A combined space discrete algorithm with a Taylor series by time for CFD

Transcription

1 A combined space discree algorihm wih a Taylor series by ime for CFD I.V. Kazachkov Hea and Power Faculy Naional Technical Universiy of Ukraine KPI Kyiv Polyekhnichna 7 B.5 56 Ukraine SUMMARY The firs order by ime parial differenial equaions are used as models in applicaions such as fluid flow hea ransfer solid deformaion elecromagneic waves and ohers. In his paper we propose he new numerical mehod o solve a class of iniial-boundary value problems for he PDEs using one of he known space discree numerical schemes and a Taylor series epansion by ime. Normally a second order discreizaion by space is applied while a firs order by ime is saisfacory. Neverheless in a number of differen problems discreizaion boh by emporal and by spaial variables is needed of highes orders which complicaes numerical soluion in some cases dramaically. Therefore i is difficul o apply he same numerical mehods for he soluion of some PDE arrays if heir parameers are varying in a wide range so ha in some of hem differen numerical schemes by ime fi he bes for precise numerical soluion. The Taylor series based soluion sraegy for he nonsaionary PDEs in CFD simulaions has been proposed here ha aemps o opimise he compuaion ime and fideliy of he numerical soluion. KEY WORDS: Firs-order by ime; Parial differenial equaion; Navier-Sokes equaions; Taylor series; Numerical algorihm; Fracional derivaive; Non-saionary soluion. INTRODUCTION The second order PDEs have found eensive applicaions in he sudy of problems in fluid mechanics flow in porous media hea conducion ec. [-4]. A large number of numerical mehods have been proposed for solving he second order PDEs which are mainly he firs order in ime in a CFD simulaion. A key issue is he need o effecively use high performance numerical mehods [5-] and compuers including parallel clusers [] o complee analysis in ime frames. In designing CFD sofware ools he auhor has aemped o build an essenially open single sofware framework ha enables arbirarily comple non-saionary -D PDE array o be represened which run efficienly on modern compuers and allow simple increasing of accuracy in numerical simulaion by ime. The feaures of he above approach are ha i employs Taylor series epansions o compue soluion of he PDE by ime wihou emporal discreizaion of he PDE. The idea of using he Taylor series epansions for numerical soluion of non-saionary boundary problems has arisen from original use of a Taylor series described in [45] as an efficien procedure for parameric sudy in comple problems where a number of ypical compuaions was replaced by Taylor series approimaions. The sraegy for he numerical soluion of non-saionary -D PDE using Taylor series by ime has been proposed as follows: Numerical soluion sars as usually wih discreizaion of numerical domain and developmen of appropriae numerical grid Discreizaion of PDE by space is done by one of he known mehods which fis he bes o he PDE given address: kazachkov@soline.ne.ua (I.V. Kazachkov.

2 The emporal derivaives are compued up o he desired order by ime for he numerical soluion sough Using compued emporal derivaives he numerical soluion sough is found from he Taylor series.. THE COMBINED NUMERICAL METHOD USING A TAYLOR SERIES BY TIME Convenionally numerical soluion of any iniial-boundary problem for PDE or PDE array wih one of he known numerical mehods is going as follows:. Discreizaion of he numerical domain and developmen of he appropriae numerical grid.. Discreizaion of PDE by space and ime wih furher ransformaion of he ougoing PDE o is approimaion for eample algebraic finie-difference equaions by space and ime.. Numerical soluion of he approimae (e.g. algebraic finie-difference equaions by space and ime. 4. Tesing of he numerical soluion obained and validaion of i agains he known daa (oher numerical soluions analyical soluions for limi cases of he PDE saed eperimenal resuls ec.. Highly imporan peculiariy of he above sraegy is discreizaion of he PDE (sep performed according o he accuracy of he numerical soluion by space and ime required as far as his predeermines furher seps and mehods seleced for he numerical soluion. If any changes o he requiremens of soluion accuracy hen he sep changes hus he numerical algorihm changes oally. Our sraegy replaces he seps of he above algorihm which becomes he following one:. Spaial discreizaion of he numerical domain and developmen of he appropriae numerical grid.. Discreizaion of PDE by space wih furher ransformaion of he ougoing PDE o is approimaion for eample algebraic finie-difference equaions by space.. Numerical soluion of he approimae (e.g. algebraic finie-difference equaions by space. 4. Compuing he emporal derivaives using he ougoing PDE or PDE array wih furher calculaion of he numerical soluion in ime based on he Taylor series epansion by ime. 5. Tesing he numerical soluion obained and is validaion agains he known daa (oher numerical soluions analyical soluions for limi cases of he PDE saed eperimenal resuls ec.. Le us sar wih a few simple eamples showing he idea of he proposed sraegy. For his firs consider he following one-dimensional non-saionary equaion (describing for eample -D flow along wih he corresponding iniial and boundary condiions: U U + U U ( ( ( U U Γ U U. ( We do no specify he boundary condiion ( because i is no maer for eplanaion of he proposed mehod. U U Supposed he equaion ( is rewrien in a more convenien form Γ

