form (19) where now

B =[a − a22

D�ta12D�t

a21D�t a − a11

D�t

], c = 1

D�t

[(a22 − a12)un

(a11 − a − 21)un

]

Now the dual time stepping scheme will reach a steady state if the roots of

det(λI − B) = 0

lie in the left half plane. Substituting the coefficients of B the roots satisfy

λ2 − λ

(2a − a11 + a22

D�t

)+ a2 − a

a11 + a22D�t

+ 1

D�t2 = 0

and using the coefficient values of the Gauss scheme (6), we now find that

λ2 − λ

(2a − 6

�t

)+ a2 − 6a

�t+ 12

�t2= 0

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yielding

λ = a − 3

�t± i

√3

�t

Accordingly both roots lie in the left half plane whenever a lies in the left half plane, estab-lishing the feasibility of the dual time stepping scheme.

In the case of the linear system

du

dt= Mu

a similar analysis may be carried out if M can be reduced to diagonal form by a similaritytransformation

M = VV−1

Then setting v = V−1u,

dv

dt= v

and the same preconditioning scheme can be used separately for each component of v.Following this approach, the proposed dual time stepping scheme for the nonlinear equa-

tions (5) is

r1 = V

�t(wn − ξξξ1) − a11R(ξξξ1) − a12R(ξξξ2)

r2 = V

�t(wn − ξξξ2) − a21R(ξξξ1) − a22R(ξξξ2)

(20)

anddξξξ1dτ

= (a22r1 − a12r2)/D

dξξξ2dτ

= (a11r2 − a21r1)/D(21)

A more general approach, which facilitates the analysis of dual time stepping for implicitRunge–Kutta schemes in general, is presented in the next paragraphs. Using vector notationa naive application of dual time stepping yields the equations

dξ

dτ= aAξ + 1

�t

(wn − ξ

)(22)

and the eigenvalues of the matrix

B = aA − 1

�tI (23)

do not necessarily lie in the left half plane. Introducing A−1 as a preconditioning matrix thedual time stepping equations become

dξ

dτ= aξ + 1

�tA−1 (

wn − ξ)

(24)

so we need the eigenvalues of

B = aI − 1

�tA−1 (25)

to lie in the left half plane for all values of a in the left half plane.

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The eigenvalues of B are

a − 1

�t

1

λk, k = 1, 2, 3

where λk are the eigenvalues of A. Thus they will lie in the left half plane for all values ofa in the left half plane if the eigenvalues of A lie in the right half plane. The characteristicpolynomials of A for the two-stage Gauss and Radau 2A schemes are

λ2 − 1

2λ + 1

12= 0

and

λ2 − 2

3λ + 1

6= 0

with roots

λ = 1

4± i

√1

48

and

λ = 1

3± i

√1

18

respectively, which in both cases lie in the right half plane. It may be determined by a ratherlengthy calculation that the characteristic polynomials for the three stage Gauss and Radau2A schemes are

λ3 − 1

2λ2 + 1

10λ − 1

120= 0 (26)

and

λ3 − 6

10λ2 + 3

20λ − 1

60= 0. (27)

Rather than calculating the roots directly, it is simpler to use the Routh–Hurwitz criterionwhich states that the roots of

a3λ3 + a2λ

2 + a1λ + a0

lie in the left half plane if all the coefficients are positive and

a2a1 > a3a0.

The roots of A will lie in the right half plane if the roots of −A be in the left half plane. Here,the characteristic polynomials of −A for the two-three stage schemes are

λ3 + 1

2λ2 + 1

10λ + 1

120

and

λ3 + 6

10λ2 + 3

20λ + 1

60

and it is easily verified that the Routh–Hurwitz condition is satisfied in both cases. Thus, itmay be concluded that the preconditioned dual time stepping scheme will work for all fourof the implicit Runge–Kutta schemes under considerations.

In the present analysis it has been directly proved that for each of the four schemes theeigenvalues of A lie in the right half plane. It may be noted, however, that for an implicit

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Runge–Kutta scheme with coefficient matrix A and row vector bT of the final stage coeffi-cients, the stability function is

r(z) = 1 + bT (I − zA)−1e,

where

eT = [1, 1, . . . ].When this is expanded as

r(z) = N (z)

D(z),

det(I − zA) appears in the denominator. For an A-stable scheme, D(z) cannot have any zerosin the left half plane. Accordingly it may be concluded the eigenvalues of A must lie in theright half plane for any A-stable implicit Runge–Kutta schemes.

The proposed dual time stepping scheme for the three stage schemes is finally

r1 = V

�t(wn − ξξξ1) − a11R(ξξξ1) − a12R(ξξξ2) − a13R(ξξξ3)

r2 = V


r3 = V


anddξξξ1dτ

= d11r1 + d12r2 + d13r3

dξξξ2dτ

= d21r1 + d22r2 + d23r3

dξξξ3dτ

= d31r1 + d32r2 + d33r3

where the coefficients d jk are the entries of A−1.

