At present, many practically important problems of the inviscid compressible fluid dynamics are solvednumerically with the use of curvilinear spatial grids. One of the basic ways for increasing the computationalefficiency of the Euler solvers on curvilinear grids is the increase in the stability robustness. Then thestationary solution of the Euler equations can be achieved faster with the aid of the pseudo-unsteadymethod at the expense of using larger time steps [1].
The stability interval of a finite-difference method for the numerical integration of the Euler equationscan be increased by using fully implicit schemes. However, in the case of the hyperbolic systems of partialdifferential equations the requirement of diagonal dominance of the matrices arising in the numericalalgorithms may impose limitations on the time steps [2] or may be violated in the splitting-up schemes [3].
2 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
The explicit difference schemes are much easier to implement, but the stability condition imposesrestrictions on the time steps. A number of researchers have proposed techniques for constructing explicitschemes with extended stability intervals. In particular, Jameson’s breakthrough [4] has been to showhow the central differencing in the space directions can be combined in time-marching explicit methodsto increase considerably the stability robustness. The schemes proposed in [4] use the explicit multistageRunge–Kutta (RK) schemes for the time integration of the Euler equations. The stability robustnessof the Runge–Kutta type schemes (we will also call them the Jameson’s schemes) increases with thenumber of stages [4–7]. Along with this property, the Jameson’s schemes possess a number of otherpositive properties [8]. This has caused a very wide acceptance of the Runge–Kutta type schemes for thenumerical solution of the aerodynamics problems; the review of relevant works may be found in [8].
Many problems of inviscid fluid dynamics involve strong discontinuities, such as shock waves andcontact discontinuities. In these cases, parasitic or spurious oscillations arise in the numerical solutionsnear the strong discontinuities. In order to suppress the spurious oscillations one usually introduces inthe schemes of the finite difference or finite volume method the extra terms representing the artificialviscosity or dissipation.
One might think that the incorporation of the artificial viscosity into the finite difference equationswould lead to the increase in the size of the stability region, because the artificial viscosity dampsspurious oscillations of the numerical solution. The available results of the stability invstigation of manydifference schemes in the presence of artificial viscosity, however, show that this extra viscosity has muchmore complex effects on the stability of difference schemes.
The first investigation of the effects of artificial viscosity on the stability of difference equations wascarried out by Neumann and Richtmyer [9] (see also [10,11]). The artificial viscosityq proposed in [9]had the form
q = −(ah)2ρ∂u
∂xL
∣∣∣∣ ∂u
∂xL
∣∣∣∣ , (1)
wherexL is the Lagrangean coordinate,u the gas velocity,ρ the gas density,h is the step of a uniformgrid along thexL-axis, anda the dimensionless constant near unity. The artificial viscosity (1) wasintroduced additively into the pressure, so that the pressurep was replaced with the termp + q in theequations governing the inviscid fluid flow in Lagrangean coordinates. In the Fourier stability analysis ofdifference equations, the authors of [9] assumed the quantityσ = 2(ah)2(ρ/ρ0)|∂u/∂xL | to be locallyconstant, whereρ0 is the initial fluid density. In the stability analysis of [9], the “normal regions” and“shock regions” were distinguished. In addition, in shock regions the effects of weak shock waves andvery strong shock waves were studied separately. The general conclusion of [9,10] was as follows: forsufficiently weak shocks, the artificial viscosity (1) does not impose additional restriction on time stepτ
as compared to the case of the absence of artificial viscosity. If there are, however, strong shocks, thenthe artificial viscosity imposes more severe limitations on the time step size in comparison with the caseof zero viscosityq.
Rusanov [12] has proposed an explicit difference scheme for the 2D Euler equations (see also [13]).In this difference scheme, a vector artificial viscosity was introduced:
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 3
wherew is the vector of dependent variables,x andy the Eulerian spatial coordinates, andϕα(w, h1,h2, τ ), α = 1,2, are the scalar coefficients of artificial viscosity, which possess the property (vanishingviscosity):
limh1→0,h2→0,τ→0
ϕα(w, h1, h2, τ ) = 0, α = 1,2.
By using the Fourier method of stability analysis it is easy to show that the vector artificial viscosityon the right-hand side of (2) gives a contribution in the form of a diagonal matrix to the amplificationmatrix. We will therefore refer to such vector artificial viscosities as to the diagonal viscosities. Rusanov[12] has shown that his vector artificial viscosity generally imposes additional restriction on time step incomparison with the usual CFL restriction
(|�u| + c)τ
√1
h21
+ 1
h22
≤ 1,
where|�u| is the absolute value of the local fluid velocity,c the local sound velocity, andh1, h2 the stepsof uniform rectangular grid along thex- andy-axes, respectively. A number of the generalizations ofRusanov’s scheme are mentioned in [13].
More general non-diagonal artificial viscosities, which have a vector matrix form, are discussed in[10,11,14,15]. In particular, it was shown in [15] that the matrix form of the numerical dissipation modelenables one to apply the appropriate scaling of the dissipation in each flow equation, yielding a reductionin the amount of dissipation being introduced and improves accuracy.
In [16] we have studied the stability of three-stage and five-stage Runge–Kutta schemes for the 2Dadvection–diffusion equation
∂u
∂t+ A
∂u
∂x+ B
∂u
∂y= ν
(∂2u
∂x2+ ∂2u
∂y2
), (3)
whereA andB are the constant components of the advection velocity vector along thex- andy-axes,respectively;ν the diffusion coefficient,ν = constant> 0. Following [16], let us introduce the non-dimensional variable
κ3 = ντ
(1
h21
+ 1
h22
).
Similarly to [17] we can also consider the case of vanishing viscosityν in (3): ν = O(h1); in this case,the term on the right-hand side of Eq. (3) is interpreted as an artificial viscosity.
The following conclusions may be drawn from the results of [16].
• The RK schemes become unstable ifκ3 > κ3max > 0, where the numerical value ofκ3maxdepends on aspecific Runge–Kutta scheme. This means that it is not desirable to use too a large artificial viscosity.
• There is a subinterval 0≤ κ ′3 ≤ κ3 ≤ κ ′′
3 within which the stability interval is larger than the stabilityinterval in the absence of artificial viscosity.
• There is a subinterval 0< κ ′′3 ≤ κ3 ≤ κ3max within which the stability interval is smaller than the
stability interval in the absence of artificial viscosity.
The energy method was used in [17] to analyze the stability of Runge–Kutta schemes involving fromone to four stages. The advantage of the energy method over the Fourier method is that it can be applied
4 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
also for the stability analysis of difference schemes with variable coefficients. In particular, the authorsof [17] have considered the stability of thes-order fully discrete RK approximations of a general systemof convection–diffusion equations ind spatial dimensions with variable coefficients:
wt =d∑
j=1
Aj(x, t)∂xjw + ε
2
d∑j,k=1
∂xk(Qjk(x, t)∂xj
w) + B(x, t)w, (4)
whereAj(x, t) ∈ C1 are the symmetric convection matrices, andQjk ∈ C1 the symmetric diffusionmatrices,ε the diffusion coefficient (which may also include the case of vanishing viscosityε = O(h)).
The analysis method of [17] can not be applied directly to the finite difference or finite volume RKschemes for the 2D Euler equations on curvilinear grids in the presence of artificial dissipation terms,because
1. the convection matricesAj(x, t) are not symmetric in the case of the Euler equations;2. the consideration in [17] is restricted to the artificial viscosity involving only the second derivatives
∂xk∂xj
w (see (4)), whereas the artificial viscosity of [4–6] involves, in particular, also the fourthderivatives∂4w/∂x4, ∂4w/∂y4;
3. the consideration in [17] is limited only to uniform rectangular spatial grids.
We now present a review of a number of further works, in which the stability of the Runge–Kutta typeschemes on uniform rectangular spatial grids has been analyzed. The stability criteria were presented in[7] for the Runge–Kutta schemes involving from 2 to 11 stages as applied to the 1D advection equationwithout artificial dissipator terms. A symbolic–numeric method was applied in [8,16] for obtaining thestability regions of the the Runge–Kutta schemes with the number of stages from three to five, whichapproximate the 2D advection–diffusion equation. Since the form of the obtained regions proved tobe rather complex, the approximate analytic formulas were obtained in [8,16] for the approximationof the stability region boundaries. The symbolic–numeric method of [8] was extended in [18] for thestability investigation of a four-stage Jameson’s scheme approximating the 3D thin-layer Navier–Stokesequations for turbulent compressible fluid flows on uniform rectangular spatial grids. The numericaldata on the stability region boundary, which were presented in [18], were fitted analytically in [19] toyield a sufficiently non-trivial formula for the stability condition of the Jameson’s scheme consideredin [18].
An optimization of the weight coefficients in the Runge–Kutta schemes for the 1D advection equationwas performed in [20] from the requirement of an efficient suppression of high-frequency componentsof the solution error described by the amplification factor obtained from the stability analysis of thenumerical scheme by the Fourier method.