3 U U U U + f ( ( where from differeniaing he las equaion by ime resuls U U fu U U U U U U U f (. (4 Now a Taylor series wih a second order accuracy by ime (inroduce as he ime sep in numerical soluion accouned ( (4 yields he following approimaion of he soluion o he equaion ( f ( fu U U + f + o +! ( ( ( (5 Where U is he known iniial daa and U U U U ( f U U + o ( (6 is easily compued by he known funcion U (. In a similar way f ( fu U U U U. (7 4 U 4 U U Then subsiuion of he equaions (6 (7 ino a Taylor series (5 resuls in U U U U U U U U + U + U + U U U o ( ( ( (8 and so on. Evidenly one can coninue his procedure o ge any desired order of accuracy by ime. The ransformaion of he procedure from any n h layer by ime o he (n+-h layer by ime is similar o he saed in he equaions (5 (8 above: f ( fu U n+ U n + fn + ( o ( + n ( (9 where n N. Thus he righ hand of he equaions (5 (9 is always a funcion of he coordinaes a he curren momen of ime. Thus neiher eplici nor implici approimaions by ime are applied; no difference equaions by ime are needed! As he equaion (9 shows he second order by ime approimaion in he equaion ( requess all spaial derivaives of he funcion sough up o he 4-h order. Adding he ne erm in he ime series requires he corresponding erm wice differeniaed by space. If compuing he highes order

4 4 i i derivaives f / is analyically complicaed i is done numerically. Consequenly insead of a soluion of a difference (or any discree equaion compuaion of he spaial derivaives wih he accuracy saed is proposed. Then he numerical soluion sough is compued from he Taylor series by ime.. EXAMPLES OF A TAYLOR SERIES IN NUMERICAL SOLUTIONS Eample. In case of he simple wave equaion wih he following iniial and boundary condiions u u u u ( ; u u( ( ; an analyical soluion of he boundary problem ( is known as a wave spreading wih he velociy counercurren o he ais u f ( +. According o our sraegy he u u + u / + O(( where from wih accoun numerical soluion of ( is ( of he above-menioned u +. The soluion obained does no change wih an increase of accuracy because he firs order soluion coincides here wih he eac analyical soluion. Eample. The one-dimensional non-linear equaion U + U U ν U ( wih he following iniial and boundary condiions U U U ; ( is solved according o he proposed sraegy as follows U U U ν U ν ν U U U U U U U U U U U ν ν U U ν U where from yields soluion o he iniial-boundary problem ( ( in a second order approach by ime U U U U U U U U + ν U ν ν U U ν U + The firs order approach by ime gives here he same resul as any higher order approaches: U. (

