Difference -Analytical Method of The One-Dimensional Convection-Diffusion Equation

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1 Diffrnc -Analytical Mthod of Th On-Dimnsional Convction-Diffusion Equation Dalabav Umurdin Dpartmnt mathmatic modlling, Univrsity of orld Economy and Diplomacy, Uzbistan Abstract. An analytical diffrncing schm solution for th on-dimnsional convction-diffusion problms is prsntd. Driving of an analytical solution is basd on a diffrntial quation diffrnc analogu. Ky words: diffrnc schm, convction-diffusion, diffrnc -analytical Introduction Th mthod of finit diffrncs [] is most oftn usd at a diffrntial quation solution. Th ida of a solution of boundary valu problms is rathr simpl: instad of drivativs in a diffrntial quation ar usd it finit diffrnc approximations. Convction-diffusion problms ar bas at modlling of problms of hydrodynamics and hat-mass transfr []. Th basic attntion at a numrical solution is givn to problms of approximation of convctiv trms [-6]. Upwind schms, cntral diffrncs, hybrid schms, a high ordr upwind schms ar widly usd. In th cas of cntral diffrncs, itrativ mthods for solving th rsulting systm of linar quations do not convrg whn th convctiv trms dominat. Schms of a low ordr for th convction-diffusion quation connct unnown quantitis in th thr nodal points (UDS, CDS, Powr law, hybrid schms). For schms of highr ordr nods mor than thr will b rquird (QUICK, SMART, MINMOD, TOPUS tc.). Usually, solution of diffrntial quations by numrical mthods is obtaind in th form of numbrs. Hr w will show a possibility of driving of a solution of diffrntial quations by diffrnc mthods in th approximatly-analytical form. This ida, which show s for a on-dimnsional convction-diffusion quation. Diffrnc-analytical mthod Lt's considr th on-dimnsional convction-diffusion quation on a finit intrval with boundary conditions: dф dф P = + P ( ) Sx () dx dx Ф ( ) = Ф, ФE ( ) = Ф () E 34

2 x E ρv( E ) whr P = - Pclt numbr, Sx ( ) a givn Г function, Ф unnown function. ta an arbitrary point x [, E] and w will fig. divid a sgmnt on two parts (fig. ). Flow dirction is pointd fig.. us upwind schm. Thn instad of () w hav th diffrnc -analytical quation U U U E U U U P = + P Sx ( ) x E E x x (3) Hr U valu of unnown function in th nodal point x ( U Ф UE ФE),. As x [, E] an arbitrary point from (3) w can dfin and rciv th approachd analytical solution of th problm ()-(): ( ) ( )( ) ( )( ) x UE + + P( E ) E xu ( x )( E x) U = + P Sx ( ). E + P( E x) + P( E x) Lt's not that, by this way of such approach som problms [7] ar solvd. x+ To improv approximat solution w ta an additional nods: x =, x+ E x =. will writ th UDS schm of typ (3) for [, x], [ x, x ] and [ xe., ] will rciv systm from thr quations. xclud th rcivd systm, U ( x ) and U ( x ) as a rsult w will obtain th improvd schm: P U U 4 UE U U U = + P Sx ( ) + x E x x ( τ E + ) ( + γ) ( + τ) ( ) τ γ 4 + P E 4 Sx ( ) Sx ( ) E + τ E + γ (4) + θ whr τ =, γ =, t = P( x ), θ = P( E x). In (3) U is improvd valu + t U Ф, U Ф. of unnown function in th nodal point x ( E E) Solving (4) rathr U w will rciv th improvd analytical solution. Again to improv solution w will arriv similarly: w will writ th schm (4) for [, x ], 35

