AN ALGORITHM FOR THE NUMERICAL SOLUTION BY THE MESH METHOD OF THE CYLINDRICAL BLAST PROBLEM, ...

8/13/2019 AN ALGORITHM FOR THE NUMERICAL SOLUTION BY THE MESH METHOD OF THE CYLINDRICAL BLAST PROBLEM, AL

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AN ALGORITHM FOR THE NUMERICAL SOLUTION BY THEMESH METHOD OF THE CYLINDRICAL BLAST PROBLEM,

ALLOWING FOR BACK PRESSURE*N. A. AKKH.L\NGEL'SKII

MoscowReceiued 22 J une 1970

AN algorithm is given for the numerical solution of the cylindrical problem by themesh method in Lagrange variables. An implicit computational scheme isemployed.

Two types of scheme are used in computations of ideal gas flows involvingjumps: schemes with artificial dissipative terms simulating the properties of theactual physical medium, where the jumps are manifested automatically, andschemes in which the isolation of singularities has to be provided for in advance.Both types of scheme have been used to solve the spherical blast problem in itsnon-linear statement D-31.

The method of integral relationships differs from the above [4]; it has beenused for solving both the spherical and the cylindrical problems [5, 61.

The present paper describes an algorithm for the numerical solution of thecylindrical problem by the mesh method in Lagrange variables. An implicitcomputational scheme is employed.

I. Formulation of the problemThe explosion of an infinite straight cord occurs at the instant t = 0 in a

perfect ideal gas at rest with constant pressure P, and constant density R,, andis accompanied by the release of a large amount of energy and a rise in thepressure and temperature. The resulting strong cylindrical shock wave travelsthrough the gas at a speed C, w ,. . separating the medium into a disturbed and anundisturbed part. The gas movement is cylindrically symmetric, i.e. onedimen-sional. The problem will be stated theoretically under the usual simplifying -.-*Z/z. u?chisl. Mat. mat. Fit.. 11, 1, 222-236, 1971.

289


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290 N. A. Arkhngelskii

assumptions; in particular, the release of finite energy E, per unit length isassumed to occur instantaneously, while changes in the gas properties at hightemperatures are ignored.

The equations of motion and continuity in Lagrange variables arePR 4 p-e-m-

(4.1) atz 5x dR9i R$= l OS,where s is the Lagrange coordinate of the particle, R (s, t) is the Euler coordinate,P (s, t) is the pressure, and % (s, t) is the density. The Lagrange coordinate ischosen in such a way that R (ss w., t) = s, w on the shock wave. Since heatexchange is ignored, the gas flow is adiabatic with the equation of state

P = S(S)W~,where 6 6) is the entropy function and y Poissons adiabatic index. Since theprocess is reversible,

(1.2) $0(S)). = +w$ = 0,where the subscript s ipdicates that the entropy S is constant for all particleswith the given Lagrange coordinate s. When the shock wave passes through aparticle, its entropy increases; this is bound up with the representation of theideal gas flow as the limiting flow of a viscous heat-conducting gas.

Letu= aR/at.

be the particle velocity. Equations (l.l), (1.2) may then be replaced byR ap(1.3) = --- seas as 7 yZRg=%$s, ;= U,g(Pe) = 0.

To avoid being concerned with specific values of the problem parameters E,, P,,R,, we transform to dimensionless variables in (1.3). according to the expressions


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R,where(l.c;j Eo = a(y) E,and the coefficient of proportionality a 0,) is determined from the solution of therelevant similarity problem 171. In these dimensionless variables, Eqs. (1.3) canbe transformed to(1.7) g+A$=O, z+R$+n=O, A%&h P

$w) = 0,where 4 = r/v: B = IA, ll = ypu / r, I = a.p (acoustic impedance), andu = T(1.p /p) is the velocity of sound. The set (1.7) describes in dimensionlessvariables a smooth cylindrically symmetric motion of the ideal gas. Beforeintegrating the set, boundary and initial conditions must be stated.