5 5 4. ALGORITHM FOR NUMERICAL SOLUTION OF THE NAVIER-STOKES EQUATIONS WITH THE PROPOSED STRATEGY Consider numerical algorihm for he soluion o he equaions of -D non-saionary moion of hea-conducing incompressible viscous fluids: div v v + v v p + ν v + f ρ where v { v v v } f { f f f } y z y z T + +Φ ( ρc v T div( λ T are he velociy and he eernal force vecors respecively. The Caresian coordinaes yz are implied here hen ρ µ λ are he densiy dynamic viscosiy coefficien and hea conduciviy coefficien correspondingly ν he coefficien of kinemaical viscosiy and с he specific hea capaciy denoe he gradien and Laplace operaors. Finally Ф is he dissipaive funcion µ Φ v ρc τ + µ τ µ v v y v v v z y v z y z z y. The parial differenial equaion array ( hus obained has o be supplemened wih he corresponding iniial condiions saed in he numerical domain Ω : v v ( y p p ( y T T ( y ( y z Ω (4 z z z as well as wih he corresponding boundary condiions (Dirichle Neumann mied ec. a he boundary ( y z Γ. They are no specified here because i has no maer for he proposed sraegy of he numerical soluion which is applicable by any boundary condiions. To apply he above described numerical sraegy o he Navier-Sokes equaion array ( wih he iniial condiions (4 rewrie hese equaions in he form: where is div v v F( v v f p ρ F( v v f ν v v v + f F ( F Fy Fz T FT ( v T T v (5 FT v T T v div T v T (6 ρc ( Φ+ ( λ is he vecor of he righ hand of he momenum equaion ecep he pressure gradien. Thus all righ hands of he equaions (6 are known a he iniial momen of ime from he iniial daa (4 and hen all emporal derivaives of he velociy vecor and of he emperaure are compued from (5. Consequenly he velociy and emperaure fields are calculaed a he ne ime sep from he Taylor series: v v v v v ( + + ( + ( + o(( (7!!

6 6 T T T T T ( + + ( + ( + o((.!! Here is he emporal sep chosen for he generaed numerical grid. The approimae numerical soluion (7 is compued wih a required order of accuracy by ime (here i is up o he hird order erms for eample. I is very imporan ha a he firs emporal sep compued by equaions (7 he pressure disribuion is unknown and coninuiy equaion has no been used ye. Surprisingly he velociy and he emperaure fields do no depend on he pressure disribuion a he firs ime sep as shown in deail below. This numerical scheme in a firs order by (when only he erms up o are kep in (7 compleely coincide wih he simples firs order eplici numerical scheme. Bu hese wo mehods differ a lo aferwards. For eample well-known numerical schemes of a second order by ime are very ime-consuming and cumbersome while in he sraegy proposed here he numerical soluion procedure in a second order by ime (as well as in any higher order by ime is nearly he same as in he firs order by ime. Moreover any highes order numerical soluion is go similarly and wha is very imporan does no reques more compuer resources han he firs order soluion. All needed for his is jus an easy compuaion by equaions (5 (6 wih furher subsiuion of he resuls ino he Taylor series (7. The numerical soluion of a second order accuracy by ime requess in conras wih he firs order approimaion calculaion of he derivaives of pressure because he equaion array ( ransforms o (div v v v F p ρ T T F T (8 where F v f v v ν + v + v FT T T v Φ+ div λ v T. ρc Here Φ µ τ Φ F + ρ c µ F F y F F F z y F z τ µ y z z y. Then furher ransformaion resuls in F f F ν F p + F p v v F p ρ + ρ ρ F T FT Φ+ div( λ FT v FT+ T F p ρc ρ (9

7 7 And now ransformaion of he firs equaion in he equaion array (8 o he form div( v / wih furher subsiuion of he componens ( v / from he equaions (5 yields div( v / divf / div( p ( ρ where from finally goes o he following Poisson equaion p ρdivf. ( The equaion ( hus obained allows compuing he pressure disribuion in he numerical domain by he known values of he vecor F which have been compued from he equaion (6 based on he firs order approimaion for he velociy field (described above. This equaion ( is solved comparably easily and furher numerical algorihm does no reques soluion of any equaion because all needed here is compuing he spaial derivaives for he funcions in he numerical domain. The approimaion of he second order accuracy by ime o he numerical soluion is go aferwards simply from he Taylor series (7. Finally he closed sysem of he equaions (8-( has been go here for he compuaion of he second order by ime numerical approimaion o he ougoing Navier- Sokes equaion. Imporanly any difference equaions are absen here ecep he one wellknown Poisson equaion ( for he pressure. Only he spaial derivaives are o be compued wih he required accuracy which is much easier han soluion of he difference equaions and does no complicae he algorihm in a second order approach. Now differeniaing he parial differenial equaions (8 by ime accouning (9 he hird order accuracy by ime is go as follows: where p v F p ρ T F T ( is compued based on he firs and on he second approimaions by ime go for he pressure and is derivaive by ime including he iniial daa as described above. Obviously he second order derivaive by ime for he pressure can be compued only wih he firs order accuracy because he firs order derivaive has been compued wih he firs order accuracy and aferwards he second order derivaives were compued also wih he firs order accuracy as derivaive from he firs order derivaive. This imporan quesion is subjec for a separae deail invesigaion. Here are F F F f F p + p v+ F p + F p F p F p F p v + + ρ ρ ρ ρ ρ F ν p f F p + v ρ + ρ p p F p + F F v + ρ ρ ρ ν ρ ρ

