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UNIVERSITY OF ILLINOIS 
 
 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 
 REPORT -NO. 69 
 
 STABILITY OF A SYSTEM OF EQUATIONS 
 DESCRIBING THE FLOW BEHIND A SHOCK 
 
 By 
 M, Suzuki 
 
 September 15, 1955 
 
 This work has been supported by 
 the National Science Foundation 
 under Grant G-1221 
 
We shall investigate the stability of a solution of the following 
 system of differential equations which arose from the investigation of shocked 
 waves by Professor Taub: 
 
 ^ = A iJ ( ^" C j } V«l*2 
 
 JL£ _ JT-_1 c ?£l 
 g T - 2 3T 
 
 fr = — (1+ 2 m Jar-— r CD 
 
 ax 
 
 $T " 
 
 95 2 
 
 COS (J. r 
 
 3s 
 
 Ba 
 
 sin u ^ 
 
 _2 
 
 ^T \ as 
 
 Here and in the sequel we shall use the same notations as in a paper "Deter- 
 mination of Flows Behind Stationary and Pseudo-stationary Shocks" by Professor 
 Taub. 
 
 To discuss the problem the most important coefficients in this system 
 are A. . (i,j = 1,2) and they are given by (5„ll): 
 
 2 
 / Bin 11 m cos ii 
 
 V v'^A i L M .J (2) 
 
 X(l - m cos uj ['-'-" — o> cos ^ sin I 1 
 
 \ m 
 
 The characteristic roots of this matrix are the solutions of 
 
 + 2 M 2 sin u 1 . n 
 
 Wn 2 2 x x * .2- 2 2 » U ' 
 
 \(1 - m cos mJ a. (1 - m cos uj 
 
 namely 
 
 sin u cos u ■/ 2 , . 
 
 a + ■- ~Z 2~T7 i — 2— TT V» - 1 (3) 
 
 — \(1 - m cos |i) \(1 - m cos p.) 
 
 -1- 
 
We may write our system of equations (l) in the following form: 
 
 3x, 
 
 a x 
 
 - — = a. . ^— u + b. 
 3n ij 3v 1 
 
 (i,j = 1,2,. ..,6), 
 
 w 
 
 where a. 4 and b. are known functions of x, , .... x c 
 Id I 7 o 
 
 x 2_ = -T ij x 2 = 2' X ? = C ' X h 
 A 
 
 m, x 5 = X, x 6 = a, u 
 
 (a..) = 
 
 11 
 A 21 
 
 2 C 11 
 
 i/, jr-i 2%, 
 
 12 
 
 22 
 
 m 
 
 11 
 
 B cA 
 
 2 ° A 12 
 
 i,_ r-i 2 U 
 
 - -(1 + -V" m )A 
 
 m 
 
 (V- 
 
 
 
 
 2 
 
 COS (J. 
 
 sin u. 
 
 12 
 
 -A, ,C . 
 
 2j J 
 
 id - >a m 2 > Al .c. 
 
 
 
 
 Our case will be 
 
 T, v = s and 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (5) 
 
 The characteristics values of the matrix (a. .) will be Ct and four 0's. 
 
 ij + 
 
 Suppose that there are variations Ox. of the solutions. Then these 
 
 dx. satisfy the equations 
 
 : 
 
 du 
 
 a& 
 
 3b. 
 
 >x . da. . dx . 
 
 = a. . -sr- 11 + -^ ^ 6x, + ^ Sx, + ~^- -^ Jx, 
 ij dv 3x k ^v k 3x k Is. ^x k 3v 
 
 da. . ddx 
 
 (6) 
 
 -2- 
 
We assume <$x. takes the form 
 
 Sx. = Jx° e^^^v) 
 
 (7) 
 
 where a is real. Our system of equations (h) will be stable if the imaginary 
 part of £ is < 0. We shall consider each x. to be fixed, so that each co- 
 efficient a. . (and b.) is a constant, and a is sufficiently large, so that 
 
 1J X 9a. . dx. db. 
 the magnitudes of ~ — ■* ^ " ■ and 5 — are negligible compared with a, 
 
 k k 
 
 Substituting (7) into (6), we get a system of linear equations between a andx 7 
 
 with constant coefficients: namely 
 
 We have used an approximation that dx, are so small that their square may be 
 neglected. From our convention a is so large, and hence (8) may become 
 
 a<fe? =/a. .£c° . (8') 
 
 Hence CK^ must be one of characteristic values of the matrix (a. .), i„e„ 
 
 + 
 In our case 
 
 A= <i\(sin (J. 4 cos M-Vm - 1 ) . (9') 
 
 Hence the system (l) _is stable if m ^ 1 . 
 In case the flow is subsonic the investigation failed to produce any definite 
 statement on the stability of equations. 
 
 In the following we shall discuss the similar problem for the 
 difference equations derived from (l). Again we shall take our equations 
 in the form (h) . 
 
 -3- 
 
dx. dx. 
 
 Replace = — and -5 — by the difference quotients 
 
 dx. x.(u +A\i,v) - x.(u,v) 
 
 du Au 
 
 dx. x.(u,v +-^v) - x.(u,v) 
 
 £V 2 V 
 
 and use the notations such as 
 
 x i'^ = x i (n4u,i^v). 
 Our basic difference equations will be 
 
 n+1,/ n,X ni+1 n,*? 
 
 x. - x. x / - x . 
 
