Numerical Solution of a Nonlinear Integro-Differential Equation

Transcription

1 EPJ Web of Conferences 18, 17 (16) DOI: 1.151/ epjconf/ C Owned by he auhors, published by EDP Sciences, 16 Numerical Soluion of a Nonlinear Inegro-Differenial Equaion Ján Buša 1,a, Michal Hnaič,3,4,b, Juha Honkonen 5,c, and Tomáš Lučivjanský,6 1 Deparmen of Mahemaics and Theoreical Informaics, FEE&I, Technical Universiy, Košice, Slovakia Faculy of Sciences, P. J. Šafarik Universiy, Košice, Slovakia 3 Insiue of Experimenal Physics SAS, Košice, Slovakia 4 Bogoliubov Laboraory of Theoreical Physics, JINR, Dubna, Moscow Region, Russia 5 Deparmen of Miliary Technology, Naional Defence Universiy, Helsinki, Finland 6 Fakulä für Physik, Universiä Duisburg-Essen, D-4748 Duisburg, Germany Absrac. A discreizaion algorihm for he numerical soluion of a nonlinear inegrodifferenial equaion modeling he emporal variaion of he mean number densiy a() in he single-species annihilaion reacion A + A is discussed. The proposed soluion for he wo-dimensional case (where he inegral enering he equaion is divergen) uses regularizaion and hen finie differences for he approximaion of he differenial operaor ogeher wih a piecewise linear approximaion of a() under he inegral. The presened numerical resuls poin o basic feaures of he behavior of he number densiy funcion a() and sugges furher improvemen of he proposed algorihm. 1 Inroducion The irreversible annihilaion reacion A + A is a funmenal model of non-equilibrium physics. The reacing A paricles are assumed o perform chaoic moion due o diffusion or some exernal advecion field such as he amospheric eddy [1]. Many reacions of his ype are observed in differen chemical, biological or physical sysems [, 3]. In [1] he advecion of a reacive scalar using a random velociy field generaed eiher by he sochasic Navier-Sokes equaion, which serves o he producion of a velociy field corresponding o hermal flucuaions [4, 5], or by a urbulen velociy field wih Kolmogorov scaling behavior [6], was sudied by hree of he presen auhors, and an inegro-differenial equaion for he number densiy was derived. No influence of he reacan on he velociy field iself was assumed. The presen paper proposes a numerical soluion of his inegro-differenial equaion in he wodimensional case using regularizaion and hen finie differences. Numerical experimens provide hins o furher ways o invesigae he problem. Problem Formulaion The inegro-differenial equaion derived in [1] (Eq. (7)) for he mean number densiy a() of he chemically acive molecules in anomalous kineics of single-species annihilaion reacion A + A a jan.busa@uke.sk b hnaic@saske.sk c juha.honkonen@helsinki.fi 4 Aricle available a hp:// or hp://dx.doi.org/1.151/epjconf/161817

2 EPJ Web of Conferences wries down as () d λuνμ Δ Z 4 a () + 4λ u ν μ 4Δ a ( )d [ 8πuν( ) ] d/, a() a. (1) The inegral in Eq. (1) diverges a he upper limi in space dimensions d. Saring from (1), a numerical soluion is derived for he Iniial Value Problem () d λda () + 4λ D which corresponds o D uν, Δ, and Z 4 1. a ( )d [ 8πD( ) ] d/, a() a, () 3 Case d For d he singulariy a in he inegral on he righ hand side of Eq. () is divergen. This is a consequence of he UV divergences in he model above he criical dimension d c. Near he criical dimension i is remedied by he UV renormalizaion of he model [1]. In his paper, anoher approach is proposed o overcome his problem. We sar wih he following regularizaion: d λda + 4λ D Then Eq. (3) is brough o he form a ( )d 8π{D( ) + l }, a() a. (3) d a ( )d αa + β + γ, (4) where α λd, β α 8πD, γ l D. The resuling Eq. (4) is solved numerically using he difference mehod. Time discreizaion is done wih a consan he ime sep Δ: k k Δ, a( k ) a k, k, 1,,... (5) The discreizaion of he firs order derivaive in he lef hand side uses k-dependen finie difference approximaions: d a k a k 1 a k 1 and k Δ d 3a k 4a k 1 + a k a k. (6) k Δ The righ-side inegral approximaion a k is discreized as a sum of k elemenary inegrals and piecewise linear approximaion of he funcion a() is used for he evaluaion of hese inegrals: k 1 i k a ( )d k 1 k u + γ i a ( )d k + γ (7) [ (a i+1 a i )/Δ + (i + 1) a i i a i+1 ] d k. + γ 17-p.