The following estimate for the maximum time stepτ ∗jk allowed by the stability of Runge–Kutta schemes
on curvilinear grids was used in [21]:τ ∗jk = Aj,k/(λj,k+µj,k), whereλj,k andµj,k are the average spectral
radii of the Jacobian matrices inj andk directions,Aj,k is the cell area. The same estimate was usedalso in [22] in the inviscid case. The work [22] also gives the formulas describing the contributions ofphysical viscosity coefficient to the expression for the local time step limit.
Thus, only the uniform rectangular spatial grids were considered in the above stability investigationsof the Runge–Kutta type schemes. There are no stability investigations of the Runge–Kutta finite volumeschemes on curvilinear grids. This is explained by an extraordinary complexity of corresponding differ-ence equations, in which the coordinates of spatial grid nodes enter along with grid solution components.
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 5
On the other hand, the exact information about the stability region of a finite volume Runge–Kuttascheme in the presence of artificial dissipators is useful at least in the following two respects:
• this information enables one to indicate the explicit limitations for the magnitudes of the artificialdissipation coefficients from the requirement of computational method stability;
• the use of the local time stepping principle is facilitated, according to which the difference schemeoperates everywhere in the flow field close to its stability limit [4,21]. As a result, the numericalpseudo-unsteady computation using the local time stepping becomes more reliable.
In view of a considerable complexity of the Euler solvers on curvilinear grids that are based on theRunge–Kutta schemes it is necessary to combine the symbolic computations on a computer with thenumerical computations in order to investigate their stability. A number of symbolic–numerical methodsof stability investigation of difference schemes on the uniform spatial grids were presented in [23],where the symbolic computations were implemented with the aid of the computer algebra system (CAS)REDUCE, and the numerical computations were implemented with the aid of FORTRAN. The use oftwo different software systems, REDUCE and FORTRAN, leads to some inconveniences for the users,which do not enable one to proceed on the way of the computer aided automation of the stability analysisof complex difference schemes.
In 1988, a new CAS Mathematica appeared, which combines the ability of performing any analyticcomputations and the numerical computations in the machine arithmetic of floating-point numbers withany accuracy. In this connection, we have made a number of successful attempts at the implementationof some of the symbolic–numeric methods presented in [23] with the aid ofMathematica 2.2[24]. Theresults of these developments have been summarized in [25].
A new symbolic–numeric method has been proposed in [26] for obtaining the stability region bound-aries for the finite volume Runge–Kutta schemes on curvilinear grids for the 2D Euler equations. Theverification of the method was carried out in [26] at the example of a three-stage Runge–Kutta scheme,for which the analytic form of the stability condition was obtained in [26] with the aid of the Fouriermethod in the absence of the artificial dissipation terms.
The objective of the present paper is to investigate the stability of a three-stage Runge–Kutta finitevolume scheme on curvilinear grids for the 2D Euler equations in the presence of artificial dissipatorswith the aid of the symbolic–numerical method of [26].
This paper is organized as follows. In Section 2, we present a three-stage finite volume Runge–Kutta scheme involving the artificial dissipation terms. In Section 3 we discuss the peculiarities ofthe implementation of the Fourier stability analysis procedure for difference schemes on curvilineargrids. We also perform a linearization of the difference equations of the Runge–Kutta method pre-sented in Section 2. We further consider three particular cases of difference equations, when it provesto be possible to obtain the closed form analytic stability conditions for the finite volume scheme underconsideration.
The considered particular cases do not cover all possible cases. In other cases, the difference equationsare too complex for an analytic stability investigation, and it is then necessary to use a symbolic–numericmethod for stability analysis. In Section 4 we present briefly the symbolic–numeric method of [26]and describe in detail his modification, which has enabled us to apply the method to the Runge–Kuttascheme involving the artificial dissipation terms. We summarize both analytic and numerical results onthe stability region boundary of the difference scheme of Section 2 and derive a sufficiently non-trivialanalytic formula for stability condition, which fits well all the data.
6 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
Section 5 is devoted to a verification of the obtained stability condition. For this purpose we have solvednumerically a number of internal and external aerodynamics problems by using the computer codes, inwhich the obtained stability condition was incorporated.
2. The 3-stage Jameson scheme
Since the Runge–Kutta schemes with four or five stages have larger stability intervals than the three-stageschemes they have gained a relatively wider acceptance in aerodynamic computations than the three-stageschemes. It should be noted, however, that the complexity of the Fourier stability analysis of Runge–Kuttaschemes increases non-linearly with the number of stages, so that already at four stages the length of thealgebraic expressions arising in the process of symbolic computations implementing the Fourier methodbecomes tremendous, which makes increased demands for computer memory.
On the other hand, our practice of applying the symbolic–numerical method [26] to the stabilityinvestigation of a three-stage Jameson’s scheme without artificial dissipators has shown the feasibilityof such an analysis. In this connection, we have undertaken an attempt at the stability investigation of athree-stage finite volume Runge–Kutta scheme augmented by artificial dissipators. This scheme is appliedfor the numerical integration of the Euler equations governing the 2D non-stationary flow of an inviscidcompressible non-heat-conducting gas:
∂w
∂t+ ∂f (w)
∂x+ ∂g(w)
∂y= 0, (5)
wheret is the time,x andy are the spatial Cartesian coordinates, and
w =
ρ
ρu
ρv
ρE
, f (w) =
ρu
ρu2 + p
ρuvρuH
, g(w) =
ρv
ρvuρv2 + p
ρvH
. (6)
Herep, ρ, u, v, E andH denote the pressure, density, Cartesian velocity components, total energy andtotal enthalpy. For a perfect gas
E = p
(γ − 1)ρ+ 1
2(u2 + v2), H = E + p
ρ, (7)
whereγ is the ratio of specific heats.Let us take an arbitrary cell of curvilinear grid. Let the values ofw at the cell center be denoted bywjk,
and letΓjk andVjk be the cell contour and the control volume bounded by contourΓjk. The Euler Eq. (5)can be written in integral form for the regionVjk with boundaryΓjk as
∂
∂t
∫ ∫Vjk
w dx dy +∮Γjk
(f dy − g dx) = 0. (8)
In the result of discretization of (8) we can obtain the semi-discrete equation [4]
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 7
Fig. 1. The centered scheme.
whereAj,k is the cell area, and the operatorQ represents an approximation to the boundary integraldefined by the second term of (8). For example, the flux balance for thex momentum component isrepresented in (9) as
∂
∂t(Aj,kρu) +
4∑m=1
(Qmρum + .ympm) = 0, (10)
where the flux velocity
Qm = .ymum − .xmvm (11)
and the sum in (6) is over four sides of the cell, see Fig. 1. The values.xm and.ym in (10) and (11) arethe increments ofx andy along sidem of the cell, with appropriate signs. Each quantity in (10) and (11)such asu2 or (ρu)2 is evaluated as the average of the values in the cells on the two sides of the face
(ρu)2 = 12[(ρu)jk + (ρu)j+1,k]. (12)
The scheme (10)–(12) reduces to a central difference scheme on a Cartesian grid, therefore, it is secondorder accurate in space. As is known, the second-order schemes produce spurious oscillations of thenumerical solution at the shock wave fronts and in their vicinity. In this connection it was proposed in[4] to introduce the artificial dissipation terms in the finite volume scheme in order to damp the spuriousoscillations. These terms are added to Eq. (9) as follows:
dw
dt+
(Qw
Aj,k
)− Dw = 0, (13)
whereD is a dissipative operator the structure of which is similar for each of the four dependent variables:
wherewn = w(x, y, tn), tn = tn−1 + τn, n = 1,2, . . . , t0 = 0, τn = τ ; Phw = (1/Aj,k)Qwj,k, andα1, α2 are the non-dimensional weight parameters, which can be chosen from the requirement of stabilityrobustness [6]. For example, the choiceα1 = α2 = 1/2 ensures the stability condition 0≤ |aτ/h| ≤ 2 inthe case, where scheme (21) is applied atκ(2) = κ(4) = 0 for the difference approximation of the scalaradvection equation [4]
ut + aux = 0. (22)
As was noted in [6], the time stepping scheme (21) damps more effectively high-frequency modes of thenumerical solution at
α1 = 0.6, α2 = 0.6. (23)
It was stated in [27] that if the scheme (21) with coefficients (22) is applied atκ(2) = κ(4) = 0 to Eq. (22),then the Courant–Friedrichs–Lewy stability criterion for the scheme has the form
|a|τh
≤ 1.8. (24)
This estimate agrees well with the result of [23], where we have shown with the aid of the Fourier methodthat in the case of general coefficientsα1 andα2 the Courant–Friedrichs–Lewy stability condition of
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 9
scheme (21) withκ(2) = κ(4) = 0 has the form
|a|τh
≤ 1
α1
[2α1 − α2 + (α2(α2 − 4α1 + 8α2
1))0.5
2α2
]0.5
. (25)
In particular, atα1 = α2 formula (25) simplifies to
|a|τh
≤ 1
α1
[1 + (8α1 − 3)0.5
2
]0.5
. (26)
At α1 = α2 = 0.6 the stability condition
|a|τh
≤ 5
3
√1
2+ 3
2√
5≈ 1.803407572 (27)
is obtained from (26).