1. Boundary conditions. a. At the centre

(I.81 u(0, T) = 0, r(0, z) = 0.b. At the shock wave (moving boundary)

(I.91 r(r s.w.4 = rl S.W.and we have the Hankin-Wugoniot relations

where the dimensionless velocity of the shock wave cs w is the required functior:L. .of r.2. Initial conditions at T T To. These are


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292 N. A. Ar khmgel skii

1.11) u = 4% TO), F = P (rl* Go3 r = rhz), (3 = P(17.O),where O


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Algorithm for the numerical solution 293

r m = h&.z(L), Pm = mm)%and we can take as the initial conditions (1.11):

(1.15) u(rhn, zO) = &I, P (%, To) = PVI, r(VW rO) = r7Jz,P (rim, To) = pm.

Let us define the parameter 7. The Rankin-Hugoniot relations represent threeconnections between four quantities. If the value of any one of them is specified,in such a way that the similarity set gives a reasonably accurate description ofthe actual physical picture of the blast, the other three can be found, then r andV o=rg.W.o.W. can be found from

which hold for the similarity motion. As a criterion for the suitability of thechosen value, we take the condition that the density p (q7,_WI), ) on the shockwave differ by a reasonably small amount from its constant value g (1) = (y+ l)/(y - 1) in th e similarity case. Such a choice was made in [31. Otherchoices are also possible [l, 81.

To sum up, to find the required functions u, p, r, p, and cs w in the regionQ={O0lCW), 9 < c < T < DO} we have to solv;? the (mixed)

boundary value problem, i.e. integrate (1.7) with boundary conditions (1.8)~(1.10)and initial conditions (1.111, obtained approximately by virtue of (1.15). Knowingthe solution as a function of n and T for a given y, all the quantities can be foundfor any other values of the parameters E,, P, and so. ,4 change in y demand anew computation of the entire problem.

Expressions will be needed later for the asymptotic behaviour of thesolution of (1.7) in the neighbourhood of the centre, at which it has a singularity.If the usual assumption is made, that a change in the boundary conditions on theshock wave does not change the type of the density expansion when account istaken of back pressure, i.e. the asymptotic behaviour of the density is assumedto hold for the non-linear problem, it can be shown that the similarity asymptoticexpression must also hold for the remaining functions [7, 81. Asymptoticexpressions were used for computing a spherical blast in 11, 31.


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294 N. A. Arkhangelskii

Starting from the density expansion

in powers of the Lagrange variable A, with constants x, n and 6, = ai tY), i1, 2,. . . ) expansions of h (A), 1 (A) and f(h) can be obtained from the similarityset (1.131, after which the asymptotic behaviour is obtained for the set (1.7). Inits final form, the asymptotic solution of (1.7) is described by the expansions

U(%Z) = Pi(1.16)

~(l+s~A+Ir,hl+...).

r(q,-c)= &WL-/~(~ + &h2 + &A +. . .),P(9,)=$(1+vlhl+y~~L+...),p(q, ?) = 6iP(13_ 62A.2 633L+ . . .))

where h = v/~T, and the coefficients have the values

2y---vi=sygs 6 VI2=--r ~*=-~YY 2 2y-1112= &(2Y---), va = 2yv&,

b2 -- E263=8&2, Es= 2 , P3=--%(4Y-1).By using expansions (1.16). the initial data (1.11) at and in the neighbourhood

of the center of symmetry can bc cvaluatcd. rhey were employed in a modifiedform, to be described in the next section. in the computations.

Assuming the existence of a generalized solution of the mixed Cauchyproblem abovcl stated (it,s uuiqucness is ensured by the condition that the entropyincrease [9])% it may bc found by the mesh method; for this purpose, the differen-tial problem is wduced to a ditfcrtwce problem.

3. Xeduction of the differential boundary valueproblem to a finite-difference boundary value

problem on a meshSuppose thilt the shock wave has travelled a distance qz.w. from the centse

at T Y TO Fig. 1 ). \\o illtroduw thr) space step


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(2.1) *q = *;where XI is an integer. Let qtlm = mA,q, m = 0 . . , M be the points ofdivision of the interval LO, T]~.~_]. Since dn,.,,_./dr = cS ,w.,

(2.2) AT =$+O[(Aq)'] .S.W.

Neglecting the second-order terms, this relationship will in fact be taken as thealgorithm for finding the time step 3~. The straight lines r = rn, n = 0, 1,. . . , N,will be call4 layers. The set of base-points (n,, rn) forms a mesh, on whichthe mesh functions Fl(q17, z) = W, will be considered. The interval oflength qr = IAq, I? < M, next to be centre, will be termed the centralinterval.

PIG. 1.