5 A Fast Solution Method for the Euler and RANS Equations

Over the past four decades multigrid methods have been developed that are particularlyeffective for the fast solution of the Euler equations. Multigrid methods have so far generallyproved less effective in calculations of turbulent viscous flows using the Reynolds averagedNavier–Stokes equations. These require highly anisotropic gridswith very finemesh intervalsnormal to the wall to resolve the boundary layers. While simple multigrid methods still yieldfast initial convergence, they tend to slow down as the calculation proceeds to a low asymp-totic rate. This has motivated the introduction of semi-coarsening and directional coarseningmethods [12–18].

In 2007 Cord Rossow proposed using several iterations of an LUSGS scheme as a pre-conditioner at each stage of an explicit RK scheme [19]. This hybrid RKSGS scheme wasfurther developed by Swanson et al. [20] and it has been found to be an effective driver ofa full approximation multigrid scheme for steady state RANS calculations, yielding rates ofconvergence comparable to those that have been achieved for Euler calculations. Multigridschemes have the advantage that they produce local and global equilibrium at about the same

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rate. They have been studied extensively by the present author [21,22], and the full approxi-mation scheme used in the present work is described in detail in reference [22], to which thereader is referred. This paper also provides details of the Jameson-Schmidt-Turkel schemewhich has been used for the formulation of the numerical flux vectors. The following para-graphs describe the RKSGS scheme which has been used to drive the multigrid procedure.The scheme is generally similar to the Swanson–Turkel–Rossow implementation but differsin some significant details.

The scheme is described for the two dimensional case with the integral form

d

dt

∫

cell

w dS +∮

cell boundary

( f dy − g dx) = 0,

where S is the area, and f and g are the flux vectors in the x and y coordinate directions. Inorder to solve the corresponding semi-discrete finite volume equation

dwdt

+ R(w) = 0 (28)

where R(w) represents the residual of the space discretization, an n stage RKSGS schemeis formulated as

w(1) = w(0) − α1�t P−1R(0),

w(2) = w(0) − α2�t P−1R(1),... = ...

w(m) = w(0) − �t P−1R(m−1),

(29)

where P denotes the LUSGS preconditioner. As in the case of the basis additive RK schemethe convective and dissipative parts of the residual are treated separately. Thus if R(k) is splitas

R(k) = Q(k) + D(k), (30)

then

Q(0) = Q(w(0)

), D(0) = D

(w(0)

),

and for k > 0,Q(k) = Q

(w(k)

),

D(k) = βkD(w(k)

) + (1 − βk)D(k−1),(31)

Two and three stage schemes which have proved effective are as follows. The coefficients ofthe two stage scheme are

α1 = 0.24, β1 = 1,α2 = 1, β2 = 2/3,

while those of the three stage scheme are

α1 = 0.15, β1 = 1,α2 = 0.40, β2 = 0.5,

α3 = 1, β3 = 0.5,

While the residual is evaluated with a second order accurate discretization, the preconditioneris based on a first order upwind discretization. For a quadrilateral cell numbered zero, with

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neighbors k = 1–4, the residual of a central difference scheme is

Rc0 = 1

S0

4∑k=1

hk0 (32)

where S0 is the cell area and

hk0 = 1

2

[(fk + f0)�yk0 − (gk + g0)�xk0

]. (33)

Introducing a Roe matrix Ak0 satisfying

Ak0(wk − w0) = (fk − f0)�yk0 − (gk − g0)�xk0, (34)

and noting that the sums

4∑k=1

�yk0 = 0,4∑

k=1

�xk0 = 0,

we can write

Rc0 = 1

2S0

4∑k=1

Ak0(wk − w0). (35)

The residual of the first order upwind scheme is obtained by subtracting |Ak0|(wk − w0)

from hk0 to produce

R0 = 1

2S0

4∑k=1

(Ak0 − |Ak0|) (wk − w0).

Also noting that we can split Ak0 as

Ak0 = A+k0 + A−

k0,

where

A±k0 = 1

2(Ak0 ± |Ak0|)

have positive and negative eigenvalues respectively, we can write

R0 = 1

S0

4∑k=1

A−k0(wk − w0). (36)

We now consider an implicit scheme of the form

wn+1 = wn − �t(εR(wn+1) + (1 − ε)R(wn)

)

and approximate Rn+10

Rn+10 = Rn

0 + 1

S0

4∑k=1

A−k0(δwk − δw0),

where δw denotes wn+1 − wn . This yields the implicit scheme(I − ε

�t

S0

4∑k=1

A−k0

)δw0 + ε

�t

S0

4∑k=1

A−k0δwk = −�tRn

0 . (37)

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The LUSGS preconditioner uses symmetric Gauss–Seidel forward and backward sweepsto approximately solve the equation using the latest available values for δwk , and startingfrom δw = 0. In practice it has been found that a single forward and backward sweepis generally sufficient, and very rapid convergence of the overall multigrid scheme can beobtained with both the two and the three stage schemes. Moreover, a choice of the coefficientε < 1 is effectively a way to over-relax the iterations, and it turns out that the fastest rate ofconvergence is obtained with ε around 0.65 for the two stage scheme, and ε less than 0.5 forthe three stage scheme.