3. Fourier symbol on curvilinear grids
A detailed comparative analysis of ten different methods of stability investigation for difference schemesapproximating the partial differential equations of the mathematical physics, which we have presentedin [23], shows that the Fourier method [1,2,10] is the most reliable and universal method for stabilityinvestigation of difference schemes.
As it follows from (10), in the case of finite volume discretizations on curvilinear grids the differenceequation coefficients also contain the coordinates of the curvilinear grid nodes. If we “freeze” the com-ponents of the solution vectorw in the difference scheme coefficients we obtain the linearized differencescheme
wn+1 = S(x, y,w0)wn, (28)
wherew0 is the “frozen”solution vector. The difference operatorS(x, y,w0) in (28) is called the stepoperator, and its structure determines to a large extent the stability properties of a difference method.
In what follows, we investigate the stability of the difference initial-value problem
where �U0(x, y) is the given initial condition att = 0. We apply the von Neumann stability analysisprocedure [1,2,10] for obtaining the necessary stability condition of difference problem (29). We willassume in the following that there exists a one-to-one transformation
x = x(ξ, η), y = y(ξ, η) (30)
from a region in the plane of the Cartesian spatial coordinates(x, y) onto a rectangle in the plane ofcurvilinear coordinatesξ, η.
Since the coefficients of scheme (28) depend onx, y, the Fourier transform cannot be applied directlyto this scheme. Instead, one considers the Fourier symbol of the operatorS, which is obtained by fixingthe values ofx andy in S and by substituting a solution of the form
10 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
into (28), where�u0 is a constant vector, and the functionsx(ξ, η) andy(ξ, η) enter the transformation(30); k1, k2 are the real components of the wave vector,ω the wave frequency, and i= √−1. In thisway we findG(ξ, η, �u0), the Fourier symbol of the operatorS; in the case of constant coefficients inS,the operatorG is called the amplification matrix of the linearized difference scheme (28). The uniformboundedness of‖G‖ is necessary [28,29] for the stability of scheme (28). For the uniform boundednessof ‖G‖ it is in turn necessary that the conditions
|λα| ≤ 1 + O(τ ), α = 1, . . . ,Mu, (32)
are satisfied [2,10], whereλα are the eigenvalues of the amplification matrixG andMu is the dimensionof matrix operatorG. Inequalities (32) represent the necessary von Neumann stability conditions. Theyshould be checked locally at each point on the curvilinear spatial grid, where the difference solutioncomponents are computed.
We now apply the above outlined von Neumann stability analysis procedure to the three-stage Jameson’sscheme (21). We at first linearize the difference Eq. (21). For this purpose, we use the formula
f (wm,q) ≈ f (wj,k) + A(wj,k)δwm,q, g(wm,q) ≈ g(wj,k) + B(wj,k)δwm,q, (33)
whereδwm,q = wm,q −wj,k, (m, q) are any values of the subscripts entering (21); the Jacobi matricesA
andB are defined by formulas
A = ∂f (w)
∂w, B = ∂g(w)
∂w. (34)
Denote the result of the linearization of operatorPh in (21) byPh. Substituting formulas (33) in (10) weobtain with regard for (11) and (13) the following expression for the operator(Phw)j,k:
For the sake of brevity we have used here the notationsA = A(wj,k), B = B(wj,k). The lineariza-tion assumes that the coefficientsA andB are constant. A direct substitution of expression (35) in thethree-stage scheme (21) leads to huge algebraic expressions at the stage of a symbolic computation ofthe Fourier symbolG. It is, however, possible to simplify significantly the expression for the operatorPh
under certain reasonable assumption. Rewrite expression (35) in the form
It follows from Fig. 1 that the conditions (39) are the conditions that the pairs of the sidesP3P4 andP1P2,P2P3 andP4P1 are parallel. Thus, the volumeVj,k proves to be a parallelogram under conditions (39).Note that these conditions are satisfied for all cells of a parallelogram grid, which is a particular case ofthe curvilinear grid. In the case of a general curvilinear grid only some cells will have the parallelogramform or will differ little from the parallelograms.
Expression (35) simplifies greatly under satisfaction of conditions (39):
Phwj,k = A1(wj,k, x, y)wj+1,k − wj−1,k
2+ A2(wj,k, x, y)
wj,k+1 − wj,k−1
2, (40)
where
A1 = 1
Aj,k
[(yj,k+1 − yj,k)A − (xj,k+1 − xj,k)B],
A2 = 1
Aj,k
[(xj+1,k − xj,k)B − (yj+1,k − yj,k)A]. (41)
We now linearize the artificial dissipation terms. In our linear stability analysis we will assume that
ε(2)j±1/2,k = ε2x = constant, ε
(4)j±1/2,k = ε4x = constant,
ε(2)j,k±1/2 = ε2y = constant, ε
(4)j,k±1/2 = ε4y = constant. (42)
We will also assume in our stability analysis that
Aj±1/2,k ≈ Aj,k, Aj,k±1/2 ≈ Aj,k
in (15)–(17). Denote byD the result of the linearization of operatorD. Then
We can now obtain from (44), the expression for the step operatorS in (28) that corresponds tolinear difference scheme (44). For this purpose, we substitute sequentially in the expression forw(3)
the expressions forw(2) andw(1). As a result we obtain the following expression for the step operatorS:
S = I − τ Ph + α2(τ Ph)2 − α1α2(τ Ph)
3 + α1α2τ3P 2
h D − α2τ2PhD + τD, (45)
whereI is the identity operator.
12 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
Let us now calculate the Fourier symbol of operator (45). Denote byF(−τ Ph) the Fourier symbolof the operator−τ Ph. In order to obtain the analytic expression forF(−τ Ph), we substitute in (40) thesolution of the form (cf. (31))
wnj,k = �u0 e−iωtn ei(k1j.ξ+k2k.η), (46)
where.ξ and.η are the steps of uniform rectangular grid in the(ξ, η) plane. Substituting solution(46) in (40), we obtain the following expression for the Fourier symbolF(−τ Ph) of the operator−τ Ph:
In order to obtain the Fourier symbolF(τD) of the linearized operatorD of artificial dissipation, wesubstitute the solution (46) into the right-hand side of Eq. (43). As a result we obtain the Fourier symbolF(τD) as
The eigenvaluesµj of the matrixZ are known to have the form [30]
µ1 = µ2 = d1u + d2v, µ3 = µ1 + c(d21 + d2
2)0.5, µ4 = µ1 − c(d2
1 + d22)
0.5, (54)
wherec is the velocity of sound in gas.
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 13
Substituting the expressions for coefficientsd1 andd2 (48) into expressions (54), we can see that theeigenvaluesµm,m = 1,2,3,4 depend on the following six non-dimensional variablesκ1, . . . , κ6:
κ1 = cτ(yj,k+1 − yj,k)
Aj,k
; κ2 = uτ(yj,k+1 − yj,k)
Aj,k
;
κ3 = vτ(yj,k+1 − yj,k)
Aj,k
; κ4 = −(yj+1,k − yj,k)
yj,k+1 − yj,k
;
κ5 = (xj+1,k − xj,k)
yj,k+1 − yj,k
; κ6 = −(xj,k+1 − xj,k)
yj,k+1 − yj,k
. (55)
We can rewrite expressions (54) in terms of the introduced variables (55) as follows:
The consideration of the artificial dissipator terms (see Eq. (50)) leads to the necessity of adding thefollowing four non-dimensional parametersκ7, κ8, κ9 andκ10 to the set (55):
κ7 = ε2x, κ8 = ε4x, κ9 = ε2y, κ10 = ε4y. (57)
Thus, the eigenvalues of matrixG (52) are the functions of ten non-dimensional variablesκ1, . . . , κ10.Despite the complexity of expressions (53) for the eigenvalues of matrixG it is possible to draw from
(52) certain conclusions about the sufficiency of the necessary von Neumann condition (32). The matrixG (52) is a polynomial in the diagonalizable matrixZ. Applying a simple sufficient stability criterionfrom Section 4.11 of [10], we arrive at a conclusion that the satisfaction of the necessary von Neumannstability condition (32) will also be sufficient for the stability of linear difference scheme (44).
Despite the fact that we have found the explicit analytic expressions for the eigenvaluesλj of theFourier symbolG, the obtaining of closed form necessary stability condition from (53) by using the vonNeumann criterion (32) proves to be impossible. The reason for this is that the trigonometric functionssin(k1 .ξ), cos(k1 .ξ), sin(k2 .η), cos(k2 .η) enter the expressions for|λj | in various combinationsof the form
with various real exponentsβ1, β2, β3, β4. As a result of this, it proves to be impossible to find analyticallythe minima of the right-hand sides of formulas (53) as the functions of the variablesk1 .ξ andk2 .η.In this connection, the only way to obtain the information about the stability region boundary in the caseof the Jameson’s scheme under consideration is the use of numerical computations by the method whichwas presented in detail in [26].
