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1 This article was downloaded by:[universidad de Malaga] [Universidad de Malaga] On: 29 May 2007 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK International Journal of Computer Mathematics Publication details, including instructions for authors and subscription information: Inverse spline-interpolation for numerically solving a piston problem with isothermic gas compression and constant velocity F. Criado a ; T. Gachechiladez b ; H. Meladze b a Department of Mathematics, Malaga University. Spain b Department of Applied Mathematics, Tbilisi State University. Georgia To cite this Article: Criado, F., Gachechiladez, T. and Meladze, H., 'Inverse spline-interpolation for numerically solving a piston problem with isothermic gas compression and constant velocity', International Journal of Computer Mathematics, 71:1, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Taylor and Francis 2007

2 Inrrrn. J. Cornpurer Math., Val. 71, pp Repnnts avdable direcily from the pbhsher Thotocopying permitted by iicenx only 1999 OPA (Overseas Publishers Asscciat~on) N.V. Yublished by iissmr mdz: the Gordon and Breach Science Publishers impmt. Pnnted in Malaysia. INVERSE SPLINE-INTERPOLATION FOR NUMERICALLY SOLVING A PISTON PROBLEM WITH ZSOTHERMIC GAS COMPRESSION AND CONSTANT VELOCITY F. CRIADO ",*, T. GACHECHILADEZ and H. MELADZE a Department of Mathemutics, Malagu University, Spain; Department of Applied Mathematics, Tbilisi State University, Georgia (Received 25 September 1997; In final form 11 November 1997J The application of an inverse spline-interpolation method for solving non-linear algebraic equation systems, obtained when changing the differential equations of gas dynamics by the corresponding difference scheme, is considered. The iteration process constructed on account of inverse spline-interpolation ~ith the aid of p-linear splines for wiving a non-linear algebraic equation system may be considered as a generalisation of a known method used for solving equation F(x) = 0, based on the contraction principle. This process is presented in such a way as to facilitate its realization on abstract parallel processing systems (APPS). Keywords: Splines; interpolation; computer applications to physics C. R. Categories: G.l.l [Numerical Analysis]. Interpolation-Spline and Piecewise Polynomial Interpolation; G. 1.5 [Numerical Analysis]. Roots of Non-linear Equations - Systems of Equations, Iterative Methods; G.1.8 [Numerical Analysis], Partial Differential Equations- Difference Methods; 5.2 [Computer Applications]. Physical Sciences * Corresponding author.

3 84 F. CRIADO et al. 1. SOLVING A XON-LINEAR ALGEBRAIC EQL'AI'IOB SYSTEM BY INVERSE SPLINE-INTERPOLATION Let us consider the following non-linear algebraic equation system: Let it be known that there is one solution to system (1.1) m a p- dimensional region D, Let us consider the following expressions: We assume that existing conditions for non-explicit functions are fulfilled [I] in a 2p-dimensional parallelopiped with centre in (x y,..., xj; u 7,..., u:). Then relations (1.2) define XI,..., x, as unique functions of ul,..., up: As an approximation to an exact solution (x r,...,xi) of system (1. l), we can take (El,..., Xp), which are calculated as shown below: where S,,,..., S,, are p-linear spline-interpolating values of pl,... cp, functions at the knots of net Au, respectively. We can write the expression for a p-linear spline such as [2]:

4 where iu,,..., up) E (u!,,-l 5 ul 5 MI,,,..., rip,-1 < up 5 up,). from (1.5) we can see that for calculating an approximate solution (Xi,..., Xp) to system (1.1) by formulae (1.4) it is necessary to know the values of nonexplicit functions pl,..., yp at the fixed knots of the net. These values are calculated using the following formulae: where jx y,.... x $) is a given point and correspondingly The deduction of formulae (1.6) is given in Chapter 2. Let us construct the iteration process based on the inverse splineinterpolation idea for solving system (1.1). Let us consider a point (xy,...,x;) E V and lei cube (-uy I: ul 5 uy,..., -uj < up 5 u j) with centre in the origin of the co-ordinate system be contained in the definition region of non-explicit functions. On such a cube we construct splines SO,,,..., Let us assume that Correspondingly we have: On cube (-u! _< ul 5 ut,...,-uj _< up _< uj) we again construct splines s;,,..., s;~ etc. At n-th step of this process we shall have Using (1.5) and (1.6) we can explicitly express this process in the following way:

