An unconditionally stable implicit difference scheme for 2D porous medium equations using four-point NEGMSOR iterative method
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1 A otolly stl plt ffr s for D poros qtos sg for-pot NEGMOR trtv to Jkl Cw V Lg Jt l * Flty of Ntrl Rsors vrst Mlys Kot Kl Mlys Astrt. I ts ppr rl to s propos for solvg svrl two-sol poros qtos (D PME). T to os Nwto Explt Grop MOR (EGMOR) trtv to ly for-pot NEGMOR. Trogot ts ppr tlory vl prol of D PME s srtz y sg t plt ft ffr s orr to for olr pproxto qto. By tkg fx r of gr pots solto o t forlt olr pproxto qto pros lrg olr syst w s solv sg t Nwto trtv to. T solto vtor of t sprs lrz syst s t opt trtvly y t pplto of t for-pot EGMOR to. For t rl xprts tr xpls of D PME r s to llstrt t ffy of t NEGMOR. T rl rslt rvls tt t NEGMOR s ttr ffy trs of r of trtos optto t x solt rror opr to t tst NG NEG NEGOR trtv tos. T stlty lyss of t plt ft ffr s s o D PME s lso prov. Itroto Ts ppr osrs t rl solto of t followg tl-ory vl prol of t two-sol poros qto (D PME): t x x y y () tt s st to t tl oto: x y 0 0 x y 0 x y () t ory otos: * Corrspog tor: t@s..y T Ators pls y EDP s. Ts s op ss rtl strt r t trs of t Crtv Coos Attrto Ls 4.0 (ttp://rtvoos.org/lss/y/4.0/).
2 0yt g yt yt g yt () x 0t g xt x t g4 xt 0 t wt so rl prtrs: 0. Eqto () kow s t grlz for of t t qto. By rfrrg to qto () t t qto ot y tkg. I prtlr s lk t Bosssq qto ot. Eqto () s typ of prtl ffrtl qto t olr grt prol lss w s wly s to sr ols for gss poros t ffso rgro fltrto poplto ys s wll s g rgy pyss []. T prs of olr tr qto () ks t xt solto fflt to ot. Grlly t rl to osr s ltrtv opto solvg t prtl ffrtl qto splly for t s wr t xt solto s lost possl to ot. T rl to yls pproxt solto tt s ffrg fro t xt solto y lss t spf tolr. Frtror trs of t pltto of t rl to ffy s o of t portt rtrsts to osrv. T slt rl to sol t o tt rqrs lss optto t oplxty otg t sr rslt []. T rl solto of ltsol PME s rv osrl trst fro r of rsrrs. For st Bor t l. [] st t rl solto of t o-sol (D) s of PME tt s oll s o t stlty poo t ol ps flow of two sl fls trog t l oogos poros. T to s s Crk-Nolso ft ffr s. Mwl Pr t l. [4] ppl t ft lt to for solvg t D PME ol tt rprstg t stlty poo t rovry of ol fro rsrvor. For t rl solto of D PMEs Ngo Hg [5] st t pplto of ptv ovg s ft lt to s provt to t xstg ovg s to. I to Cw l [6] strt t vstgto o t ffy of rl to tt os Nwto MOR (NMOR) trtv to for solvg D PMEs. T Cw l [7] xt tr sty wt t pltto of t Explt Grop (EG) to [8] to prov t ffy of t NMOR trtv to solvg t D PMEs. T of ts ppr s to sty t ffy of t Nwto Explt Grop MOR trtv to or NEGMOR for solvg svrl xpls of D PME. T otl of ts ppr s rrg s follows. to sows t srtzto of D PME y sg t plt ft ffr s to olr pproxto qto t forto of sprs lrg olr syst. to prsts t solto of t olr syst y t pltto of t Nwto EGMOR trtv tos. to 4 llstrts t rl xprts to 5 ols t fg of ts sty. Iplt ft ffr pproxto qto To ot t rl soltos w osr rtglr o tt s rk off t t sqr sg t gr ls s follows: x y. (4) M for t t stp w f