It should, however, be noted that a direct implementation of the numerical computations in varioussections of the 10D Euclidean space of(κ1, κ2, . . . , κ10) points would lead to a tremendous amount ofnumerical computations. In order to reduce this computational work we have used a reasonable com-bination of the analytical results on the stability of scheme (44), which can be obtained at a number ofparticular values of the non-dimensional parametersκ1, . . . , κ10, and the direct numerical computations bymethod [26] in several sectionsκm = constant, which were chosen in such a way that these computationscomplement the analytic investigations and finally enable us to obtain the needed stability condition.
14 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
In accordance with this strategy, we now enumerate those particular cases, in which it proves to bepossible to obtain the exact analytic information about the stability region boundary of scheme (44).
Particular case 1. There are no artificial dissipation terms in the three-stage scheme (21), that isD = 0.Thend3 = 0 in (51), and the expression for the Fourier symbolG simplifies greatly:
G = I + Z + α2Z2 + α1α2Z
3.
It follows from (56) that the expressions forµj involve only two trigonometric functions sin(k1 .ξ) andsin(k2 .η). In addition, the eigenvaluesµj are purely imaginary. This has enabled us to obtain in [26] aclosed form stability condition from the von Neumann criterion (32):
where the constantC is called the Courant number. In the case of scheme (21), the numberC has theform (cf. (25)):
C = 1
α1
[2α1 − α2 + (α2(α2 − 4α1 + 8α2
1))0.5
2α2
]0.5
. (59)
It can be shown (see also [7,8]) that the left-hand side of the von Neumann stability condition (58) remainsthe same for Runge–Kutta type schemes with a different numberm of intermediate stages (m ≥ 1). Onlythe value of the Courant numberC on the right hand side of (58) is different for different Runge–Kuttaschemes in the absence of added artificial dissipators; for example,C = 4 for the optimal five-stageRunge–Kutta scheme from [6] (see also [8]).
Particular case 2. Let us assume that there is a stagnation subregion in the flow problem, that isu =v = 0 in this subregion. Then we obtain from (55) thatκ2 = κ3 = 0. Let us further assume that the soundvelocity c in this subregion has the following order of smallness:
c = O(maxm,q
|δwm,q |), (60)
where the incrementsδwm,q have the same meaning as in (33). Applying the linearization procedure toexpressions (55), we should linearize the expression forκ1 to κ1 = 0. Then it follows from (56) thatµ1 = µ2 = µ3 = µ4 = 0, and the eigenvaluesλj , j = 1, . . . ,4 of the Fourier symbolG (52) degenerateinto a single eigenvalueλj = 1+ d3 of multiplicity four. The von Neumann necessary stability condition(32) then leads to inequalities
−1 ≤ 1 + d3 ≤ 1. (61)
Let us consider the particular caseε2x �= 0, ε4x = ε2y = ε4y = 0. Then we have from (50) thatd3 = ε2x(2 cos(k1 δξ) − 2). The substitution of this formula into (61) yields the inequality
−ε2x(2 cosξ1 − 2) ≤ 2,
or
ε2x ≤ 1
1 − cosξ1. (62)
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 15
The right-hand side of inequality (62) achieves its minimum atk1 .ξ = π , and we obtain the followinglimitations forε2x :
0 ≤ ε2x ≤ 12. (63)
Particular case 3. We again assume thatu = v = 0 and that the estimate (60) holds. We further assumethat ε4x �= 0, ε2x = ε2y = ε4y = 0. Then it follows from (50) thatd3 = −4ε4x(cos(k1 .ξ) − 1)2.The substitution of this formula into inequalities (61) leads similarly to (62) to the following limitationsfor ε2x :
0 ≤ ε4x ≤ 18. (64)
A similar consideration of the particular case
ε2x = ε4x = ε2y = 0, κ1 = κ2 = κ3 = 0 (65)
yields the inequalities
0 ≤ ε4y ≤ 18. (66)
In accordance with (18),(19) and (42), the coefficientε2x depends onκ(2) and the pressure valuespnk,l.
We now determine the limitations for the coefficientsκ(2) andκ(4). For this purpose, we investigate theextrema of the function (18). We consider the following cases.
1. The points(j −1, k), (j, k), (j +1, k) (the cell centers, see Fig. 1 are located in a subregion of smoothflow. Denoted byStj,k the stencil applied in scheme (10)–(18) and (20) for the computation of thenumerical solution valuewn+1
j,k . According to (20), this stencil varies from point to point. Leth be areference size of those grid cells whose centers belong to the stncilStj,k. We can calculateh as in [23]by formula:
h = 1
Nj,k
∑(k,m)∈Stj,k,(k,m)�=(j,k)
[(xk,m − xj,k)2 + (yk,m − yj,k)]
2
0.5
,
where(xk,m, yk,m) are the coordinates of the center of the(k,m) cell, andNj,k is the total numberof stencilStj,k points except for the central point(xj,k, yj,k). Following [23,25] let us present thecoordinates(xk,m, yk,m) as the deviations from the central point(xj,k, yj,k):
xk,m = xj,k + cxk,m · h, yk,m = yj,k + cy
k,m · h, (k,m) ∈ Stj,k, (k,m) �= (j, k), (67)
wherecxk,m = O(1), cy
k,m = O(1). Let us expand the quantitiespj−1,k, pj+1,k entering formula(18) into the Taylor series with respect to(xj,k, yj,k) point. As a result we obtain with regard for(67):
pj±1,k =pj,k + ∂p
∂x· cxj±1,kh + ∂p
∂y· cyj±1,kh
+ h2
2
[∂2p
∂x2(cxj±1,k)
2 + 2∂2p
∂x∂ycxj±1,kc
y
j±1,k + ∂2p
∂y2(c
y
j±1,k)2
]+ O(h3).
16 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
All the derivatives∂p/∂x, ∂p/∂y, ∂2p/∂x2, ∂2p/∂x∂y, and∂2p/∂y2 are assumed to be calculatedhere at central point(xj,k, yj,k). Substituting these expansions into formula (18), we obtain:
νj,k =
|∂p/∂x(cxj+1,k + cxj−1,k)h + ∂p/∂y(cy
j+1,k + cy
j−1,k)h
+h2/2[∂2p/∂x2((cxj+1,k)2 + (cxj−1,k)
2) + 2∂2p/∂x∂y(cxj+1,kcy
j+1,k
+cxj−1,kcy
j−1,k) + ∂2p/∂y2((cy
j+1,k)2 + (c
y
j−1,k)2))t ] + O(h3)|
{4pj,k + ∂p/∂x(cxj+1,k + cy
j−1,k)h + ∂p/∂y(cy
j+1,k + cy
j−1,k)h
+h2/2[∂2p/∂x2((cxj+1,k)2 + (cxj−1,k)
2) + 2∂2p/∂x∂y(cxj+1,kcy
j+1,k
+cxj−1,kcy
j−1,k)] + ∂2p/∂y2((cy
j+1,k)2 + (c
y
j−1,k)2))] + O(h3)}
. (68)
In the particular case of quadratic grid in the(x, y) plane, we must specify the following values ofthe coefficientscxk,m andcyk,m in (67):
In the general case, when the curvilinear grid lines in the neighborhood of central point(xj,k, yj,k)
are not parallel with theOx andOy axes, we will assume that the first and second derivatives withrespect to spatial coordinatesx andy are bounded at the(xj,k, yj,k) point. Then it follows from (68)thatνj,k = O(h). From formula (63) we obtain with regard for (42) and (19) the following limitationfor κ(2):
0 ≤ κ(2) ≤ 1
2O(h).
Thus, at the computations of problems without shock waves the limitations forκ(2) dictated by stabilityprove to be very weak.
2. The(j, k) point lies in the zone of shock wave smearing. As follows from the computational resultspresented in [6], in the case of a stationary shock wave with constant gas states to the right and tothe left of shock this shock is smeared over a single cell of spatial grid. However, in cases of shockwaves with more complex solution profiles on different sides of the discontinuity, the shock wave issmeared over a larger interval including from three to five grid cells [4–7]. In addition, the profile ofthe numerical solution obtained by the Runge–Kutta type schemes with artificial dissipator (14)–(19)proves to be monotonous.