5 Let US now consider the following functions: where A,, j = 1,...,p are constants and values Fop satisfy condition: in this case, the sclutio~ to system is equivalent to the solution to system (1.1). In [3] it is proven that if we conveniently choose constants XI,..., A, then the iteration process will converge. If we apply this to (1.8) and take the following expression for rap then condition (1.10) will be fulfiiled because it is a consequence of the existence condition of non-explicit functions. Hence, the iteration process for solving system (1.l) can be written in the following way: The solution to system (1.1) is reduced to solving the equivalent system (1.1 I), which in operator representation can be written as and the corresponding iteration process (1.12) is

6 IhVERSE SPLINE-INTERPOLATION 87 For investigating the convergence of this process we can therefore use the T operator's fixed point Theorem [3] and [4]. The estimation of parameter X =maxi (see (1.l4)) in [3] is obtained in the following way: Likewise, it is shown that the convergence rate of process (1.14) is no greater than CAiCLZATiNG NON-EXPLICIT FUNCTION VALUES Let x y,..., x: E V and corresponding values, in accordance with Eq. (1. I), are given in the following way: Let point (u?,..., u:) belong to the existence region of non-explicit functio ns. In this case, we have: Let us calculate the values of functions PI,..., pp at point (u y,..., u 0 uk, u k+l,..., u:), which also belong to the existence region of non-explicit tunctions. We have: XI xp =(pp(u1i...iu~-1iukiuk+li...iu;). Considering xl,..., xp as functions of one variable and using the finite increment formula, we can write 0 j Where i:= ujj + Oj(uk - u:), loj/ < 1, j= 1,...,p. For h,, j = 1,...,p let us use the following known formulae:

7 88 F CRIADO et 01 Using these expressions in (2.1 j, we obtain j j In these formulae, points (11,... 1,), j = 1,...,p correspond to points - 0 (u!,..., u!-,, u k, u :,!,..., U: j, j = 1,..., p. We can evidently write the following: Where xl,... xp are defined from formulae (2.3). Introducing these expressions into (2.5), we can write: = -(uk - ul), k = I,...,p (2.6) Considering values as unknown, we define them using system (2.6): the determinant of this system is not equal to zero owing to the existence assumption of non-explicit

8 TNVERSF SPLINE-INTERPOLATION functions. Taking into account that we can finally write Substituting these expressions into (2.3), we obtain 3. PARALLEL ALGORITHM OF A NUMERICAL SOLUTION OF DIFFERENCE EQUATIONS OF GAS DYNAMICS Let us consider a problem in which a piston is moving into gas with a constant velocity (isothermic case) [5]. The structure of the resulting shock wave in an ideal gas is described by the following differential equation system: Where c > 0 is a sound isothermic velocity, v a suitable pseudoviscosity coefficient, 77 a specific volume, v the movement velocity of gas particles, p pressure, w the linear pseudo-viscosity and S the Lagrange mass variable. The initial conditions for all unknown functions are the following: and the boundary condition is

9 90 F. CRIADO el a1 The non-explicit completely conservative uniform difference scheme for equations (3.1) has the following form [J]: Let us consider the case a = I. (3.4) is a non-linear algebraic equation system. Using the known designations of the difference scheme theory [5], we can write system (3.q in the foliowing expanded form: where h is a net step at axis S and T the net step time. Conditions (3.2) and (3.3) will be written in the following way: 7: = 70, p: =PO, 11: = YF, v~+l = Vo respectively. System (3.5) has a form (1.1). For writing it in an equivalent form (1.1 l), we choose values rap from uncoupled system (3.5). Uncoupling is performed in the following way: substitute all the quantities with a subscript different from i and superscript j + 1 by their values from the j-th layer, i.e., we shall consider quantitites vpl, T$', i = 1,..., M as unknowns. So, as a result of uncoupling the system (3.5) with 2M unknowns we obtain M systems, all with two unknowns. In this case, matrix l/l?apl/ has a cell structure and non-zero cells are located on the diagonal. Therefore, det III'aPII is equal to the product i= 1,...,M, of each cell's determinants ir $ill