At each interfaceA− ismodified by the introduction of an entropy fix to bound the absolutevalues of the eigenvalues away from zero. Denoting the components of the unit normal tothe edge by nx and ny , the normal velocity is

qn = nxu + nyv

and the eigenvalues are

λ(1) = qn,λ(2) = qn + c,λ(3) = qn − c,

where c is the speed of sound. Velocity components, pressure p and density ρ at the interface

may be calculated by arithmetic averaging and then c =√

γ p

ρ.

The absolute eigenvalues |λ(k)| are then replaced by

ek =

⎧⎪⎨⎪⎩

|λ(k)|, |λ(k)| > a

1

2

(a + λ(k)2

a

), |λ(k)| ≤ a

where a is set as a fraction of the speed of sound

a = d1c.

Without this modification the scheme diverges. In numerical experiments it has been foundthat the overall scheme converges reliably for transonic flow with d1 ≈ 0.10. The modifiedeigenvalues of A− are then

λ(1) = qn − e1,λ(2) = qn + c − e2,λ(3) = qn − c − e3.

It also proves helpful to further augment the diagonal coefficients by a term proportional toa fraction d2 of the normal velocity qn . The interface matrix is also modified to provide forcontributions from the viscous Jacobian. The final interface matrix A± can be convenientlyrepresented via a transformation to the symmetrizing variables

dwT =[dp

c2,

ρ

cdu,

ρ

cdv,

dp

c2− dρ

].

Let s be the edge length. then the interface matrix can be expressed as

AI = sM(Ac + s

ρS0Av − d2qn I

)M−1,

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where Ac and Av are the convective and viscous contributions, d2qn I is the diagonal augmen-tation term, and M is the transformation matrix from the conservative to the symmetrizingvariables. Here

M =

⎡⎢⎢⎣

1 0 0 −1u c 0 −uv 0 c −v

H uc vc − q2

2

⎤⎥⎥⎦

and

M−1 =

⎡⎢⎢⎢⎣

γq2

2 −γ u −γ v γ

− uc

1c 0 0

− vc 0 1

c 0

γq2

2 − 1 −γ u −γ v γ

⎤⎥⎥⎥⎦

where

q2 = u2 + v2, γ = γ − 1

c2.

Define

r1 = 1

2(q2 + q3 − 1),

r2 = 1

2(q2 − q3).

Then the modified convective Jacobian is

Ac =

⎡⎢⎢⎣r1 + q1 nxr2 nyr2 0nxr2 n2xr1 + q1 nxnyr1 0nyr2 nxnyr1 + q1 n2yr1 + q1 00 0 0 q1

⎤⎥⎥⎦ .

Also let u, λ and κ be the viscosity, bulk viscosity and conductivity coefficients. Then theviscous Jacobian is

A =

⎡⎢⎢⎣

−(γ − 1)κ 0 0 −κ

0 −(μ + n2xμ∗) −nxnyμ∗ 0

0 −nxnyμ∗ −(μ + n2yμ∗) 0

−(γ − 1)κ 0 0 −μ

⎤⎥⎥⎦

where

μ∗ = μ + γ,

and typically λ = − 23μ. Here it is assumed that the Reynolds stress is modeled by an eddy

viscosity. Then if μm and μt are the molecular and eddy viscosity,

μ = μm + μt ,

while it is the common practice to take

κ = μm

Pr+ μt

Prt,

where Pr and Prt are the molecular and turbulent Prandtl numbers.

123

J Sci Comput

An alternative implementation can be derived from a first order flux vector split schemewith the interface flux

hk0 = (f+0 + f−k )�yk0 − (g+0 + g−

k )�xk0,

where f and g are split as

f = f+ + f−, g = g+ + g−

and f± and g± have positive and negative eigenvalues respectively. In this case the implicitequation (37) should be replaced by

(I + ε�t

S0

4∑k=1

A+k0

)δw0 + ε�t

S0

4∑k=1

A−k0δwk = −�tRn

0 .

where

A+k0 = �yk0

∂f+

∂w− �xk0

∂g+

∂w

and

A−k0 = �yk0

∂f−

∂w− �xk0

∂g−

∂w.