In this connection, we investigate in what follows the question about the extrema of function (18)for the following two particular cases:
Case 2a.The stationary shock wave is smeared over a single cell of spatial grid (see Fig. 2(a)). Let anormal shock wave propagate in the(x, y) plane along thex-axis so that its straight front is normal to thex-axis. Then the profile of the pressurepn
i,k in the neighborhood of the(j, k) point is described by formula:
pni,k =
{P1, i ≤ j
P2, i > j,(69)
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 17
Fig. 2. Shock wave is smeared: (a) over a single cell; (b) over two cells.
where the constant quantitiesP1 andP2 are the gas pressure values behind the shock front and ahead ofthe shock front, respectively, so thatP1 > P2 (see Fig. 2(a)). Substituting the solution (69) into formula(18) we obtain:
νi,k =
0, i ≤ j − 1(ζ − 1)/(3ζ + 1), i = j
(ζ − 1)/(ζ + 3), i = j + 10, i > j + 1,
(70)
whereζ = P1/P2 > 1.The shock waves, which may arise in transonic flows, are weak, so thatP1/P2 = 1 + ε, whereε is a
small positive quantity. Then we obtain from (70) thatνj,k = O(ε) at i = j, i = j + 1. With regard for(19) and (73) this implies the following limitation forκ(2):
0 ≤ κ(2) ≤ 1
2+ 2
ε. (71)
Thus, the limitation forκ(2) dictated by stability is again very weak in the transonic flow case.As we have shown above, the estimateνj,k = O(h) is valid in the subregions of smooth flow. It follows
from here that if a comparatively crude curvilinear spatial grid is used in transonic flow computations,so thath = O(ε), then the quantityνj,k proves to have approximately the same order of smallness inthe overall flow field. This observation may serve as a justification for “freezing” the coefficientνj,k in alinear stability analysis.
As is known [11], in the case of an infinitely strong normal shock wave the relationζ = P1/P2 = ∞holds. Then we obtain from (70):
limζ→∞
νi,k =
0, i ≤ j − 113, i = j
1, i = j + 10, i > j + 1.
(72)
In this case, we obtain from (63) the following limitation for the coefficientκ(2) in (19) ati = j withregard for (72):
0 ≤ κ(2) ≤ 12. (73)
18 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
At i = j we obtain from (19) with regard for (72) that
ε(2)j+1/2,k = κ(2)max(|νj+1,k|, |νj,k|) = κ(2).
Therefore, we have with regard for (64):
ε(4)j+1/2,k = max(0, κ(4) − ε
(2)j+1/2,k) = max(0, κ(4) − κ(2)) ≤ 1
8. (74)
It follows from (74) that the inequality (74) is satisfied atκ(4) < κ(2).According to (15) and (17), the quantityε(2)j−1/2,k is also involved in the computation of the quantity
wn+1j,k . We obtain the following expression forε(2)j−1/2,k from (72):
ε(2)j−1/2,k = κ(2)max(|νj−1,k|, |νj,k|) = 1
3κ(2).
Therefore, we obtain the following limitation forκ(2) with regard for (63):
0 ≤ κ(2) ≤ 32.
We have further
ε(4)j−1/2,k = max(0, κ(4) − ε
(2)j−1/2,k) = max(0, κ(4) − 1
3κ(2)) ≤ 1
8.
It follows from here that ifκ(4) > 1/3κ(2), thenκ(4) must satisfy the inequality
κ(4) ≤ 18 + 1
3κ(2). (75)
Case 2b.The stationary shock wave is smeared over two cells of spatial grid (Fig. 2(b)). In this case,we set
pni,k =
P1, i ≤ j
P2 + a(P1 − P2), i = j + 1P2, i > j + 1.
(76)
Herea is a parameter, which must satisfy the inequalities 0< a < 1 to ensure a monotonous pressureprofile in a smeared shock wave. The substitution of the solution (76) into (18) yields
νi,k =
0, i ≤ j − 1
(1 − a)(ζ − 1)
1 − a + (3 + a)ζ, i = j
(ζ − 1)|1 − 2a|3 − 2a + (2a + 1)ζ
, i = j + 1
a(ζ − 1)
4 + a(ζ − 1), i = j + 2
0, i > j + 2.
(77)
The pressure profile (76) includes as a particular case a shock wave smeared over a single cell ata = 0.In this case, formulas (77) are easily seen to go over into formulas (70).
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 19
Despite the complication of formulas forνi,k in comparison with the case of the shock smearing overa single cell, it is easy to see from (77) that in the case of weak shock waves, whenζ = P1/P2 = 1 + ε,the estimateνi,k = O(ε) again holds ati = j, j + 1, j + 2. Then we again obtain from (19) and (63) thelimitations forκ(2) of the form
0 ≤ κ(2) ≤ O
(1
ε
). (78)
In the case of an infinitely strong shock we obtain from (77):
limζ→∞
νi,k =
0, i ≤ j − 1
1 − a
3 + a, i = j
|1 − 2a|1 + 2a
, i = j + 1
1, i = j + 2
0, i > j + 2.
(79)
In this case we again obtain from (63) the limitations forκ(2) of the form (73) with regard for (79).We have not carried out a similar analysis for the cases of monotonous smearing of a stationary shock
wave over 3, 4, and 5 grid cells. We believe that also in these cases the general conclusions about theorder of smallness of the quantityνj,k will remain the same as in the above considered cases of shockwave smearing over one and two cells.
The inequalities (63),(64) and (66) were obtained above atk1 .ξ = π ork2 .η = π . We now substitutethe valuesk1 .ξ = π, k2 .η = π in the general Fourier symbolG (51). With regard for expressions (53)and (56) we obtain thatG = λ0I , where
λ0 = 1 − 4ε2x − 4ε2y − 16ε4x − 16ε4y, (80)
andI is the 4×4 identity matrix. From the von Neumann condition (32) we obtain the inequality|λ0| ≤ 1,which leads to the following inequalities:
0 ≤ ε2x + ε2y + 4ε4x + 4ε4y ≤ 12. (81)
It is easy to see that the above inequalities (63),(64) and (66) may be obtained as the particular cases ofinequalities (81). With regard for (19),(72) and (79) there are the following relations betweenε2x , ε2y andκ(2); betweenε4x, ε4y andκ(4) in the case of very strong shock waves:
Therefore, the coefficientsκ(2) andκ(4) must satisfy with regard for (81) the following inequalities:
0 ≤ κ(2) + 4 max(0, κ(4) − 13κ
(2)) ≤ 14. (82)
These inequalities are constructive, one can easily choose with their aid the coefficientsκ(2) andκ(4) fromthe stability requirement. We can now check whether the values (20) recommended in [4] satisfy or donot satisfy the inequalities (82):
κ(2) + 4 max(0, κ(4) − 13κ
(2)) = 14 + 0 = 1
4.
20 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
Thus, the three-stage Runge–Kutta scheme (21) is stable for these values of the coefficientsκ(2) andκ(4).In the case of weak shock waves, as we have seen above from (71) and (78), the limitations for
the coefficientsκ(2) and κ(4) prove to be much weaker than the limitations dictated by inequalities(82).
4. Symbolic–numerical method for stability investigation
In this Section, we briefly present a symbolic–numeric method of [26], which implements on a computerthe von Neumann stability analysis procedure described in the foregoing Section. The peculiarity ofthe finite volume scheme (21) under consideration is that the corresponding Fourier symbol (52) is apolynomial in a diagonalizable matrixZ. This has enabled us to obtain the explicit expressions (53)for the eigenvalues of matrixG. In this case it is possible to use a very efficient numerical method forcomputation of the coordinates of the stability region boundary, which was presented in [23] and whichuses directly the analytic expressions for the eigenvalues of the amplification matrix.
It should, however, be noted that the amplification matrices corresponding to many well-known Eulersolvers do not have the form of a matrix polynomial in some matrix. The example of such an Euler solver isthe MacCormack scheme of 1969 [23]. Further examples of such “non-polynomial” schemes are presentedin [23]. In this connection, we have presented in [23] a number of universal symbolic–numerical methodsfor stability investigation, which solve directly the characteristic equation
det(λI − G) =Mu∑j=0
cjλm−j = 0, (83)
whereMu is the dimension of matrix operatorG. We have extended the symbolic–numeric method of [23]for the case of finite volume schemes on curvilinear grids in [26]. We describe this extension in detail in[26] and compare the implementation of our symbolic–numeric method in theMathematicaenvironmentwith the implementation of [23] that was based on a combined use of CAS REDUCE and FORTRAN.
Therefore, we restrict in the present paper to a short description of the method of [26] and discuss in moredetail a modification of the method of [26] to adapt it for the stability investigation of the Runge–Kuttafinite volume schemes with artificial dissipators.
At the symbolic stages of the symbolic–numerical method [26], the entries of the amplification matrixG entering (83) are computed in symbolic form with CASMathematica. To save the needed memory atthis stage we at first substitute the entries of the Jacobi matricesA andB asa21= a21, b43= b43, etc.,where we have denoted byaj,k, bj,k, j, k = 1, . . . ,4 the entries of matricesA andB, respectively. Wefind thereafter the expressions for the first row ofG by replacingajk andbjk with their specific entriescorresponding to the gasdynamic Jacobi matricesA andB (the expressions for these matrices may befound in [23]).