10 where INVERSE SPLINE-INTERPOLATION 9 1 As det Ilr(i)a~II # 0 in the definition region of non-explicit functions, then det llraplj also will not be equal to zero. We can therefore use the general formulae (1.14). An iteration process bsaed on these formulae for solving system (3.5) will have the following form: i=2,..., (M- I), j=o! 1,... And for i = 1 we shall have: (3.8)

11 -. 1 his process is presented In a form wh~ch 1s directly applicable in parallel prmessmg. Iterations were performed for each time layer we took values of net functions from rhe previous iayer as the initial approximation when passing to the next layer. The test calculations were performed on the condition that the gas finite mass 0 < S 5 M was bounded from the left side by a piston moving at a constant velocity. The initial state of the gas was The value of a sound isothermic velocity was c = 0,5. The values of quantitites h, r and v are shown in Figures 1-4, along with velocity profiles. In our calculations, the values of all parameters coincided with those given in [5] and [6], where the system (3.4) was solved by the % Newton method. On account of our iteration process, the results of test calculations showed that the solution to system (3.1) for the considered parameter values was satisfactorily reproduced. These results coincided with those obtained using the Newton method. It should be noted that the same amount of computer time was spent on both cases. Let us finally deduce the maximal values for X1 and X2 for which the iteration process is convergent. For these values, the number of iterations necessary to attain a given precision is minimal. For r = 0,001 and r = 0,Ol: XI = X2 = 1; for r = 0,25: XI = X2 = 0,95; and for r = 0,4: X1 = X2 = 0,45. Let us consider the realisation of the iteration process (3.8), (3.9) on abstracta parallel processing system (APPS) [7]. (4,. In our case, the parallel calculation of values q, and v,, z = 1,..., (M- 1) on piston co-ordinate knots is performed. Hence, the number of parallel processors (pp) necessary will be P = Ad- I (i.e., we calculate the

12 INVERSE SPLIXE-INTERPOLATION 93 FIGURE 1 FIGURE 2 values of the two aforementioned quantities on each pp). According to formulae (3.8), (3.9), the number of binary operations necessary for calculating unknown quantities on one pp will be m = = 90.

13 F CRIADO er a1 FIGURE FIGURE 4 Thus, the computation time on a sequential computer will be tl =90.(M- 1) (3.10) and we estimate the computation time on APPS by the Munro and

14 Peterson's Theorem [7, 81, such as INVERSE SPLINE-INTERPOLATION where jxl denotes the smallest integer greater or equal to x. The estimation, based on formulae (3.10): (3.1 I), of the highperformance rising coefficient a [7] is the following: References tl 9q'F-i) p = - 2 /' i~o~~(~~-l,l+ 2iw~-1); < 90 - i ~!-''l~~(~-lj1~ (3.12) 1 12,82(M - l)> in the other cases. [I] Cartan, H. (1967). Calcul. Differentiel. Hemann, Paris. [2] Zavialov, J., Kvasov, B. and Miroshnichenko, B. (1980). Methods of spline-functions (Nauka, Eds.), Moscow. [3] Gachechiladze, J. (1969). Solution of non-linear algebraic equation system by inverse spline-interpolation, Reports of Tbilisi University, Cybernetics an Applied Mathematics, Moscow, Mir. [4] Samarski, A. and Popov, Yu. (1980). Difference methods for solving gas dynamics problems (Nauka, Eds.), Moscow. [5] Popov, Yu. and Samarskaya, E. (1979). Feature of the gas dynamics equations numerical solution on the rough nets, Ins. Appl. Math. of Acad. Sci., URSS, No. 103 (to appear). [6] U'allach, Y. (1982). Alternating Sequentiai~Parallel Processing (Springer-Veriag, Eds.j, Berlin, Heidelberg, New York. [7] Borodin and Muro (1975). The computational complexity of algebraic and numerical problems (American Elsevier, Eds.).