This formula is similar to the scheme proposed by Swanson, Turkel and Rossow [20], butthe consistent procedure is then to evaluate A+

k0 at cell zero, and A−k0 at cell k. Numerical

tests indicate that the alternative formulations work about equally well. In any case thesepreconditioners may be applied with alternative second order schemes for the residual R0

on the right hand side. It turns out that the scheme works very well when R0 is evaluatedby the Jameson-Schmidt-Turkel (JST) scheme [22,23], provided that the artificial viscositycoefficients k(2) and k(4) and the blend factor r for the spectral radii in the different coordinatedirections are all carefully tuned.

RAE 2822 AIRFOIL Mach 0.676 Alpha 1.930CL 0.5823 CD 0.0038 CM -0.0863Grid 512x 64 Ncyc 100 Res 0.939E-09

1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

++++++++++++++++++++++++++++++++

+++++++++++

+++++++++

++++++++

+++++++

++++++

+++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+

+

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

RAE 2822 AIRFOIL

Mach 0.676 Alpha 1.930

Resid1 0.127E+05 Resid2 0.939E-09

Work 99.00 Rate 0.7369

Grid 512x 64

0.00 50.00 100.00 150.00 200.00 250.00 300.00

Work

-24.

000

-20.

000

-16.

000

-12.

000

-8.

000

-4.

000

0.0

00 4

.000

8.0

00

Log

(err

or)

Nsu

p -

0.20

0 0

.000

0.2

00 0

.400

0.6

00 0

.800

1.0

00 1

.200

1.4

00

(b)(a)

Fig. 1 RAE 2822 airfoil, Ma = 0.676, α = 1.930. RKSGS scheme using the Baldwin–Lomax turbulencemodel and the JST scheme

123

J Sci Comput

RAE 2822 AIRFOIL


Resid1 0.155E+05 Resid2 0.756E-09

Work 99.00 Rate 0.7337

Grid 512x 64

0.00 50.00 100.00 150.00 200.00 250.00 300.00

Work

-24.

000

-20.

000

-16.

000

-12.

000

-8.

000

-4.

000

0.0

00 4

.000

8.0

00

Log

(err

or)

-0.

200

0.0

00 0

.200

0.4

00 0

.600

0.8

00 1

.000

1.2

00 1

.400


1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

+++++++++++++++++++++++++++++++++

++++++++++++

+++++++++

++++++++

+++++++

++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++

+

+

+

+

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++

++++++

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

(b)(a)

Nsu

p



1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

++++++++++++++++++++++++++++++++++

+++++++++++

+++++++++

++++++++

++++++

++++++

+++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++

++++++++++++++

++++++++++++

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++

RAE 2822 AIRFOIL


Resid1 0.153E+05 Resid2 0.223E-07

Work 99.00 Rate 0.7593

Grid 512x 64

0.00 50.00 100.00 150.00 200.00 250.00 300.00

Work

-12.

000

-10.

000

-8.

000

-6.

000

-4.

000

-2.

000

0.0

00 2

.000

4.0

00

Log

(err

or)

Nsu

b -

0.20

0 0

.000

0.2

00 0

.400

0.6

00 0

.800

1.0

00 1

.200

1.4

00

(b)(a)


Figures 1, 2 and 3 show the results of the RKSGS scheme for three standard benchmarkcases: the RAE 2822 airfoil cases 1, 9 and 10. These calculations were performed with theBaldwin-Lomax turbulence model and the JST scheme.

6 Results for a Pitching Airfoil

The flow past a pitching airfoil has been used as a test case for the new dual time steppingimplicit Runge–Kutta schemes. All the calculations presented in this section used the JSTscheme [22,23] for spatial discretization. The selected case is the AGARD case CT-6 [24],which was also studied in the author’s original paper on dual time stepping [11]. This is a

123

J Sci Comput

NACA 64A010 grid 160 x 32

Fig. 4 O-mesh used in pitching airfoil calculation

pitching NACA 64A010 airfoil at a Mach number of 0.796. The airfoil is symmetric andthe mean angle of attack is zero, leading to a flow in which shock waves appear alternatelyon the upper and lower surfaces. The pitching amplitude is ±1.01 degrees, and the reducedfrequency, defined as

k = ωchord

2q∞

where ω is the pitching rate, has a value of 0.202.Simulations using the Euler equations were performed on an O-mesh with 160 × 32 cells

(displayed in Fig. 4), which has a very tight spacing at the trailing edge. An initial steadystate was established using 40 multigrid cycles. These were sufficient to reduce the densityresidual to a value less than 10−8. Then 5 pitching cycles were calculated with the dual timestepping scheme. This is sufficient to reach an almost steady periodic state, and the results ofdifferent schemes are compared during the fifth cycle. Typically the implicit time step was

123

J Sci Comput

NACA 64A010 - 2 stage Radau scheme - 18 steps per period Nstep 81 Phase 1620.0 Mach 0.796 Alpha 0.000CL 0.0386 CD 0.0016 CM 0.0082Grid 160x 32 Ncyc 20 Res 0.348E-10