At the next stage of the symbolic computations by our method [26], the entries of the matrixG areexpressed in terms of the parametersκ1, . . . , κ10 given by formulas (55) and (57). This is done in anefficient way in ourMathematicacode by using the transformation rules. Let, for example,cp1 = κ1.Then we can introduce the notationcp1 into the entrygg[[1,1]] of the matrixG with the aid oftransformation rule
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 21
whereAjk = Ajk, y[[j,k]] = yj,k, etc. The remaining parametersκ2, . . . , κ6 are introduced in theentries ofG in a similar way. The first row ofG thus obtained is then stored in the filerow1.m. Afterthat we compute in a similar way the next rows ofG. Such a strategy saves a lot of computer memory,so that only about 2 Mb are needed to compute one row ofG. Once we have computed all the rows ofmatrix G, we can assemble them into a 4× 4 matrix. The resulting matrixG takes 5529 lines of text,with 65 symbols in each line on the average, in the case o using a general ratio of the gas specific heatsγ in the equation of statep = (γ − 1)ρε. We have reduced considerably the length of the expressionfor G by substituting the valueγ = 7/5. The corresponding matrixG takes only 2503 lines. Note thata similar amplification matrixG, which was obtained in [26] in the absence of the artificial dissipationterms, takes only 1898 lines of text, with 65 symbols in each line on the average.
At the numerical stages of our method, the zeroes of characteristic Eq. (83) are computed numericallywith the aid of theMathematicafunctionSolve[. . .]. It is well known [23] that these zeroes are verysensitive to roundoff errors when the machine arithmetic of floating-point numbers is used to compute thenumerical values of coefficients of Eq. (83). On the other hand, it is well known that the computer algebrasystemMathematicaperforms exact arithmetic operations on rational numbers. In accordance with (48)and (50) the coefficientscj in (83) depend on cosζm and sinζm,m = 1,2, whereζ1 = k1 .ξ, ζ2 = k2 .η.In order to avoid the introduction of any roundoff errors when computing the numerical values of thesefunctions we have determined these values as the following rational numbers:
cosζm =
−1, ζm = π
1 − R2(tm, ε)
1 + R2(tm, ε), ζm �= π
; sinζm =
0, ζm = π2R(tm, ε)
1 + R2(tm, ε), ζm �= π
, (84)
whereR(tm, ε) =Rationalize[N[ tm, e+1], ε ],m = 1,2 andtm = tan(ζm/2); ε = 10−(e+1), e ≥0, is the user-specified accuracy with which the built-in Mathematica functionRationalize[. . .]converts a floating-point numbertm into a rational number. The Mathematica functionN[a,n] computesthe numbera as a floating-point machine number ton-digit precision. It is important that the calculationof cosζm and sinζm by formulas (84) always ensures the satisfaction of the relation cos2ζm + sin2ζm =1,m = 1,2.
It should be noted that we have used in [26] a different rational approximation for sinζm and cosζm:
sinζm = R1(ζm, ε) = Rationalize[N[Sin[ζm], e + 1], ε]
Such an approximation is indeed acceptable only in those cases, where the amplification matrix underconsideration involves the values of only sinζm or only cosζm. The reason for this is that the equation
R1(ζm, ε)2 + R2(ζm, ε)
2 = 1
is violated for some values of argumentζm. Example:
R21(
18(π),10−3) + R2
2(18(π),10−3) = (
1334
)2 + (1213
)2 = 195025195364. (86)
On the other hand, the use of the rational approximations (84) yields the following result atζm = π/8andε = 10−3:
sin2ζm + cos2ζm = (1376435965
)2 + (3322735965
)2 = 1. (87)
22 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
One can show by simple algebraic computations that the equation cos2ζm+sin2ζm = 1 is always satisfiedin the case of using the rational approximation (84) at any positiveε.
On the other hand, we should mention that the rational approximations obtained for a givenε from(84) lead to the rational numbers with much longer numerators and denominators than in the case ofusing the simpler rational approximation (85) (cf. formulas (86) and (87)). The computer time needed forperforming the arithmetic operations on rational numbers withMathematicaincreases with the length ofthe numerators and denominators of the processed rational numbers.
Therefore, the most optimal strategy of using the rational approximations (84) and (85) is as follows:if the entries of the amplification matrix under consideration involve only sinζm or only cosζm, then therational approximation (85) is used; otherwise a more complex approximation (84) is used. The abovecheck-up of the availability of only sinζm or only cosζm in the entries ofG can be made easily withMathematica, and we have implemented this check-up in ourMathematicacode.
The values of non-dimensional parameters (55) and (57) were also computed in (83) as the rationalnumbers. As a result, the coefficientscj of Eq. (83) are exact for any complex numerical method. Therefore,we can compute the zeroes of (83) exactly, because theMathematicafunctionSolve[...] implementsthe exact solution formulas atMu = 4.
We have used a symbolic–numerical method in [26] for the check-up of the analytical stability con-dition (58) and (59) of scheme (21). Since the symbolic–numerical method of [26] is based on a directcomputation of the zeroes of characteristic Eq. (83), such a check-up is indeed independent of formulas(53) and (56). The numerical results presented in [26] confirm the correctness of the analytic formulas(58) and (59).
It now remains to consider those particular sections of the 10D Euclidean space of(κ1, . . . , κ10) points,in which it is impossible to obtain the closed-form analytical stability conditions because of the complexityof coefficients in Eq. (83). We at first consider the following particular case.
in (53) and (56). In this case we can obtain the points of the curve
ε2x = ϕ1(κ1), −∞ < κ1 < ∞. (89)
The point where the curve (89) intersects theOκ1-axis, is the point withε2x = 0. According to the above
considered Particular case 1, we have that the relationε2x = 0 is satisfied forκ1 = 2/√
1 + κ25 = √
2.In the numerical computation by the method described above in this section, this value was determinedwith a small error< 0.0001.
We know the exact solution for one more point of curve (89): this is the point at which the curveintersects theOε2x axis. According to (63), the equationϕ1(0) = 1/2 should hold at this point. Rightthis value was obtained numerically, see Table 1. The numerical values ofε2x presented in this table wereobtained atε = 10−3 in (84).
Let us denote byP4(λ; κ1, ε2x, �ζ ) the characteristic polynomial of scheme (21) for the case under con-sideration, where�ζ = (ζ1, ζ2). This polynomial was computed in symbolic form and printed. An analysishas shown that the polynomialP4(·) contains only the even powers ofκ1. Therefore,P4(λ; −κ1, ε2x, �ζ ) =P4(λ; κ1, ε2x, �ζ ), that is the functionP4(·) is an even function of the variableκ1. This means that
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 23
Table 1Theε2x values on the stability region boundary in the sectionκ2 = κ3 = 0, κ5 = 1, κ6 = 0, ε4x = ε2y = ε4y = 0
the curveε2x = ϕ1(κ1) in (89) is symmetric with respect to theOε2x axis, see also Fig. 3. For amore efficient use of the data of Table 1 in gasdynamic computations it is desirable to find an analyticdependence approximating these data. Assume that we have found numericallyN points(κ1j , ε2xj), j =1, . . . , N(N > 1). The approximate functionϕ1(κ1) in (89) must be chosen in such a way that no(κ1, ϕ1(κ1)) point lies outside the stability region. This leads in our case to the following inequalityconstraints:
ε2xj ≥ ϕ1(κ1j ), j = 1, . . . , N. (90)
At first we have tried by analogy with [8] to approximate the data of Table 1 by the following elliptic arc(see also Fig. 3):
(ε2x
0.5
)2+
(κ1√
2
)2
= 1. (91)
One can see that the approximation (91) underestimates significantly the size of the stability region inthe interval 1< |κ1| <
√2. In this connection we consider the following approximation for the function
ε2x = ϕ1(κ1):(ε2x
0.5
)α
+(
κ1√2
)4
= 1, (92)
whereα is an indeterminate parameter. To determineα we make use of the data of Table 1. For the givenvalues ofε2x andκ1 we obtain from (92) the following formula forα:
α = ln [1 − (κ1/√
2)4]
ln [ε2x/0.5].
Fig. 3. The stability region lies below the curveε2x = ϕ1(κ1). (—) the curve (89) found numerically;(· · · ) the curve (91); (– – –)the approximation (92) atα = 2.7.
24 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
Table 2Theε4x values on the stability region boundary in the sectionκ2 = κ3 = 0, κ5 = 1, κ6 = 0, ε2x = ε2y = ε4y = 0.