When the time step is given by (2.2). the shock wave will always pass, withan acceptable error, through the computational base-points of the mesh. Thistype of algorithm was originated by M. V. Keldysh and realized earlier in [3].

The mesh region Q(*) = (0 < qrn < nnrl,., T? < tn < -C}, in whichthe solution is sought, thus splits into two regions: the central region Q,nn,consisting of the central intervals, and the interior region s i,T) = fi2(9- $11). In Q, tn the values of functions at points other than base-points willbe found by interpolation, while in QL*n extrapolation of their values at thebase-points r, r + 1, r t 2 by means of asymptotic expressions is used. Let usnow reduce the differential boundary value problem to a finite-difference problemfor the mesh functions (u, p, r, p ,, in Q*n; for this, the differential equationsand boundary conditions are approximated by finite-difference analogues. Thedifference scheme will be written with respect to four points, while the actualderivative is taken at the centre of the rectangle:


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296 N. A. Arkhngelskii

(2.3)

(WZ - w:+) + (WZ+1-Km)2Arl ,

(2.4) n+i-Wm+J (w,+ - Wm)28-r

W n+lm+% - l/&(Wi=: wi + WC.1 + Wmn).This scheme is well known to be stable whatever the ratio Ar/Aq, nd is ascheme of the second order of accuracy. On replacing the partial derivatives in

FIG. 2.

the first two equations of (1.7) by finite-differences in accordance with theserules, we obtain the difference set

- U,,l)+ (u;+i z&n)$,_-_2A2p$ (PZ - P:+i ) + (P?z+1- Pm") _ ()

2Atl 9


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n+ln+h+ Bm+m(u*+* - z&E++ (d,, - Urn)2611= _ $y

m+ *

Using (1.2), the fourth equation of 1.7) becomes on the mesh

(2.7)

The third equation of (1.7) may be written on the (n + l)_th layer as

(2.8) (CI) 2=2gi,where the integral on the rigi& will be evaluated by a quadrature formula.Approximation of the third equation of (1.7) by a difference equation in accordancewith the rule (2.3) leads to the entire scheme becoming unstable, since theresulting explicit difference equation demands for computational stability a ratioArjhrl, which violates condition (2.2). It follows from Tsemplens theorem that,if condition (2.2) holds, then the Courant stability criterion is certainly violatedon the wave, i.e. a stable scheme cannot he explicit [91.

Let us turn to the approximation on the shock wave. Take the parallelogramcell on the wave (Fig. 2) and replace the partial derivatives in accordance withthe rules

which have the same order of accuracy as (2.3) and (2.4). The first two ofequations (1.7) on the wave then be written as

n+1 II(2.*1) UM /1, uhf +&+% (PZ - PM 2Arl)S(pj: -pL) = o

,

a+1 ni2.12) pM A; pM + B,+ (u,,:-uM +)+(u~-u~-- L) = _n,*2Arl


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298 N. A. Arklrmg~~lskii

With these equations we associate the conditions (1.9) and (1.10) on the shockwave, which are written in an obvious way on the mesh, and also the set (2.5)-(2.8) with m -: Al - 1. On eliminating $1 and pltl from these equations bymeans of a pivotal condensation relationship, which will be obtained below,nine equations are obtained for the nine quantities (u, p, r, p)$i M and c:;.

nt l . .Knowing (u, p, r, plM , their values at all other base-points of the layer con-sidered can be found. A boundary value problem is thus obtained for the quasi-linear difference set (2.5H2.8) with linear boundary conditions at m = 0 andnon-linear conditions at m = 111. It has to be determined whether this problem iswell posed and what method can be used for solving it. First consider themethod of solution.

We find u,$: and u: from (2.5)-(2.6):

(2.13)

(2.14)

where

Replacing m by m - 1 in (2.13) and eliminating u+I from (2.14), the differencemequation(2.15) i, +- ,: n+i + (& + F;+-;)p,:+i+ y::ti;,p:,_: = S:,+_ h,18, ,5p,,t;iis obtained, where

a,::: = e$p:+, +(r:,E + r:%: + &JL -2[U,,:;-1 - ZL + (Y,+SI+ +~:?,z)An].