1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

+++++++

++++++

++++++

++++++++++++++++++++

++

+++++++++++++++++++++++++++++++++++

+

+

+

+++

+

+

+

+++++++++

++++

+++++++++++++++++++++++

+

+

+

+++++++++++++++++++++++++++++++++++++

NACA 64A010 - 2 stage Radau scheme - 18 steps per period Nstep 83 Phase 1660.0 Mach 0.796 Alpha -0.649CL -0.0328 CD 0.0014 CM 0.0123Grid 160x 32 Ncyc 20 Res 0.144E-10

1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

+++++++

++++++

++++++

+++++++++++++++++++

+

+

+++++

+++++++++++++++++++++++++++++++

+

+

+

+++

+

+

+

++++++++++

+++++

++++++++++++++++

++++

+

+

++++++++++++++++++++++++++++++++++++

+++


1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

+++++++

++++++

++++++

++++++++++++++++++

+

+

++++++++++++++++++++++++++++++++++

+++

+

+

+

+++

+

+

+

++++++++++

++++

++++++++++++++++++++

++

++++++++++++++++++++++++++++++++++++++++


1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

+++++++

++++++

++++++

++++++++++++++++++

+

+

+++++++++++++++++++++++++++++++++++++

+

+

+

+++

+

+

+

++++++++++

++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Fig. 5 Snapshot of the solution at several phase angles during the fifth pitching cycle. a Phase angle 1. bPhase angle 2. c Phase angle 3. d Phase angle 4

selected such that each pitching cycle was calculated with 9, 18 or 36 steps, correspondingto a shift of 10, 20 or 40 degrees in the phase angle per step.

Figure 5 displays a snapshot of the solution calculated by the two stage Radau schemeat several phase angles during the fifth pitching cycle using 18 steps per period. The pres-sure distribution is displayed by the pressure coefficient Cp with the negative axis upward,using + and × symbols for the upper and lower surfaces. Due to the phase lag betweenthe pitching motion and the evolution of the flow, plots of the lift and drag coefficientsCL and CD against the angle of attack α follow an oval and butterfly curve respec-tively.

In order to evaluate the accuracy of the various schemes a reference solutionwas calculatedusing the nominally fifth order accurate 3 stage Radau schemewith 360 time steps per period,corresponding to a phase shift of 1 degree during each step, which should be very close toa fully converged solution. The results with smaller numbers of steps per period are thenplotted against the reference curve. Figures 6 and 7 show the results of the 2 stage Radauscheme forCL andCD versus α, using 18 steps per period, and it can be seen that they almostexactly overplot the reference curves represented by the solid lines. This calculation used 20multigrid iterations to solve the coupled implicit equations during each time step. Figure 8shows the convergence history for the last time step. In all the calculations convergence was

123

J Sci Comput

NACA 64A010 - 2 stage Radau scheme - 18 steps per period Grid 160 x 32

-0.

250

-0.

200

-0.

150

-0.

100

-0.

050

0.0

00 0

.050

0.1

00 0

.150

0.2

00 0

.250

CL

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Alpha

+

+

+

+

+++

+

+

+

+

+

+

++ +

+

+

+

Fig. 6 NACA 64A010, 2-stage Radau scheme, 18 steps per period

measured by the sum of the absolute values of the residuals of equations 9 or 11 over allstages and all mesh cells divided by the number of stages and the number of cells. In thiscase the average absolute residual was reduced to 0.118 × 10−10, with a convergence rate of0.388 per multigrid cycle.

The largest CFL number in this calculation measured as �t/�t∗ where �t∗ is a nominaltime step corresponding to a CFL number of unity, exceeds 5,500 in the vicinity of the trailingedge. Here the nominal time step was estimated as

�t∗ = V

(qn + c)i�li + (qn + c) j�l j

123

J Sci Comput

NACA 64A010 - 2 stage Radau scheme - 18 steps per periodGrid 160 x 32

-0.

006

-0.

004

-0.

002

0.0

00 0

.002

0.0

04 0

.006

0.0

08 0

.010

CD

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Alpha

++ +

+

+++

+

+

+++

+

++ +

+

+

+


where V is the cell area, qn is the normal velocity to an edge of length �l, c is the speed ofsound, and subscripts i and j denote edges in the i and j directions.

In order to assess the accuracy of the various schemes the computed values of CL andCD during the fifth pitching cycle were compared with the values computed in the referencesolution by the following measures:

CL err =∑ |CL − CL reference|

nstep

CDerr =∑ |CD − CD reference|

nstep

123

J Sci Comput

NACA 64A010 - 2 stage Radau scheme - 18 steps per period


Resid1 0.772E-03 Resid2 0.118E-10

Work 19.00 Rate 0.3878

Grid 160x 32

0.00 50.00 100.00 150.00 200.00 250.00 300.00

Work

-12.

000

-10.

000

-8.

000

-6.

000

-4.

000

-2.