The substitution of the valuesκ1 = 7/5, ε2x = 155/1024 yieldsα = 2.702226. Let us round this valueand takeα = 2.7 in (92)). In this way we obtain in section (88) the stability condition of the form
(ε2x
0.5
)2.7+
(κ1√
2
)4
≤ 1. (93)
The curveε2x = ϕ1(κ1) obtained from (92) atα = 2.7 is shown by a dashed line in Fig. 3. One can seethat this curve fits much more exactly the data of Table 1 than the ellipse (91). The constraints (90) arenot violated.
in (53) and (56). In this case we can obtain the points of the curve
ε4x = ϕ2(κ1), −∞ < κ1 < ∞. (95)
From the foregoing consideration (see Eqs. (58), (59) and (64) we already know two points of thiscurve:
ϕ2(0) = 18, ϕ2(
√2) = 0. (96)
The numerical values ofε4x presented in this Table were obtained atε = 10−3 in (84). The(κ1j , ε4xj)
points in the neighborhood of the valueκ1 = √2 were obtained with the aid of a bisection process in
the intervals, which are parallel with theOκ1 axis for the purpose of obtaining a higher accuracy. Inthis particular case, the characteristic polynomial of scheme (21) is also an even function ofκ1. Theapproximation of the form(
ε4x
1/8
)8
+(
κ1√2
)4
= 1 (97)
proved to be very good to fit the numerical data of Table 2, see also Fig. 4.
in (53) and (56). Denote byP4(λ; κ1, ε2y, ζ1, ζ2) the characteristic polynomial of scheme (21), whichcorresponds to this particular case. If we exchange the variablesζ1 andζ2 in this polynomial, then thenew polynomialP4(λ; κ1, ε2y , ζ2, ζ1) coincides with the polynomialP4(·) considered above in particular
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 25
Fig. 4. The stability region lies below the curveε4x = ϕ1(κ1). (—) the curve (95) found numerically; (– – –) the curve (97).
case 4 if we use the notationε2x instead ofε2y . Therefore, the curveε2y = ε2y(κ1) has the same form asa solid line in Fig. 3. Therefore, we can use here by analogy with (93) the following analytic fitting:
in (53) and (56). The corresponding characteristic polynomialP4(λ; κ1, ε4y, ζ1, ζ2) coincides after thesubstitutions
ξ1 → ξ2, ε4y → ε4x
with the polynomialP4(·) for the above particular case 5. Therefore, one can use here similarly to (97)the approximation of the form(
ε4y
1/8
)8
+(
κ1√2
)4
= 1 (99)
for the stability region boundary.The general case.The information about the stability region boundary, which was accumulated above
at the consideration of the particular cases 1 through 7, enables us to obtain the approximation of thestability region of scheme (21) in the 10D space of the(κ1, . . . , κ10) points.
The limitation|κ1| ≤ √2 indeed follows from a more general formula (58), in whichC is the Courant
In the case where there are no artificial dissipator terms in scheme (21) formula (103) goes over intoformula (58).
SinceC1 does not depend onτ , we can easily find from (103) an explicit limitation for the time stepτ .By using formulas (55) we obtain the following formula from (103):
whereθ is a safety factor, which should satisfy the inequlities 0< θ ≤ 1 in order to meet the stabilityrequirement (103). The numerator on the right-hand side of (104) is different from zero, because|Aj,k|represents the area of the(j, k) cell. The denominator is also different from zero, because otherwise thiswould lead to a contraction of the(j, k) cell into a single point(xj,k, yj,k).
Let the coefficientsε2x, ε2y, ε4x, ε4y be chosen in such a way that the inequalities
0 < 1 −(ε2x
0.5
)2.7−
(ε2y
0.5
)2.7−
(ε4x
1/8
)8
−(
ε4y
1/8
)8
< 1 (105)
are satisfied. Then 0< C1 < 1. The maximum time step sizeτmax allowed by the stability of scheme(21) is determined in accordance with (104) by the formula
Based on inequalities (105) and (106) we can study the effect of artificial dissipators on the stability ofthe three-stage Runge–Kutta scheme (21). If, indeed, at least one of the artificial dissipation coefficientsε2x, ε4x, ε2y, ε4y is positive, then inequality (105) is satisfied, and in view of inequality (106) we obtainfrom (104) a smaller value of the maximum time step than in the case of the absence of the artificialdissipation terms. Thus, the incorporation of the artificial dissipation terms in the three-stage Runge–Kutta
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 27
scheme in accordance with Eq. (21) may lead to a more severe limitation for the maximum time stepτ
from the stability requirement than in the case of the absence of the artificial dissipation terms. As followsfrom our analysis in Section 3, this additional limitation for time step size generally arises only whensolving flow problems which involve very strong shock waves.
5. Verification of stability condition
In order to validate the obtained stability condition (103) we have solved numerically a number ofinternal and external aerodynamics problems involving various flow regimes: subsonic, transonic, andsupersonic flow regimes. Formula (104) for the computation of the time step was incorporated in ouraerodynamic codes for the computation of the local time step in each cell of a curvilinear computing meshto implement the principle of local time stepping, which accelerates significantly the convergence of nu-merical solution to a steady state. All the aerodynamic computations were performed in theMathematica4.0environment [31]. The same CAS was used for the numerical generation of curvilinear grids.
The stability condition (103) was obtained under the assumption that the cells of curvilinear grid are theparallelograms. One may expect that formula (103) will work well also on curvilinear grids whose cellsdeviate little from the parallelogram form. In this connection, it is important to have some quantitativemeasure characterizing the deviation of the curvilinear grid cells from the parallelogram form. Thismeasure will enable us to predict the applicability of the stability condition (103) for a given curvilineargrid.
Consider a grid cell one of the vertices of which has the indices(j, k). Let α1jk be the angle betweentwo opposite sides formed by the line segments{(j, k), (j + 1, k)} and{(j, k + 1), (j + 1, k + 1)} (seeFig. 5). Letα2jk be the angle between the sides{(j, k), (j, k + 1)} and{(j + 1, k), (j + 1, k + 1)}. Let usdefine the total deviation of the(j, k) cell from a parallelogram shape as.αj,k = α1jk +α2jk. If the (j, k)
cell has a parallelogram shape, then it is obvious that.αjk = 0. Let us define the quantitative measurefor a deviation of a given curvilinear grid from a parallelogram grid as the arithmetic mean:
.α = 1
(J − 1)(K − 1)
K−1∑k=1
J−1∑j=1
.αjk, (107)
where(J − 1)(K − 1) is the total number of the grid cells in a given grid. The anglesα1jk andα2jk werecomputed by using the definition of a scalar product of two vectors.
Fig. 5. The definitions of the anglesα1jk andα2jk.
28 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
5.1. Internal flow problems
As an internal flow problem we have chosen a problem of inviscid gas flow in a channel whose lowerwall has a circular arc bump. Since Euler equations are solved, slip conditions are prescribed at walls.This flow problem was proposed as a test case in a GAMM workshop of 1981 and is now often used toassess the accuracy of numerical schemes [32].
The curvilinear grid of quadrilateral cells was generated in the(x, y) plane numerically by aMathemat-icaprogram, in which we have implemented the multi-surface method [33] for numerical grid generation.
We have solved numerically the chosen flow problem for a wide range of the freestream Mach numbers:the subsonic, transonic, and supersonic flow regimes. The thickness-to-chord ratio of the circular arc is10% for subsonic and transonic cases and 4% for the supersonic case. Uniform inlet flow at Mach numbersM∞ = 0.5 (subsonic), 0.675 (transonic) and 1.65 (supersonic) is specified.
As a criterion for the numerical solution convergence to a stationary limit we have checked the inequalityRn ≤ δ, whereRn = maxj,k|(ρj,kn − ρn−1
j,k )/τ n−1j,k | is the maximum solution residual,n = 1,2, . . . , and
δ is a user-specified small positive number.In the case of Figs. 6 and 7, the valueθ = 0.95 was specified in (104). The valuesα1 = α2 = 0.5 were
taken in (21). The deviation.α proved to be small in the cases of curvilinear grids of Figs. 6 and 7, (a)and was equal to 0.560 and 0.127◦, respectively.
In order to check the accuracy of the obtained stability condition (103) we have performed the runsalso at the valuesθ > 1, which correspond theoretically to an unstable regime of computation by scheme(21). At the free stream Mach numberM∞ = 0.5 andθ = 1.1 the computation still was stable. However,at θ = 1.2 there began a rapid growth of the solution residualRn at n ≥ 5, and already atn = 11 theresidualRn exceeded the initial valueR0 by a factor of about 3. Thus, the stability condition (103) issufficiently accurate despite the fact that it was obtained from the linearized finite difference equations.
Fig. 6. Inviscid flow through a channel with a circular arc bump in lower wall: subsonic flow atM∞ = 0.5: (a) curvilinear gridof 50× 15 nodes; (b) predicted Mach number contours; (c) predicted Mach number profiles along lower wall (solid line) andupper wall (dashed line); (d) convergence history.
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 29
Fig. 7. Inviscid flow through a channel with a circular arc bump in lower wall: supersonic flow atM∞ = 1.65: (a) curvilineargrid of 80× 40 nodes; (b) predicted Mach number contours; (c) predicted Mach number profiles along lower wall (solid line)and upper wall (dashed line); (d) convergence history.