8 8 FT F T Φ + div F F T+ F F + v F ρc ( λ T ( T T where Φ Φ µ F ρ c µ τ + F F y F F F z y F z τ µ y z z y. F ( F Fy Fz F T are he new righ hands of he equaions according o (9. Finally he equaions for he soluion of a hird order accuracy by ime ( resul in v T F p F T ρ FT Φ + div( λ FT ( F T+ F FT+ v FT ρc f p p p p F F v+ F F + v F ρ ρ ρ ρ p + ν F ρ. ( Here f / is known as a derivaive from an eernal force saed. The mos ofen i is consan (e.g. graviaion or known (saed elecromagneic force in a conducive media acceleraion due o vibraion ec. Consequenly he equaions ( give he numerical soluion for he ougoing equaions (5 which is a hird order accuracy by ime. The pressure disribuion in a hird order accuracy is go from soluion of he Poisson equaion ( afer subsiuion of he compued velociy field. Obviously here is no problem o implemen he proposed mehod o he case of variable physical properies of coninua and o some oher more general cases. I mus be noed ha an increase of accuracy of he approimaions by ime requess compuing he emporal derivaives of he pressure which needs o keep in compuer memory a few emporal layers. Therefore his quesion needs a separae deep sudy. Neverheless his preliminary analysis shown high efficiency and simpliciy of he sraegy proposed for he soluion of he non-saionary non-isohermal equaions of he Navier- Sokes ype as well as many oher firs orders by ime PDE arrays. The same sraegy is applied consequenly a each and every ime sep and he ransformaion from any n-h layer by ime o he (n+ -h layer by ime ( n + + n is similar o he saed in he equaions (7: v v v v v ( ( ( o(( n+ n n!! n n T T T T T ( ( ( o((. ( n+ n n!! n n

9 9 where n N is. Cerainly he ime seps and he number of ieraions a each ime sep may vary from one ime sep o anoher ime sep; herefore here is also he separae subjec for addiional invesigaion concerning he opimizaion of he numerical algorihm. 5. COMPARISON OF THE PROPOSED STRATEGY WITH THE METHOD OF FRACTIONAL DIFFERENTIALS Le us consider an eample from [6] on compuaion of a hea flu by he known emperaure disribuion using an analyical mehod of he fracional differenials. The ask on heaing of he semi-infinie domain is modeled by he following equaion wih he corresponding iniial and boundary condiions: < < < < T T Ts ( T T. (4 The hea flu is qs ( T /. The differenial operaor in (4 may be represened in he form [6]: / / / / + T (5 / / T T T supposed ha. / / muliplayer of (5: Now consider he equaion presened by he righ / / + T. (6 The soluions o he equaion (6 are also soluions o he equaions (5 because he operaor applied o a zero resuls in zero. Thus soluions o (6 are soluions o (4 as well. The equaion (4 wrien for х gives immediaely soluion of he ask saed namely emperaure gradien a he boundary of he domain: T ( ( qs dτ d d. (7 τ / d Ts d Ts / π Noe ha he emperaure gradien (7 has been found wihou soluion of he ask for emperaure disribuion (4. This is why he mehod of he fracional differenials is also called he non-field mehod. I allows compuing analyically a hea flu a he boundary of a domain direcly hrough such comparably simple ransformaion of he ougoing differenial equaion. Le us analyze he simple linear equaion o compare he mehod of fracional differenials wih he mehod proposed here. For his ecep soluion (7 consider also a general soluion o he boundary problem (4 [7]: T T T d ( ( τ τ s τ τ (8