Equation (2 15) is in fact a set of III - 1 linear algebraic equations in the MunkI10w11s pon- l .i 11 L%I. I... , pA\I-- . To complete the set, we satisfy the first boundarycondition (1.8) by setting m 0 in (2.14), as a result of which we obtain thecondition


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Al gorit hm for the numeri cal solut ion 299

Equations :2.15), (2.16) have a triangular matrix and may be solved by the pivotalcondensation method in the form described in [lOI. We introduce into (2.16)

n+li6% n+%= pin - poay2 -

and consider

(2.17)

instead of (2.16). Applying induction, PC-: = a,z_;zp, n+l+ pi:; is obtainedfrom (2.17). On eliminating p,_rtl from this and (2.15) and solving the resultingequation for pk+ l, recurrence relations are obtained for computing the pivotalcondensation coefficients (direct pivotal condensation):

(2.18)

where CI{ and p, nt s are defined above, and the reverse pivotal condensationexpression

(2.19) pi+, = a~ -bZ + S2 , m = M - 1, M - 2,. . . , I + 2.

Computation by this method is stable because all the C&Z are uniformly boundedon the layer, in fact, Ju~+~i Oandq>OinQ(*n). Let Ial_.s( < 1, then


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since 1 y / salt-55 < 1. Assuming that CL,+,/ depends only on m and not onthe required functions, (2.18) can be varied. Up to first order terms in thevariation &z,,,_~, we obtain

Hence the error m computrng a,,,+ % does not increase with m and the sweep isstable.

Now consider the computation on the wave. Eliminating us; from (2.11)and (2.12), the result can be written as

(2.20)

K, = (2x - /zq) ,+,Putting m = m - 1 in (2.13) and (2.19), and recalling (2.11), a set of equations isobtained from which z$ l and pi_: can be eliminated. The result is(2.21) - q:+p::i + La&+ = T,,

From the conditions (1.10) on the shock wave,

(2.22)7l+iII+1 pBf+t - 1

U+i = -0 (Y + l)J572 +(y - I)/21


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Algori thm for the numerical solution 301

where the radical is taken with the + sign, since we are concerned with acompression shock wave (ps w > 1). Equations (2.2OH2.22) form a non-linear. .set of equations in u;yi, p,,+:, pMfl. It can be reduced to a single equation inp+iS.W. by using the first of Eqs. (1.10); the result is the cubic equation

(2.23) Ga~c;.,) LMC~;W_ (Y+1)&+(y---)Gx c2 8 .wT YLM = 0,

where

G += (L ,/2 + K,) ,= qna H = Q M - Tar.I t will be shown that this equation has a single, physically meaningful, positiveroot. Considering the coefficients I ,,, G, and H, in Q*n), we have

L, = Xlr_%(i + aBe-yJ+QM--h(i a-a)+ fly >min {XM-5, QM--ya}+ Qna> 0,

Descartes theorem on the number of,positive roots is now applicable to (2.23),whence it follows that cS w is in fact such a root.. .

Equation (2.231 will be solved by the method of tangents. Writing cz:d. = x,and using the tangents formula we have

(2.24) zi+ = 2(GJ L&r? + xiz + Y3(G,/L,)~+2zi-[(y+~)H~+(v--)~~]/~~~i=O,,l,...,

where x, = ci w . The fact that it is well posed in the neighbourhood of the. .required root follows from the following inequality for the denominator:(y + l)H, +(Y - ~)GM39,: + 2Xi-- =M 2LM

:[3x, - (c, :jz] + 2% - i;;;j- j- > 27%.M 8 WIt will be assumed in this case that


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= qM -I- Aq from (1 .lO). For u$ * and pit l we have the linear set (2.1 I ), (2.21).which is always solvable in view of the above bound for L,,. rhis gives U

I(225) ;+I= TN+ t+p;:: .

Nnt k

2.26) u;+~= una -Q::: - p:+ + pMn pL),

after which pbItl can be found from (2.7). Using (2.19), (2.14) and (2.71,(n, P* P) $5 n =M-&M--2 ,..., r+2. can now be obtained.