000

0.0

00 2

.000

4.0

00

Log

(err

or)

-0.

200

0.0

00 0

.200

0.4

00 0

.600

0.8

00 1

.000

1.2

00 1

.400


where nstep is the number of steps per pitching cycle.Considering first the Gauss schemes, it was found that they require the equations to be

solved to a very small tolerance in each step. In the case of the two stage scheme the requiredtolerance is typically smaller than 10−6 as measured by the average stage residual. This

123

J Sci Comput

Table 1 Error with 18 steps perperiod

Scheme CL err CDerr

BDF2 0.1070 × 10−2 0.9154 × 10−4

BDF3 0.2844 × 10−3 0.6425 × 10−4

SDIRK2 0.1111 × 10−3 0.1501 × 10−4

SDIRK3 0.5425 × 10−4 0.6215 × 10−5

2 stage Radau 0.1781 × 10−4 0.3966 × 10−5

3 stage Radau 0.3447 × 10−6 0.7833 × 10−7

Table 2 Convergence ofSDIRK2

Steps/period CL err CDerr

9 0.4418 × 10−3 0.5554 × 10−4

18 0.1111 × 10−3 0.1501 × 10−4

36 0.2741 × 10−4 0.3768 × 10−4

72 0.6411 × 10−5 0.9352 × 10−6

Table 3 Convergence ofSDIRK3


9 0.2544 × 10−3 0.3065 × 10−4

18 0.5425 × 10−4 0.6215 × 10−5

36 0.9540 × 10−5 0.9207 × 10−6

72 0.1344 × 10−5 0.1231 × 10−6

Table 4 Convergence of 2 stageRadau scheme


9 0.1112 × 10−3 0.2410 × 10−4

18 0.1781 × 10−4 0.3966 × 10−5

36 0.2245 × 10−5 0.5418 × 10−6

72 0.3592 × 10−6 0.6843 × 10−7

Table 5 Convergence of 3 stageRadau scheme

Steps/Period CL err CDerr

9 0.4332 × 10−5 0.1218 × 10−5

18 0.3447 × 10−6 0.7833 × 10−7

36 0.1018 × 10−6 0.1945 × 10−7

72 0.4384 × 10−7 0.3431 × 10−8

tolerance was achieved with 15 inner iterations of the dual time stepping scheme in eachimplicit step. This is the absolute minimum number of inner iterations required for this case,and the calculation fails when 14 inner iterations are used. The failure mode is a failure tosatisfy the Kutta condition leading to the sudden appearance of a low pressure spike andpossibly a vacuum state at the trailing edge. It appears that the most dangerous operation inthe Gauss schemes is the calculation of the final updated value from the stage values, which

123

J Sci Comput


-0.

250

-0.

200

-0.

150

-0.

100

-0.

050

0.0

00 0

.050

0.1

00 0

.150

0.2

00 0

.250

CL

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Alpha

+

++

+

+

+

Fig. 9 NACA 64A010, 5 cycles per step

involves multiplication by�t/V . Here�t may be quite large, while the cell area V becomesvery small in the vicinity of the trailing edge. This step is eliminated in the Radau schemesbecause the last stage value is the final updated value of the time step.

Very similar results were obtained with the three stage Gauss scheme. However, it wasfound necessary to converge the inner iterations to an even smaller tolerance, typically smallerthan 10−7 as measured by the density residual, and the number of inner iterations needed to

123

J Sci Comput


-0.

006

-0.

004

-0.

002

0.0

00 0

.002

0.0

04 0

.006

0.0

08 0

.010

CD

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Alpha

+

++

+

+

+

Fig. 10 NACA 64A010, 5 cycles per step

be increased from 15 to 25 in order to reliably solve the AGARD CT6 test case. Since threecoupled equations have to be solved instead of two, the computational cost of the three stagesixth order accurate scheme is about 2 1

2 that of the two stage fourth order scheme for thisapplication.

123

J Sci Comput


-0.

250

-0.

200

-0.

150

-0.

100

-0.

050

0.0

00 0

.050

0.1

00 0

.150

0.2

00 0

.250

CL

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Alpha

+

+

+

+

Fig. 11 NACA 64A010, 3 stage Radau scheme, 3 steps per period

It was found, however, that the Gauss schemes did not exhibit the expected accuracy mea-sured by CL err and CDerr as defined above. This may also be due to the amplification frommultiplying by �t/V . Because of their lack of robustness and failure to attain the desiredaccuracy they were discarded from further evaluation.

The remaining numerical tests are focused on the accuracy and computational costs ofthe SDIRK and Radau schemes, all of which are L-stable. They are also compared with

123

J Sci Comput


-0.

006

-0.

004

-0.

002

0.0

00 0

.002

0.0

04 0

.006

0.0

08 0

.010

CD

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Alpha

+

+

++


the second order backward difference formula (BDF2) and the third order backward differ-ence formula (BDF3). Table 1 shows the results for simulations using 18 steps per pitchingcycle.