3 Mahemaical Modeling and Compuaional Physics 15 The inegrals in he righ hand side of (7) are calculaed analyically under he assumpion ha γ Δ : (8) [ (a i+1 a i )/Δ + (i + 1) a i i a i+1 ] d k + γ [( k Δ Δ) (a i+1 a i )/Δ + (k + ) (a i+1 a i ) + (i + 1) a i i a i+1 ] d k Δ + Δ [(k + i) a i+1 (k + i 1)a i (k Δ + Δ) (a i+1 a i )/Δ] d k Δ + Δ [(k + i)a i+1 (k + i 1)a i ] d k Δ + Δ [(k + i)a i+1 (k + i 1)a i ] (a i+1 a i ) (i+1) Δ (k Δ + Δ)d Δ (k Δ + + Δ) + (a i+1 a i ) (Δ) (k Δ + Δ) d (k Δ + Δ) [(k + i)a i+1 (k + i 1)a i ] [ ln(k Δ + Δ) ] (i+1) Δ [(k + i)a i+1 (k + i 1)a i ] (a i+1 a i ) Δ + (a i+1 a i ) [ (k Δ + Δ) ] (i+1) Δ Δ (Δ) { [ ] k + i (k + i)ai+1 (k + i 1)a i ln k + i 1 ( k + i + 1 ) (ai+1 a i ) a i (a i+1 a i ) }, (9) i,1,...,k 1. Subsiuing he approximaions (6), (7), and (9) ino Eq. (4) we arrive a he quadraic equaions Eq. (11) and Eq. (1) below wih respec o a k : k β i {[ (k + i)ai+1 (k + i 1)a i ] ln 3a k 4a k 1 + a k Δ + α a k (1) k + i k + i 1 ( k + i + 1 ) (ai+1 a i ) a i (a i+1 a i ) } β {[ ] (1 + )ak a k 1 ln 1 + ( ) (ak a k 1 ) a k 1 (a k a k 1 ) } or in he sanrd form [ α β {(1 + ) ln 1 + ( 3 ] [ 3 + )} a k + Δ β a k 1 { + 1 (1 + ) ln 1 + )}] a k 4a k 1 a k β a k 1 Δ {1 + ln 1 + } (11) {[ ] (k+ i)(ai+1 a i )+a i ln k β i For he firs sep we ge a 1 a Δ + αa 1 β{ [ (1 + )a1 a ] ln 1 + k + i k + i 1 (a i+1 a i ) [ (k+ i)(a i+1 a i )+a i + a i+1 + a i ( ) (a1 a ) a (a 1 a ) } ]}. 17-p.3

4 EPJ Web of Conferences 4 log(a()), < a() < log( a()-a/(+c) B ), A6.58, B.56, C Figure 1. Logarihms of he number densiy a() ogeher wih is fi (above), and he fiing error (below) or in he sanrd form [ α β {(1 + ) ln 1 + ( 3 ] + )} [ 1 a 1 + Δ β a { + 1 (1 + ) ln 1 + )}] a 1 a Δ β a {1 + ln 1 + }. (1) The algorihm sars wih he known iniial value a a() o ge a 1 from he Eq. (1) and hen derives successive values a k, k 1,,...,K by recurrence from Eq. (11). 4 Numerical resuls Below he resuls for a() 5, λ.1, D.4 and l.1 are presened. All calculaions were done wih a predefined uniform ime sep Δ. Figure 1 up plos he variaion of a() in logarihmic scale on he inerval [;1] for Δ 1/3 ogeher wih he logarihm of is asympoic approximaion on he inerval [7.5;1] a() (13) (.4389) p.4

5 Mahemaical Modeling and Compuaional Physics 15 Table 1. Comparison of he a() values a seleced poins for differen values Δ Δ In figure 1 down, he error associaed o he approximaion (13) on he inerval [7.5;1] is shown. The approximaion error associaed o a() is found o be smaller han 1 5. Table 1 compares he resuls a seleced poins for he increasing precision obained a decreasing values Δ from.1 o.315. Even a he fines ime sep Δ 1/3.315 he resuls can no be considered o be sufficienly precise. If he sep number is increasing wice, hen he CPU ime is increasing four imes. Assume ha he difference beween he numerical value a Δ () a he sep Δ and he exac value a() is of he order p, i.e., a Δ () a() C (Δ) p. (14) Wriing down similar equaions a Δ/ and Δ/4 as well, he following empirical esimae is obained for he precision order p(): a Δ () a Δ/ () p() log a Δ/ () a Δ/4 (). (15) If his expression resuls in roughly consan values of p() under he variaion of boh and he sepsize discreizaion Δ, hen we may infer ha he soluion of () was obained wih he fine enough discreizaion sepsize Δ. Figure plos he precision order p() calculaed for four grid sequences:.1.5.5, , , and The plo of he righ poins o he occurrence of he smooh p() behavior a >.5, wih orders of cca..8, 1., 1., and 1.4, respecively. The plo a he lef shows ha for smaller -values he p() behavior follows unsable complicaed paerns. The repored numerical resuls sugges ha i would be reasonable o sudy analyically he behavior of he funcion a() over small values of, and o sar he numerical compuaion from some poin >. Also he use of a non-uniform ime grid could be considered. 5 Conclusions The numerical resuls presened above evidence basic feaures of he behavior of he number densiy funcion a(). However, furher improvemens of he algorihm are desirable. I will be also ineresing o ry o solve he renormalized inegro-differenial equaion also presened in [1] () d { λ [ λuνμ Δ a () + λuνμ Δ a () γ + ln(uνμ ) ] } + 4π + λ uνμ Δ [ a ( ) a () ] d π, (16) 17-p.5

6 EPJ Web of Conferences Figure. The -dependence of he empirical precision order p() for four cases:.1.5.5,..., where Δ(d )/ and γ.5771 is he Euler consan, and o compare he resuls. Anoher possibiliy is o sudy he behavior of he soluion of he problem (1) hrough a sequence of soluions go a uniformly increasing d values, d. Acknowledgemens The work was suppored by VEGA Gran 1//13 of he Minisry of Educaion, Science, Research and Spor of he Slovak Republic. References [1] J. Honkonen, M. Hnaič and T. Lučivjanský, EPJ B, 86 : 14 (13) [] B. Derri, V. Hakim and V. Pasquier, Phys. Rev. Le. 75, 751 (1995) [3] R. Kroon, H. Fleuren and R. Sprik, Phys. Rev. E 47, 46 (1993) [4] D. Forser, D.R. Nelson and M.J. Sephen, Phys. Rev. Le. 36, 867 (1976) [5] D. Forser, D.R. Nelson and M.J. Sephen, Phys. Rev. A 16, 73 (1977) [6] L.T. Adzhemyan, A.N. Vasil ev and Y.M. Pis mak, Teor. Ma. Fiz. 57, 68 (1983) 17-p.6