Let us now compute rz+l. We write (2.8) in the recurrence form1mt1(r:+y = (r:f:1)225 fdq, f=InI

With m = M, the following quadrature formula is used:2.27) riti) = r;:i)- Aq[f.sr+fw - (fi+i - 2fx + fs-l)].At the remaining base-points of the layer, the integral on the right is evaluatedfrom the improved trapezoid formula [ll]2.28) r,? ) = (rz-?d - Aq[fm -I- fm+l -i-%2(-fm-i + fnti + fm f743 m=M- ...,I?+2

Now consider the computation at the base-point r + 1, inside the centralinterval at base-points r, . . ., 1, and at the centre. On the n-th layer theasymptotic expressions (1.16) are(2.29) W(q,t.) = $W{U(T) + b(z)$ + c(GnN + - - *I*Consequently, in the neighbourhood of the centre, Wn-kWcan be written on thelayer as a polynomial in T]. Retaining only the first three terms in (1.16), andapplying the Lagrange interpolation formula, we obtain for the base-points nv,7vtl and v,+~:


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.An algorithm for the numerical solution 303

where the notation J: = $, k, = k, = (y - 1) / y, kp ==:2 / yp k, = 0.has been used for brevity. In particular, on putting kp = 0, then x = 0, the valuePOS+of the pressure at the centre is obtained.

FIG. 3.

Let us now describe how the required quantities are computed on the (n +1Mh layer, when their values on the n-th layer are known. Since the coefficientsof the difference equations depend on the required functions, the solution of thedifference set has to be found by iterations. At the first iterative step, thequantities are taken from the n-th layer in accordance with the scheme illustratedin Fig. 3, where x denotes linear interpolation at the mid-point, and 6~ denotesthat averaging has been ~rforrn~ in accordance with (2.30) with I,> : r. The firstapproximation &- is determined from C&2), then the pivotal condensation coeffi-cients ;im+ % and grn+% are evaluated. The values of q,,, uy2, and I,,,~ are foundfrom the following, obtained from (1.16) in which one term is retained:

We find ??:A from the equation (2.23) on the wave, and then (Z, p, p) lz:,M.Next, (2.19j, i2.14) and (2.7) are used to find their values at the interior base-points up to and including r + 2, I- + 1 and r. The quantity r, is computed inthe first iterative step from the centre to the shock wave as follows: on integra-ting (2.8) with m = r, r +- 1, r + 2 by means of (2.30) with k = 1 - 2/y, the


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expression

P(x)=Y{(x/xr+d (x/xr+2)+ czx/xr+i + x/xr+2) + cs x(Q/G+1 - 1) (Zr/Zr+z - 1) ( )i (-$ r +(2.31)G (x/n) (hk+J + G(x/xr + x/xr+z) + CJ

(Xr+z/Xr - 1) (Xr+zlXr+ - 1) ( &)(sNVr+2}is obtained, where C, = 1/(3y - I ), Cz = - 1/(2y - 1), Cz = l/(y - I ),V = X/P = rl//p. From this, taking x = xr, xr+r, ~r+~ we obtain respectivelyT?, rg+, r +* We then find Tcgl ;;I;, . . ., -$? from (2.28), after solvingit for r i. This concludes the computation of the first iterative step.

The second iterative step starts with averaging of the functions, i.+. findingtheir values at the mid-points of the computational cells in accordance with thescheme of Fig. 3; the + sign denotes the arithmetic mean of the four points, @the arithmetic mean of two points, and e the values at the mid-points of the cells,obtained from (2.30) with v = r + h.

We improve the time step:

and findzn+j = 7 + AZ.

The rest of the computation follows the same lines as in the first iterative step,except that the coefficients of the difference equations are now found in terms ofthe averaged values of the required functions. We compute rz+r from the shockwave to the centre, in accordance with (2.27) and (2.28). Knowing the values ofthe required functions at the base-points r + 4, I + 3, and r + 2, we can use(2.30) with v = r + 2 to find their values in the-central interval*up to and inclu-ding the centre, and thus terminate the computation of the (n + l)_th layer.

Now consider the check computation. Since the initial equations are writtenin Lagrange variables, the law of conservation of mass is satisfied automaticallyto the required accuracy in the finite-difference approximation; this follows fromthe expression d ($12) = qdq for the differential of the mass in Lagrangevariables.


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An algorithm for the numerical solut ion 305

Ihc energv check is performed according to the expression

a(r)I , + I2 = - +q 2.-x Y :;*Let us clarify this. According to the law of conservation of energy [71

The integral on the left is written as the sum of an integral I, from 0 to n,- andf, from nr to Q~.~.~ and the second integral is evaluated from the trapezoidformula

where

while the first is evaluated as foliows from the asymptotic expressions: theintegrand can be written in RLAn) in the form (2.29) with k = 1 - 2/y. Hence

is equal to the right-hand side of 12.31) with x = x,- and J = (u / 2 + p I (v -l)PIr12.