The results presented in Fig. 6(b), (c) and Fig. 6(b) and (c) agree well with those of [32]. In particular,the maximum value of the Mach number on the lower wall is 0.698 in our case atM∞ = 0.5; and in [32]it is equal to 0.700 on a mesh of 224× 56 nodes.
In the case of supersonic flow atM∞ = 1.65 (see Fig. 7, two oblique shocks are generated at bothleading and trailing edges of the airfoil; these shocks are well visible in Fig. 7(b) as the subregions,in which the different Mach number contours coalesce. An expansion fan caused by the convex airfoilsurface is formed between the leading-and trailing-edge shock waves.
5.2. External flow problem
As an external flow problem we have chosen the problem of transonic flow around the airfoil NACA0012. The functiony = f (x) specifying the form of this airfoil may be found in [1]. We have used aC-type curvilinear grid for this flow problem. We have used two different methods for numerical gridgeneration.
In the case of the grid shown in Fig. 8 we have used the method of [34], in which the C-type grid wasgenerated with the aid of the transformation
x = B + A cos h(η) cosξ, y = A sin h(η) sinξ. (108)
HereA,B are constants. The coordinate linesη = constant wrap around the airfoil, whereas the linesξ = constant are approximately normal to the linesη = constant.
The grid shown in Fig. 9 was obtained by the multi-surface method [33]. The both grids of Figs. 8and 9 have the same number of grid nodes in each coordinate direction; in addition, they have the same
30 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
Fig. 8. The curvilinear grid around the NACA 0012 airfoil obtained with the aid of transformation (108): (a) the complete C-typegrid of 65× 15 nodes; (b) the partial view of the grid.
number of grid nodes on the cut line downstream of the airfoil. The deviation.α (Eq. (107)) proved tobe smaller in the case of the grid of Fig. 8:.α = 4.461◦; and.α = 5.651◦ in the case of Fig. 9.
The far field boundary conditions were implemented as in [4,5]. Several variants of the treatment of theboundary conditions on the airfoil surface were discussed in [5]. We have implemented for the velocitythe slip conditionun = 0, whereun is a velocity component normal to the airfoil surface. For the bodypressurepb we have used the extrapolation formula [5](pb)j = pj,1.
Similar to [4] we have considered the problem of transonic flow around the NACA 0012 airfoil atzero degree angle of attack and the free stream Mach numberM∞ = 0.80. We have taken the valuesα1 = α2 = 0.6 in (21);κ(2) = 1
5, κ(4) = 1/200 in (21); andθ = 1.0 in (104). The chosen value ofκ(2)
obviously satisfies the inequalities (71) and (78).
Fig. 9. The curvilinear grid around the NACA 0012 airfoil obtained by the multi-surface method: (a) the complete C-type gridof 65× 15 nodes; (b) the partial view of the grid.
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 31
Fig. 10. The computational results for the problem of transonic flow around the airfoil NACA 0012 atM∞ = 0.80 and zeroangle of attack: (a) the distributions of the pressure coefficientCp along the upper (—) and lower (· · · ·) airfoil surfaces; (b)convergence history; (c) the Mach number contours; (d) the surface|(ρn
j,k − ρn−1j,k )/τ n−1
j,k |, n = 800.
We show in Fig. 10 the numerical results obtained by scheme (21) on the Rizzi mesh of Fig. 8. Themaximum residualRn dropped by five orders of magnitude during the first 800 time steps: from thevalueR1 = 6.29× 101 to R800 = 3.90× 10−4 (see also Fig. 10(b)). The results agree well with thoseobtained in [4] on a 64× 32 mesh. Let us denote byx the non-dimensional coordinate masured alongthe airfoil chord from the airfoil leading edge, wherex = 0, to its trailing edge, wherex = 1. Let usfurther denote byxs the shock location determined as the point where the gradient|∂Cp/∂x| reachesits maximum. We have found in this way from Fig. 4 of [4] thatxs ≈ 0.48. According to Fig. 10(a)xs ≈ 0.50.
It can be seen from Fig. 10(d) that the main source of the numerical solution error is caused by theshock waves near the upper and lower airfoil surfaces. These errors decay with the increasing distancefrom the airfoil. This may be explained by the fact that the shock strength decreases as the distance fromthe airfoil increases. As a result, the numerical solution gradients within the zone of smeared shock wavebecome smaller, which leads to the diminution of the residual|(ρn
j,k − ρn−1j,k )/τ n−1
j,k |.In order to check the accuracy of the stability condition (103) we have performed the runs also at the
valueθ = 1.17, which corresponds to an unstable regime of computation. It is clearly seen in Fig. 11 that
32 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
Fig. 11. The Mach number contours after the first 15 time steps (α1 = α2 = 0.6, κ(2) = 1/5, κ(4) = 1/200): (a)θ = 1.0; (b)θ = 1.17.
Fig. 12. The computational results for the problem of transonic flow around the airfoil NACA 0012 atM∞ = 0.80 and zeroangle of attack: (a) the distributions of the pressure coefficientCp along the upper (—) and lower (· · · ·) airfoil surfaces; (b)convergence history.
V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35 33
the numerical instability develops atθ = 1.17. This instability causes the numerical solution blow-up atan attempt at a further continuation of computation.
Note that the computation atθ = 1.1 is still stable, although some of the Mach number contours arealready not so smooth as in the case whereθ = 1.0. Thus, the minimum value of the safety factorθ ,at which the numerical instability begins, lies in the interval 1.1 < θ < 1.2. This is exactly the sameinterval, which we have obtained at the computations of internal flow problems considered above in theforegoing Subsection.
In Fig. 12 we show the numerical results of the solution of the same flow problem as in the case ofFig. 10, but this time we have used the mesh of Fig. 9. We have taken the valuesα1 = α2 = 0.6 in (21);κ(2) = 1/5, κ(4) = 1/200 in (19); andθ = 0.9 in (104). It can be seen from Fig. 12(b) that the maximumresidual again reduces by five orders of magnitude after the first 800 time steps. The pressure coefficientdistribution of Fig. 12(a) is similar to that of Fig. 9(a).
At θ = 1.0 in (104), the computation was stable untiln ≈ 150. After that, a growth ofRn was observeduntil n ≈ 200. Atn > 200, the residualRn oscillated near a certain constant value. We have printed out theindices(j, k) of a grid cell, in which the solution residual achieved its maximum. These proved to be theoutflow boundary cellsj = 1 andj = J . On these boundaries we have used for the pressure computationformula (33) from [4], which involves the user-specified parameterα. We have made several attemptsat varying this parameter, and we have found that its value generally affects the stability: the increasingvalues ofα improved the stability robustness. One more conclusion is that the stability of computationnear the outflow boundary depends on the local grid geometry near this boundary. For example, in thecase of the mesh of Fig. 8 the computation near the outflow boundaries was more stable than in the caseof the mesh of Fig. 9. Taking the valueθ = 0.9 made the computation stable also in the case of the meshof Fig. 9. The fact that the computation on the mesh of Fig. 8 near the outflow boundaries atθ = 1.0was not so stable as in case of the mesh of Fig. 8 may be explained by the fact that the grid cells near theoutflow boundaries deviate more significantly from the parallelogram shape in case of the mesh of Fig. 9than in the case of Rizzi’s mesh of Fig. 8. And the stability condition (103) has been obtained under theassumption that the grid cells have the shape of parallelograms.
6. Conclusions
We have investigated the stability of a Runge–Kutta finite volume method in the presence of artificialdissipation terms. Our conclusion is that the incorporation of artificial dissipation terms according toscheme (21) leads to a reduction of the maximum time step allowed by stability if there are strongshock waves in the flow problem. In cases when there are no shocks or the available shocks are weak theincorporation of artificial viscosity into the Runge–Kutta finite volume scheme does not impose additionalrestrictions on time step in comparison with the case of the absence of artificial viscosity.
Since the stability condition (103) was obtained under the assumption that the curvilinear grid cells arethe parallelograms, it would be natural to generalize formula (103) for the case of non-parallelogram cells.Our analysis shows that in this case, the set of six non-dimensional variables (55) should be augmentedby three more variables of similar form. Together with the four parameters (57), one has then to obtainthe points of the stability region boundary in the 13D space of(κ1, . . . , κ13) points.
One more direction of future extension of the stability analysis method presented in [26] and in thepresent paper is to study the effect of the implicit averaging of the difference solution residuals [5]. This
34 V.G. Ganzha, E.V. Vorozhtsov / Mathematics and Computers in Simulation 58 (2001) 1–35
averaging enables one to relax significantly the limitation for time steps. In terms of the Fourier stabilityanalysis, the incorporation of this averaging into the Runge–Kutta Euler solvers will result in a morecomplex Fourier symbolG. It is important to stress that the symbolic–numeric method developed by usis universal because it can be applied to the finite difference or finite volume schemes involving not onlydiagonal artificial viscosity but also much more complex non-diagonal vector matrix viscosities.
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