10 -τ α 4( -τ e dα e -τ π (9 Tτ Θ τ τ τ π - ( τ he emperaure T s ( is kep by for all τ from ill. From (8 (9 follows T ( ( s τ T e d. π ( τ Inroducing in ( he new variable ξ resuls in τ 4( -τ τ ( / T ( T + e d π ξ s ξ 4ξ ( which gives soluion o he ask (4. By х he soluion ( saisfies o he boundary condiion (4. Now q s can be go from ( or direcly from ( wih differeniaion of he inegral by parameer [8]: where from comes T Ts ( τ 4( τ e dτ / π ( τ ( τ T T ( τ T ( τ dτ qs e d s s ( τ / / π ( τ ( π τ. ( Comparing ( wih (7 one can see ha hey coincide because from (7 follows T d Ts ( τ Ts ( τ dτ Ts ( τ qs dτ + / π d ( τ π τ τ τ Ts ( τ dτ / π ( ( τ where Ts ( τ unil τ. Thus ( and ( coincide. The soluions obained by he mehod of fracional differenials and he eac analyical soluion of he boundary problem (4 compleely coincide. Now his hea flu a he boundary will be go ones more following o our algorihm. For his firs he emperaure profile is compued hrough he Taylor series (7: T T T T + ( + ( +...!! (4

11 where T T ( according o (4. Replacing here accordance wih (4 resuls T hrough T in 4 4 T T Ts Ts T ( ( ! (5! Then he Taylor series epansion (4 wih accoun of (5 can be rewrien up o an arbirary order by : n T n : T ( n n n! n n T T n : n ( T ( + ( n n n n! n n! n n n T T T m : Tm T ( m... ( n + n + + n n n! m n (6 for he consan ime sep saring from (acually ime sep may be chosen variable which does no affec he proposed mehod. The Taylor series (6 represens he algorihm for numerical soluion of he boundary problem (4 using approimaions by ime up o he desired accuracy. All needed for his is spaial derivaives by in he domain. According o (5: 5 T T T qs +...! 5 (7 Now compue derivaives by using he eac analyical soluion ( of he ougoing boundary problem (4 and subsiue ino (7. Differeniaing by yields from (: T T s ( τ 4( τ e dτ / π ( τ ( τ ( τ T T s ( τ 4( τ e dτ / π + ( τ 4( τ ( τ 4( τ ( τ 4 4 T T s ( τ 4( 5 τ e + 5 dτ 4 / π ( τ 4( τ 4( τ τ (8 5 4 T T s ( τ 4( τ 5 e 45 5 / d + τ π ( τ 8( τ 4( τ ( τ T T s ( τ 4( τ 5 e 45 dτ 6 / π ( τ 4( τ + 6( τ 8( τ 4( τ

12 T T s ( τ 4( τ 8 e 7 / π ( τ 8( τ 64( τ ( τ + + dτ 6( τ 4( τ The approimae numerical soluion is given by recurren formulae (6 herefore he analyic epression for derivaives (8 in equaion (7 are saisfacory only for an iniial small ime sep: T ( τ T ( τ 5 dτ ( τ ( τ τ 5 T s T s dτ 5/ 5 5/ 4 π 8 π. (9 Now he hea flues in a firs and second order approaches are respecively: T qs 4 π Ts ( τ ( τ 5/ dτ Ts ( τ Ts ( τ τ 5/ 5/ 4 π 6 π T 5 qs d dτ ( τ ( τ τ (4 where is small. To compare (4 wih he eac soluion le us ake also he hird order approimaion by τ τ 5 τ 4 qs dτ + dτ + dτ 4 π τ 6 π τ τ 64 π τ τ Ts ( Ts ( Ts ( 5/ 5/ 7/ ( ( (. (4 The approimae soluion (4 wih accuracy of a hird order erms by ime is as follows: 4( τ / Ts ( τ T ( ( sgn Ts ( e π ( ( τ τ ( + i τ 4( τ i ] d τ 4( τ τ ( τ 6( τ 8( τ 4( τ (4 where sgn( is a signum funcion sgn sgn( by >. To esimae he approimae soluion (4 obained by our sraegy and compare i wih he eac analyical soluion (7 he new variable ξ is inroduced in (4 τ which go o ξ ξ ξ 4 T T + e 4 6 ξ + ( 4ξ ξ π 4ξ 4 8 ξ ( 4ξ 4ξ + 5ξ 45 dξ (4