3 t kk N (5) T rl soltos tt r rprst y t pots of trsto tw t two gr ls r ot y x y t. For t sk of splty w s x y t wt M N By sg t plt ft ffr s w ot t plt ft ffr pproxto qto of qto () tt wrtt s f A A A A 4 (6) wr A A k / A A4 k / 4. Eqto (6) s t fto t tror gr pot ovr t solto o tt lks wt goroo gr pots to ot x- y-rtos. As rslt sprs lrg olr syst s for grlly wrtt s wr F 0 (7) T F f f f f fm M M M. Lr stlty lyss Grlly t pplto of t lr Forr lyss to t olr prtl ffrtl qto s s D PME ot rgorosly stf. Nvrtlss t s fo to fftv prt [9-]. To sow t stlty of t plt ft ffr s s o D PME w ss tt t solto fto xy t s o t sptotporl rgo of 0 x y 0 t. Now t olr tr froz t gr pot t solto o gvs wr v x t x v x. Eqto (8) rwrtt to t v x Fro r t stlty of t plt ft ffr s s ts ppr sow y pltg t vo N to ovr qto (9) s sow Tor. y v y y (8) (9)
4 Tor. T flly plt ft ffr s of qto (9) o t ft o 0 x y wt zro ory otos for ll t 0 s otolly stl. Proof. To g w f t orrspog plt ft ffr oprtors s follows. D t pq pq pq k (0) D xd x pq p q pq p q () D yd y pq pq pq pq () By sstttg qto (0) - () to qto (9) ftr so rrrgg t orrspog qto ot s pq p q pq p q pq pq pq pq () wr k v. To pply t vo N to lt pq p q (4) By sg qto (4) qto () trsfor to. (5) 4s 4s w ot t plto ftor (6) 4 s 4 s Bs o qto (6) t s tt t plt ft ffr s to D PME s otolly stl wt 0 for ll postv vls of for. 4 Drvto of t for-pot NEGMOR trtv to I t otxt of rg t opttol oplxty t ovrg rt of t solto pross s to rs. Tr r svrl trtv tos tt v propos orr to r t opttol oplxty ts ppr osrs t pplto of t Explt Grop (EG) to w ws tro y Evs [8]. Ts to ss t opt of rkg t orgl trx to svrl sll fx-sz grops of pots. By ts wy t opttol oplxty r y solvg svrl sll grops of pots st of lrg lr syst pr trto. T EG to s s xtsvly togtr wt so fft trtv tos s s OR MOR AOR [-5]. 4
5 5 Bfor t EG to s ppl qto (7) trsfor to lr syst y sg t Nwto to s follows:... F H J F (7) wr F J s pt-gol Jo trx wt so of M M H F r t ol trs wt so of M for t trtv x w ot t s s osr to t xt t lvl. T qto (7) s v to svrl oplt grops of for pots so grop tt trt s grop of two pots /or sgl pot [8]. Bs o y oplt grop of for gr pots t grl for of t EG trtv to s gv y 4 (8) wr f f f 4 f. Bss tt K Yog [6] tro Mof ssv Ovr-Rlxto (MOR) to w osr s pot trtv to tt s two rlxto prtrs. T s of t MOR to s to ssot t ffrt prtr wt row ol of sprs lrg lr syst. Trfor to rv t NEGMOR trtv to two ffrt rlxto prtrs r to qto (8) fors 4 (9) 4 (0) Bs o qtos (9) (0) t NEG NEGOR trtv tos ot y oosg wt t rg of rsptvly. T pltto of t NEGMOR trtv to sow Fgr.
6 Fgr. T pltto of NEGMOR trtv to. By rfrrg to Fgr t rl solto of D PME y sg NEGMOR trtv to srz to Algort. T opt vls of r tr w t lst r of trtos s ot ftr fw xtos. 0 Drg t xtos w osr tolr rror of 0. Algort : NEGMOR trtv to. t p t vls for A A A A 4 0. t t tl gss. 0. Forlto of t sqr trx J t ol trx v. Itrto of qtos (9) (0) F F v. Covrg tst : f H H t H v. v. Covrg tst : f Dsply t pproxt soltos. t go to t xt t stp 6