The ulocaln check was performed according to the relationship 8r q,, T) /fh = = vr, 4, where the time derivative was approximated by

lb- n( 1 1dz *= at1 + a)h (7-Y + a2 rln - r:- ) - riiz>h = z - ,f--, ah = T+l - g_

Since the number of computation base-points increases by unity on moving to thenext layer, as soon as the number of them on a layer has doubled the space stepis doubled and the length of the central interval is increased. The fact that


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r I can be evaluated in two ways (from (2.31) and from (2.30)) enables us bothto check the computational accuracy as regards the Euler coordinate, and toassess the degree to which the increase in the central interval is acceptable.

The fact that the difference boundary value problem stated at the beginningof this section is well posed means that it is uniquely solvable for (2.5)--(2.6),(1.8), and (2.25). Conditions for the unique solvability of a difference boundaryvalue problem obtained by approximation of any hyperbolic set were found in [12].All these conditions are satisfied for the case in question.

Since the number of iterative steps is finite when solving the difference set,it is actually a question of dealing with a new difference set, the stability ofwhich needs to be investigated afresh. It can be shown [12l that, starting fromthe second iterative step, the second order of accuracy is maintained, so that itis not necessary to make more than two iterations.

The algorithm here described was used for numerical solution by high-speedcomputer of the cylindrical problem with y = 1.4 and y = 4.

Translated by D. E. Browr

REFERENCES

1. GOLDSTI NE , H. and NE UMANN, J . Blast wave calculation, Communs. Pure andAPP~. Rfath., 8, 327-345, 1955.

2. BRODE. H. Numerical solutions of spherical blast waves, J . Appl. Ph.ys.. 26, 6,766-775, 1955.

3. OKHOTSIMSKI I , D. E., KONDRASHEVA, I . L., VLASOVA, Z. P. and KAZAKOVA,R. K . Calculation of a point explosion, allowing for back pressure, Tr. MIANSSSR, 50. 1957.

4. BELOTSERKOVSK II , 0. M. and CHUSHK IN, P. I . A numerical method of integralrelations, Zh. u3;chisl. M at. mat . Fiz., 2, 5, 731-759. 1962.5. KOROBE INI KOV, V. P. and CHUSHK IN, P . I . P lane, cylindrical and spherical

blasts in a gas with back-pressure, Tr. AfI AN SSSR, 87, 196ti.6. KOROBEINI KOV. V. P., CHI J SHKIN , P. I. and SHAROV~\ TOVA, K . V. Cas-

dynamical functions of a point explosion, VTs AN SSSR, Moscow. 1969.7. SEDOV, L. I . Dimensional and similarity methods in mechanics, (Metody podobiya

i razmernosti v mekhanike), 4th ed., Costekhizdat, Moscow, 1957.8. KOROBEINI KOV, V. P., WEL NI KOVA, N. S. and RYAZANOV, E. V. Theory of

the point explosion (Teoriya tochechnogo vzryva), Fizmatgiz, Moscow, 1961.


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An algorithm for thv nrrmeric*aL .soltrtion 307

9. RC)%llDESTVENSKII, B. L. and YANENKO, N. N. Stats of quasi-linvnr equationsand their applications to gas dynarnlcs (Sistcmy kvazillncAin,ykh urnvntnnil i ikhprilozheniya k gnzovoi dinamike), Nauka, Moscow, 1968.

10. GELFANL), I. M. and LOKUTSIEVSKII, 0. V. The pivotal condensation method forsolving diffcrvncc equations , ,\ppendix II to the book: CODUNOV, S. K. andRYABENKII, V. S. Introduction to Theory of Finite-diff6,rcnce Schemes(Vvedrnir v teoriyu raznostnykh skhem). Fizmatgiz. Moscow, 1962.

II. MILNE, W. E. Numerical cnl~ulus; approximations, interpolation, finite differences,numerical integration and curw fitting, Princeton Ilniv. Press, 1949.

12. BABENKO, K. I., VOSKRESENSKII, G. P., LYUBIMOV, A. N- and RUSANOV, V. V.,Space F ow of an Idcal (;as past Smooth Bodies tP&transtvennoe obtekanirgladkikh tel idealnym gazom), Nauka, RIoscow, 1964.

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