13 where T T( T( T ( from (4 ~ is he approimae soluion (4. In a firs order by ime T ( sgn Ts ( + ( sgn Ts ( + e (44 4 wih a small eponenial inaccuracy. By T. If he muliplayer T S π ( TS ma TS is go ou of inegral in he epression (4 hen i goes o 4 6 ξ ξ ξ 4 8 ξ 6 4 e ( 6 4ξ 4 ( 4ξ ξ ( 4ξ 4ξ + 5ξ 45 dξ Then in a firs order accuracy by ime : 4 (45 ξ 4 ξ 4 (46 sgn( e dξ e e dξ e 4 4 where from by resuls eacly zero: -- and --. Aferwards accouning he formulae (45 and (44 in a firs order approach by ime he following deficiency of he approimae soluion is go: ξ ξ T e dξ e dξ sgn( e 4. (47 4 In a second order approach by accordingly: T e dξ e dξ sgn( e 5 + e 4 6 ξ ξ 4 4 (48 where from follows ha inaccuracy of he numerical soluion is eceeds by small an order of dramaically insance by - i is ~ ( / e which ( / e imes eponenially. For e e Which is huge value ecep small where small value due o small. 4 e e or and hen T 4 4 4

14 4 By T hen by T and we go complee coincide wih he eac soluion. The second approach (48 decreases accuracy by adding he erm ( / e of order / comparing o he firs order soluion. Obviously by fied and small his can decrease accuracy which is good in a firs order approach. In (45 he following inegrals were compued: ξ ξ ξ ξ 4 ξ ξ e dξ ξe d ( ξ e dξ ξe e + e dξ 4 ξ ξ 4 4 ξ ξ e dξ e ξ dξ e + e e dξ e ξ dξ ξ de e e dξ ξ ξ 5 4 ξ ξ 5 4 ξ e ξ dξ e e dξ 4 ξ ξ 4 4 e 4 ( ξ ξ + dξ + e Surprisingly all erms of a negaive power by are muually omied he same as he inegrals hen only he eponen and he argumens / / are kep. By he numerical soluion by our algorihm coincides compleely wih he eac analyical soluion in a firs order accuracy by ime. By fied he accuracy is high (deficiency decreases eponenially by ime. By small inaccuracy may grow herefore i is imporan o choose righ seps by and nearby he boundary. 6. NUMERICAL RESULTS TO SUPPORT THE EFFICACY OF THE METHOD Now he numerical soluion by he mehod proposed here is applied o he nonsaionary wo-dimensional hea ransfer problem for he Sokes flow around he sphere: Pe cgθ Pe sinθ cosθ T + + τ ρ ρ ρ ρ ρ ρ θ ρ ρ 4ρ 4ρ θ (49 < < < < ; ( θ τ ρ θ π τ T T ; ; s T T ρ ρ τ wih a symmery condiions by θ where are ρ r τ a UR R Pe. R a Numerical simulaion performed for a number of a differen boundary emperaure disribuion ( T θ τ has shown ha he second and he hird order by ime soluions nearly s coincide so ha for his problem no higher han a second order by ime is needed. A compuaion ime among he firs-fifh order by ime does no reveal remarkable difference bu by he sih order by ime i sars o grow subsanially.