In these simulations enough inner iterations were used at each time step to reduce theresiduals to small values of the order of 10−9 or less. For example the simulations with

123

J Sci Comput

NACA 64A010 - 2 stage DIRK scheme - 18 steps per period


Resid1 0.128E-02 Resid2 0.133E-06

Work 19.00 Rate 0.6170

Grid 160x 32

0.00 50.00 100.00 150.00 200.00 250.00 300.00

Work

-12.

000

-10.

000

-8.

000

-6.

000

-4.

000

-2.

000

0.0

00 2

.000

4.0

00

Log

(err

or)

-0.

200

0.0

00 0

.200

0.4

00 0

.600

0.8

00 1

.000

1.2

00 1

.400

Fig. 13 NACA 64A010, 2 stage SDIRK scheme, 18 steps per period

the two and three stage simulations were performed with 20 and 25 iterations per time steprespectively. It can be seen from Table 1 that the SDIRK2 and SDIRK3 schemes exhibitedbetter accuracy than the backward difference formulas BDF2 and BDF3. While the twostage Radau and the SDIRK3 schemes are both nominally third order accurate, the two

123

J Sci Comput

NACA 64A010 Grid 512 x 64

Fig. 14 NACA 64A010, C-mesh for URANS calculation

stage Radau scheme exhibited smaller errors, and finally the three stage Radau schemewas the most accurate, as would be expected. Tables 2, 3, 4 and 5 show the results ofconvergence studies with the SDIRK2, SDIRK3, two stage Radau and three stage Radauschemes.

The results in Table 2 confirm the second order accuracy of the SDIRK2 scheme, witherror reductions very close to 4 in bothCL err andCDerr each time the time step is halved. Theresults in Tables 3 and 4 confirm better than second order accuracy for the SDIRK3 and twostage Radau schemes, but with error reductions less than 8 when the time step is halved. Forthe two stage Radau scheme it ranged between 6 and 8. Finally Table 5 confirms that the threestage Radau scheme is the most accurate of the tested schemes, but the error reductions arenot consistent with the nominal fifth order accuracy. For the larger time steps the simulationsmay not have reached the asymptotic range in which the nominal order of accuracy shouldbe attained. For the smaller time steps the errors are so small that they may be affected byround off error. However, when the simulations were repeated in quadruple precision they

123

J Sci Comput

NACA 64A010 - 2 stage Radau scheme - 18 steps per period Nstep 81 Phase 1620.0 Mach 0.796 Alpha 0.000CL 0.0311 CD 0.0088 CM 0.0085Grid 512x 64 Ncyc 10 Res 0.551E-01

1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

++++++++

+++++++++++

++++++++++++

+++++++++++

++++++++++

++++++++

++++++++

++++++

++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

++++

+

+

+

+

+

+++++++++++++++++++++++

+++++++++

+++++++++++

++++++++++++

++++++++++

++++++++

+++++++++

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

++++++++

+++++++++++

++++++++++++

++++++++++

++++++++++

++++++++

++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++

+

+

+

+

++++

+

+

+

+

+

++++++++++++++++++++++++

+++++++++

+++++++++++

++++++++++++

++++++++++

+++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

++++++++++

+++++++++++

++++++++++++

++++++++++

++++++++++

+++++++++

++++++

++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

++++

+

+

+

+

+++++++++++++++++++++++++

+++++++++

++++++++++

+++++++++++

+++++++++++

++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


1.2

00 0

.800

0.4

00 -

0.00

0 -

0.40

0 -

0.80

0 -

1.20

0 -

1.60

0 -

2.00

0

Cp

++++++++++

++++++++++++

+++++++++++++

++++++++++

++++++++++

+++++++++

+++++

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++

+

+

+

+

+

+++

+

+

+

+

+

+++++++++++++++++++++++++

+++++++++

++++++++++

++++++++++++

++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Fig. 15 Snapshot of the solution during the last pitching cycle of the URANS calculation. a Phase angle 1.b Phase angle 2. c Phase angle 3. d Phase angle 4

produced almost identical results. Since the simulated flows contain moving shock wavesthey may not have the smoothness in time that would support the error expansions which areused to estimate the order of accuracy.

The L-stability of the Radau schemes might be expected to improve their capability totreat stiff equations. In this application, however, they proved to be robust in a different sense.First, they showed good accuracy with remarkably few time steps per pitching cycle. Second,they maintained accuracy using very small numbers of iterations to solve the equations ateach time step. Figures 9 and 10 show CL and CD versus α for a simulation using the twostage Radau scheme with 5 steps per period. Figures 11 and 12 show CL and CD versus α

for a simulation using the three stage Radau scheme with only 3 steps per period. With 3steps per pitching cycle, the nominal CFL number in the vicinity of the trailing edge exceeds33,000. Both these simulations were performed using 5 inner iterations for each time step.More generally, it was found that 5 inner iterations per time step suffice to produce resultsthat are indistinguishable from the fully converged results of simulations with 9 or more timesteps per period.