3 [ x, x ] and [ xe,, ] and w will xclud unnowns in points x and x, tc. Continuing this procss as a rsult w will hav + U U UE U U U P = F( x) E + (5) x τ E x γ x τ τ γ τ + θ whr τ =, γ =, + t ( ) ( τ ) + P E x Fx ( ) = P Sx ( ) + S+ j + j i τ E τ j i = = + j ( γ ) i E x ( ). γ S x+ j E γ j i = = In improvd (5) U valu of unnown function in a nodal point x U Ф, U Ф. Solving th quation (5) rathr U w will rciv th ( E E) approachd analytical solution of an initial problm. If in (5) = 0 and summation to suppos qual to zro (an uppr bound of th sum it is lss than a limit infrior) thn w will rciv th schm (3), at = th schm (4) tc. turns out. Lt's not that if from (5) w will dfin U and w will calculat that a limu givs an xact solution of corrsponding problms. Analytical xampls In ordr to vrify th thory, th numrical computations wr carrid out with svral Sx ( ). On fig. solutions of a problm () ar rducd at R= 0, Sx ( ) = 0 on [0;] with boundary conditions Ф = 0, ФE =. On fig. solutions of a problm () ar rducd at R= 50, Sx ( ) = 5cos 4x on [0;] with boundary conditions Ф = 0, ФE =. From th graph it is visibl that, in procss of incras to approximat solutions coms narr to th xact. From graphs it is visibl that, sinc = 7 xact and approximat solution ar visually coincids. 36

4 fig. fig.3 Continuous lin xact, point wis = 0, dashd =, point wis-dottd =, long dashd = 3, rar dashd = 7 Numrical xampls Th analytical schm (5) allows not only rciving th approachd analytical solution, but also givs th chanc in cration of th qualitativ schm. For xampl, from (5) at in rgular intrvals disposition of nods of a grid with h w hav th schm: h cu i i = bu i i+ + au i i + F i (6) whr P g R =, =, g = + R, b =, a = g, c = a + b, h i i i i i + R g ( g ) ( ) j n h i ( i) i h j n = = F = P Sx + + P S x + j h j n h ( ) g S xi + j j n = =. g Hr Ph = ρvh / Г so-calld grid Pclt numbr, xi = + ih nodal points. Comparisons of xact and diffrnc solutions ar on fig. 4 and 5 rducd. 37

5 fig. 4 fig. 5 Comparison of various schms Solution of problm () at P = 50 ar on fig. 3 and 4 graphs on [0;] with boundary conditions Ф = 0, ФE =. Fig. 4 corrsponds at Sx ( ) = 5cos4x and graphs on fig. 5 ar rcivd at 0 50x if х 0,3 S( x) = 50x 0 if 0,3 < x < 0,4 0 if 0,4 < x Numrical rsults ar rcivd at h = 0, and P h = 5. Solid lins of a drawing of xact solutions of a problm. Circl numrals ar rcivd for UDS schm, rctangls undr th schm of Patanar, an astris on (6) at =, диамант at = and circls at = 7. From fig. 4, 5 appars that, th offrd schms yild good rsults. Conclusions By an xampl of a on-dimnsional problm th nw st of analytical schms is offrd. By ths schms it is possibl to rciv an analytical solution. Tsting of th offrd schms by xampl of stationary modlling boundary valu problms is hld. Tst calculations confirm fficincy of application of th diffrnc -analytical schm for solution convction-diffusion problms. 38

6 Rfrncs [] Samarsii, A. A. Th Thory of Diffrnc Schms. Monographs and Txtboos in Pur and Applid Mathmatics, 40. Marcl Dr, Inc., Nw Yor, (00). [] Patanar S.V. Numrical hat transfr and fluid flows. Nw Yor: Hmisphr Publishing Corporation; (980). [3] Lonard B.P. A stabl and accurat convctiv modlling procdur basd on quadratic upstram intrpolation. Comput Mthods Appl Mch Eng 9: (990) [4] Lonard, B.P., S. Mohtari, Byond first-ordr upwinding: th ultra sharp altrnativ for non-oscillatory stady-stat simulation of convction, Int. J. Numr. Mthods Eng., 30, 79(990) [5] Lonard, B.P. Th ULTIMATE consrvativ diffrnc schm applid to unstady on-dimnsional advction. Comp. mthods applid mch. ng. 99. Vol. 88. pp [6] Frrira V.G., d Quiroz R.A.B., Lima G.A.B., Cunca R.G., Oishi C.M., Azvdo J.L.F., McK S.. A boundd upwinding schm for computing convction-dominatd transport problms. Computrs and Fluids 57(0) pp [7] U.Dalabav About on mod of avrag diffrntial th quation and its application for a solution of problms of a mchanics of a liquid, Modrn problms of a mchanics, lctur and thss, Octobr, 9-3th 00, Tashnt, pp