7 5 Nrl xprts As oprtv lyss tr xpls of D PME r s. T ffy of t NEGMOR trtv to s vstgt s opr to t NG NEG NEGOR trtv tos. Nr of trtos xto t sos x solt rror (rror) o t sts of oprso. Expl [7]: 0. () t x x y y wt xt solto xyt x y ( 0. 4)t. Expl [7]: 0. t x x y y () wt xt solto xyt 5 x y t /. Expl [8]: 5 5 t x x y / 4 wt xt solto xyt 4 x y t / 5. I t rl xprts w osr fv ffrt orrs of t lr syst M w t ffrt vls of s sz M r T rl rslts r t Tl to t oprtv lyss tw t NEGMOR trto fly gst t NG tos s o y lltg t rto prtg t r of trtos t optto t s follows: Agst otr trtv tos: s s % NG NEGMOR 00 % s % NG NEGMOR 00% (4) NG sng wr s r rto prtg of r of trtos xto t for NEGMOR to rsptvly. y () Tl. Coprso trs of r of trtos s o ll trtv tos for tr xpls. Expl Mto Ms z NG NEG NEGOR NEGMOR NG NEG NEGOR NEGMOR NG NEG NEGOR NEGMOR
8 Tl. Coprso trs of xto t (sos) s o ll trtv tos for tr xpls. Expl Mto Ms z NG NEG NEGOR NEGMOR NG NEG NEGOR NEGMOR NG NEG NEGOR NEGMOR Tl. Coprso trs of x solt rror s o ll trtv tos for tr xpls. Expl Mto Ms z NG NEG NEGOR NEGMOR NG NEG NEGOR NEGMOR NG NEG NEGOR NEGMOR Fro t rl xprt rslts Tl - t rto prtg of t NEGMOR trto fly s opr t G trtv to for tr xpls r srz Tl 4. Tl 4. Rto prtg of t r of trtos t xto t for NMEGOR trto fly opr to NG to Expl Mtos Nr of Itrtos (%) Exto T (%) NEG % % NEGOR % % NEGMOR % % NEG % % NEGOR % % NEGMOR % % NEG %.4-5.8% NEGOR % % NEGMOR % % 8
9 Rfrrg to t rslts of rl xprts for tr tst xpls Tl 4 t pot ot tt t prtg rto of r of trtos for t NEGMOR trtv to v l pproxtly y % % % rsptvly s opr wt t NG to. I ft plttos of xto t for NEGMOR trtv to r fstr ot % % % rsptvly t t NG to. 6 Colso Ts ppr propos t otolly stl plt ffr s togtr wt t NEGMOR trtv to for solvg svrl xpls of D PME. T plt ft ffr s s sssflly ppl to srtz t grl for of D PME. I to to tt t sprs lrg lr syst s solv sg t NEGMOR trtv to t rl rslts rvl tt t propos to s or fft trs of r of trtos xto t s opr to t tst NG NEG NEGOR trtv tos. Ovrll tr of t gt of x solt rror t ry of t propos NEGMOR trtv to t tr tst rl tos r oprl. T tors grtflly kowlg tt ts ppr ws prtlly spport y t rsr grt s (GG00-G-M-/06) fro vrst Mlys Mlys.. Rfrs. J.L. Vzqz. T Mttl Tors of Dffso: Nolr Frtol Dffso. Ltr Nots Mtts 86 (06).. P.B. Ptl.P. Vr. Nrl Copttol Mtos. Rvs E. Oxfor: Alp Itrtol Lt. (009). R.N. Bor V.H. Pr M.N. Mt. Itr. J. of Iovtv Rsr Egrg Tology. (04) 4. V.H. Pr M.N. Mt T. Ptl. Itr. J. of Av Egrg Tology (0). 5. C. Ngo W.Z. Hg. J. Cop. Pyss. (07) 6. J.V.L. Cw. J. l. T Itrtol Cofr o Mttl s ttsts (ICM06) AIP Cof. Pro. 79 (06). 7. J.V.L. Cw. J. l. Jorl of Appl Mtts Copttol Ms 5 (06) 8. D.J. Evs Itr. J. Cop Mt. 7 (985) 9. R.D. Rtyr K.W. Morto. Dffr tos for tl-vl prols.. Nw York: Jo Wly & os I. (967) 0. B.F. Fg T. Mts. J. Copttol Appl Mtts. 90 (998). D. K W. Proskrowsk Itr. J. Mt. & Mttl s. 7 (004). J. trkwr Ft ffr ss prtl ffrtl qtos. Pllp: IAM. (004). J. l M.K. Hs M. Ot.A.A. Kr. Jorl of Mtts yst. (0) 4. A. J. l. Itrtol Jorl of Coptr Appltos. 5 6 (0) 9
10 5. M.K.M. Akr J. l. Appl Mttl s. 9 (05 6. D.R. K D.M. Yog. 97. Mtts of Coptto. 6 9 (97) 7. A.D. Poly V.F. Ztsv. Hook of Nolr Prtl Dffrtl Eqtos. Bo Rto: Cp Hll/CRC Prss. (004) 8. B.P. or P.J. v r How. ACM Trstos o Mttl oftwr. 0 4 (984). 0
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