The convergence rates of the inner iterations of the SDIRK schemes were found to be nofaster than those of the fully implicit schemes. For example, Fig. 13 shows the convergencerate of the second stage of the last step in a simulation with the SDIRK2 scheme using 18steps per period and 20 inner iterations per time step. The average absolute value of the stage

123

J Sci Comput


-0.

250

-0.

200

-0.

150

-0.

100

-0.

050

0.0

00 0

.050

0.1

00 0

.150

0.2

00 0

.250

CL

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Alpha

+

+

+

+

+++

+

+

+

+

+

+

++ +

+

+

+


residuals was reduced to 0.133 × 10−6. with a convergence rate of 0.617, slower than thatobtained with the two stage Radau scheme shown in Fig. 8.

The computational cost of the two stage Radau scheme, as measured by CPU time wasfound to be slightly greater than that of the SDIRK2 scheme and significantly less thanthat of the SDIRK3 scheme. The times on a MacPro laptop of simulations using 18 time

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0.0

00 0

.002

0.0

04 0

.006

0.0

08 0

.010

0.0

12 0

.014

0.0

16

CD

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50

Alpha

++

+

+

+++

+

+

++

+

+

++ +

+

+

+


steps per period and 20 iterations per time step were 94.4, 77.3 and 123.4 s for the threeschemes respectively. Considering their robustness, accuracy and moderate computationalcost, it is concluded that the Radau schemes are the preferred choice for simulations of thistype.

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NACA 64A010 - 2 stage Radau scheme - 18 steps per period

Mach 0.796 Alpha -0.000

Resid1 0.437E-04 Resid2 0.245E-06

Work 9.00 Rate 0.5621

Grid 512x 64

0.00 50.00 100.00 150.00 200.00 250.00 300.00

Work

-12.

000

-10.

000

-8.

000

-6.

000

-4.

000

-2.

000

0.0

00 2

.000

4.0

00

Log

(err

or)

-0.

200

0.0

00 0

.200

0.4

00 0

.600

0.8

00 1

.000

1.2

00 1

.400


Finally the two stage Radau scheme has been used for simulations of the same pitchingairfoil using the unsteady RANS equations with a Baldwin-Lomax algebraic turbulencemodel at a Reynolds number of 6 million. The Baldwin-Lomax model was kindly suppliedby R. K. Swanson of the NASA Langley Research Laboratory. These calculations were

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performed on a C-mesh with 512 × 64 cells, with 32 cells concentrated in the region ofthe boundary layer (Fig. 14). A reference solution was calculated using 180 time steps perpitching cycle and 20 inner iterations per time step. Figures 15, 16, 17 and 18 show theresults of a simulation using 18 time steps per period and 10 inner iterations per time step.Figure 15 displays snapshots of the flow at four phase angles. Figures 16 and 17 showcomparisons of CL and CD versus α with the reference solution represented by the solidlines. It can be seen that the results overplot the reference solution. Finally Fig. 18 shows theconvergence history of the average absolute value of the stage residuals during the last timestep. This was reduced to 0.245 × 10−6 at a convergence rate of 0.562. in these calculationsthe largest CFL number exceeds 57,000 in cells adjacent to the wall in the boundary layer.These results provide a further confirmation of the robustness and accuracy of the Radauschemes.

7 Conclusion

The first main conclusion of this study is that dual time stepping is a feasible approach forsolving the coupled residual equations of fully implicit Runge Kutta schemes for unsteadyflows. The second main conclusion is that the Gauss schemes are not sufficiently robustand are excessively expensive because they require the residual equations to be solved withextreme levels of accuracy. The third main conclusion is that the Radau schemes outperformthe SDIRK schemes, and are the preferred method for unsteady flow calculations of the typeconsidered in this study. A similar conclusion has recently been reached for some quite dif-ferent test problems in a newly released paper by Pazner and Persson [25]. A fourth mainconclusion is that these schemes can yield remarkably accurate results even when the localCFL number exceeds 10,000 in the smallest mesh cells. Finally it would be desirable to usestrong stability preserving (SSP) schemes [26], which are among the pioneering contributionsof Chi-Wang Shu, for simulations of flows containing moving shock waves. Unfortunatelyimplicit SSP schemes with m steps still have SSP coefficients less than 2m, which leads to atime step restriction that is still several orders of magnitude smaller than that needed for thetype of simulations considered in this paper, and accordingly it may be necessary to settle forL-stability.

Acknowledgements In recent years the author’s research has benefited greatly from the continuing supportof the AFOSR Computational Mathematics Program through grant number FA9550-14-1-0186, under thedirection of Dr. Fariba Fahroo and Dr. Jean-Luc Cambier.

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