15 5 A few seleced simulaion resuls o suppor he efficacy of he mehod are given in he Table below: A picure around he sphere is symmerical herefore he hree poins afer θ 4.98 are omied jus o save a place in he Table. For he iniial-boundary value problem (49 here were no found any remarkable difference beween compuaion resuls in he second and in he hird order by ime (for higher orders as well despie diverse boundary condiions proven in he numerical simulaion. 7. CONCLUSIONS The eamples considered here have clearly demonsraed an effeciveness and simpliciy of he sraegy proposed for he numerical soluion of he non-saionary nonisohermal Navier-Sokes equaions as well as any oher firs order by ime parial differenial equaions. An order of he equaions by spaial variables has no maer for his algorihm. The sraegy is based on applicaion of a Taylor series by ime for he compuaion of he soluion sough by is emporal derivaives. These emporal derivaives of he funcions are epressed from he ougoing equaions hrough heir spaial derivaives. The highes order emporal derivaives used in he Taylor series for compuaion of he approimae soluion are go differeniaing by ime he ougoing PDEs. Only he one Poisson equaion for he pressure disribuion has o be solved numerically in case of he full Navier-Sokes equaion array. The mehod is applicable boh for incompressible as well as for compressible Navier- Sokes Equaions. I allows comparably easily increasing he fideliy by ime up o a fifh order reducing soluion of he ougoing non-saionary problem o a soluion of a consequence of he saionary problems a each emporal sep. The sraegy of numerical soluion of he boundary problems for he PDEs has been considered for a few diverse eamples. An efficiency and simpliciy of he mehod is achieved hrough a replacemen of a soluion of he finie-difference (or finie-elemen ec. equaions in he known numerical mehods wih a simpler procedure of he spaial differeniaion and furher compuaion of he Taylor series. The sraegy is going o be proved on differen CFD problems o sudy all he pros and cons for is furher implemenaion (e.g. resricion abou he ime sep of he mehod opimizaion beween he ime seps and he order of a numerical soluion by ime ec.. REFERENCES. Joseph D.D. Renardy Y.Y. Fundamenals of Two-fluid Dynamics Springer-Verlag 99.. Nigmaulin R.I. Shagapov V.Sh. Galeeva G.Ya. Lhuillier D. Eplosive flow of boiling and degassing liquids ou of pipes and reservoirs: he influence of wall fricion //Inernaional Journal of Muliphase Flow 7 ( Kazachkov I.V. and Konovalikhin M.J. A Model of a Seam Flow hrough he Volumerically Heaed Paricle Bed In. J. of Thermal Sciences.-.- Vol Kazachkov I.V. Konovalikhin M.J. and Sehgal B.R. Dryou Locaion in a Low-porosiy Volumerically Heaed Paricle Bed J. of Enhanced Hea Transfer..- Vol.8 no.6 p Chorin A.J. A Numerical Mehod for Solving Incompressible Viscous Flow Problems// J. Compu. Phys V..- P Janenko N.N. The Spli-Sep Mehod for he Soluion of Muli-Dimensional Mahemaical Physics Problems.- Novosibirsk: Nauka pp. (in Russian. 7. MacCormack R.W. Efficien Numerical Mehod for Solving Time-Dependen Compressible Navier-Sokes Equaions a High Reynolds Number NASA TM X Warming R.F Beam R.M. On he Consrucion and Applicaion of Implici Facored Schemes for Conservaion Laws // SIAM AMS Proceedings V..- P Samarskii А.А. The Theory of Finie Difference Schemes.- Мoscow: Nauka pp.

16 . Anderson D.A. Tannehill J.C Plecher R.H. Compuaional Fluid Mechanics and Hea Transfer: McGraw-Hill.- New York Wesseling P. An Inroducion o Muligrid Mehods.- John Wiley & Sons Tolsykh A.I. High Accuracy Non-Cenered Compac Difference Schemes for Fluid Dynamics Applicaions: World Scienific Williams A.J. Crof T.N. Cross M. A group based soluion sraegy for muli-physics simulaions in parallel Applied Mahemaical Modelling ( Auber S. Tournier J. Rochee M. Blanche J. N Diaye M. Melen S. Till M. and Ferrand P. Opimizaion of a Gas Mier Using a New Parameric Flow Solver. Proceedings of he ECCOMAS Compuaional Fluid Dynamics Conference Swansea UK. 5. Moreau S. Auber S. Grondin G. and Casalino D. Geomeric Parameric Sudy of a Fan Blade Cascade Using he New Parameric Flow Solver Turb Opy. Proc. ASME Hea Transfer/Fluid Engineering Summer Conf. FEDSM4-568 Charloe USA Babenko Yu.I. Hea and mass ransfer Leningrad Khimiya pp. 7. Koshlyakov N.S. Gliner E.B. Smirnov M.M. Parial differenial equaions of mahemaical physics 8. Fichengolz G.М. The course of differenial and inegral calculus Vol. Moscow.- Nauka pp. 6

17 Table. Compuaion of a emperaure disribuion T ( r θ τ in a Sokes flow around a sphere for a ime dependen emperaure on he sphere (r T ( θ τ cosθ ep( τ. s τ. order θ r nd θ rd θ s 7