 — — 71 — = a, . J . A ^— +b.. (10) 
 
 We shall consider the variations ox. of the form 
 
 1 
 
 %* = c(x°s n f *, where ■ = f 1 aCa , /= e^ 1 ^ v , (11) 
 
 a is real and is to be assumed sufficiently large. The equations (10) will be 
 stable if | J I ^ 1 for all (large) values of a. Substituting (il) into (10) we 
 get 
 
 da. 
 
 6^>* - bx n ' X = a. .tf x y +1 - 6^)^ + ^ ^^(x n ^ +1 - x n ' 7 ) ^ 
 i i iJ J J ^v 5x k k j j Av 
 
 3a.. /? „/?.-, ,„^yi <^b 
 
 -^ dx' {6x' K - d~x. ) r— + o— x,,' /\u. 
 
 Jx^ k j j Av o?x k k 
 
 The principal term of the right hand side is the first one as^du,^Jv-> keeping 
 <du/dv in a finite value. Hence again using (ll) 
 
 -4- 
 
Zlv s - 1 ( o ( o 
 Z)u £ - 1 ^ 1 U^j 
 
 (12) 
 
 This relation may be obtained from (8') since (8') gives an approximation of 
 variations when a is large. The relation (12) shows that —r~ is a 
 
 characteristic value of the matrix (a. .) 
 
 z 1 %^-7«a 
 
 /}u J- 1 + 
 
 Au ^- 1 
 
 The cond 
 
 ition for the stability is £ <. 1 for all s with s = 1 
 
 Now 
 
 5 = 1 h — t^(s - l), so that the above condition is satisfied if and only if 
 
 a S> andZ)v/du< a . Hence our difference equation is stable if m^l and 
 
 i 
 
 Zlv ^ sinu. - cosu. \( m -1 
 
 ^ u ^ ' yfi 2 2 \ 
 
 a.(1 - m cos \i) 
 
 (13) 
 
 The positiveness of the right hand side is guaranteed for a shock of non-zero 
 strength, since in that case we have 
 
 2 2 I 2 
 
 1 - m cos u. = (sinu- + cosu. Vm -l)( 
 
 sinu - cos 
 
 \i \]m -l) 
 
 >0, 
 
 If the flow is subsonic: m < 1, then the system (10) of difference equations is 
 unstable, i.e. the solution does not converge to a solution of our original 
 equations (l) whenZlu, Av — * 0. 
 
 There are many ways to set up the difference equations corresponding 
 differential equations. Integrating both sides of (k) with respect to u and 
 applying the trapezoid rule to the right hand side, we get 
 
 n+1 n <4u 
 x. - x. = — 5 a. . 
 
 2 /Lij3v 
 
 n+1 
 
 9x 
 
 Au/. n+1 
 ~(b. 
 
 /,n+l ,riv 
 (b- * " *>,), 
 
 r a. . . -. c 
 
 L i«j <5v J J 
 where x. = x.(n£u,v), b. = b.(x, , ..., x, ) etc. Considering cfx. of the form: 
 
 (i*0 
 
 -5- 
 
v-1 a u 
 
 dx. = ox.s e where s = e ' , we get a difference equation for <£x. such as 
 
 o ^u 
 
 (s - 1) 6x° = f*(X.. +/Y..)£c°, 
 
 i 2 i,i ^ i,i ,w y 
 
 (15) 
 
 where X. . = 
 
 5b 
 
 3x, 
 
 n+1 
 
 s + 
 
 3b." 
 
 <px. 
 
 '9a. . 9x, 
 
 LI ^ 
 
 <9x £v 
 
 J 
 
 n+1 
 
 s + 
 
 ^ik^k 
 £x^ ^V _ 
 
 n 
 
 ■■ • - W 1 
 
 n+1 
 
 s + 
 
 ij 
 
 n 
 
 In deriving (15) we assumed that dx. are so small that their square may be neglected, 
 The stability condition for the system (l^+) is that the real part of^ is less than 
 or equal to whenever | s| =1. In the actual form of (15) it seems difficult to 
 give an explicit condition, so we assume here that^u has been taken so small that 
 we may replace (15) by 
 
 2(s - l) ^x.^Z^uCs + l)a. .qx 
 
 
 (15') 
 
 If this approximation is permissible, then we can solve A as a function of s: 
 
 X> = 2. ± E - i 
 <° ^u a s + 1 ' 
 
 where a are characteristic values of the matrix (a. . ). Since 
 + • U 
 
 ( s - l)/( s + l) is purely imaginary „ In fact 
 
 = 1, 
 
 s - 1 i | 1 - s| 
 s + 1 " 2 T(s) 
 
 Thus the real part of^ ^ for all | s| = 1, if and only if a are real, i.e. 
 m >.l. 
 
 -6- 
 
Summary 
 
 The system (l) of equations is stable if the flow is supersonic. 
 I do not know whether stable or not in the other case. The similar stability 
 conditions for the corresponding difference equations are examined for two 
 kinds of systems. In both cases the systems are stable if the flow is super- 
 sonic, but unstable in case of subsonic flows. 
 
 MS /he 
 
 
 -7-