On Strongly Consistent Finite Dierence Approximations

Transcription

1 D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 O Strogly Cosistet Fiite Dierece Approximatios Domiik Michels 1, Vladimir Gerdt 2, Dmitry Lyakhov 1, ad Yuri Blikov 3 1 Visual Computig Ceter Kig Abdullah Uiversity of Sciece ad Techology, Saudi Arabia 2 Laboratory of Iformatio Techologies Joit Istitute for Nuclear Research, Duba, Russia 3 Departmet of Mathematics ad Mechaics Saratov State Uiversity, Saratov, Russia PCA-2018, St. Petersburg, April 17, 2018

2 D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Cotets 1 Itroductio 2 Fiite Dierece Approximatios 3 Cosistecy Aalysis 4 Numerical Tests 5 Coclusios

3 Itroductio Numerical solvig PDEs Solvig PDEs i practice PDE(s) + IC(s) or/ad BC(s) Discretizatio (FDM, FEM, FVM) Algebraic (dierece) equatios Numerical solvig Approximate solutio I the ite dierece method (FDM) partial dieretial equatios (PDE(s)) are replaced with their ite dierece approximatio (FDA) o a grid with spacigs h := {h 1,..., h }. PDE(s) = FDA The iitial coditios (ICs) ad/or boudary coditios (BCs) are also discretized. The, together with FDA it gives a ite dierece scheme. D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25

4 Itroductio D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Requiremets for FDA Covergece of a approximate solutio to a solutio to PDE(s) at h 0. Challege: d FDA whose solutios coverge to solutios to PDE(s). Such FDA must iherit at the discrete level all algebraic properties of PDE(s) such as coservatio laws, symmetries, maximum priciple, etc.). For polyomially oliear PDE(s) s(trog)-cosistecy of FDA (Gerdt'12). S-cosistecy FDA is s-cosistet with PDE(s) if ay dierece cosequece of FDA i the limit h 0 is reduced to a dieretial cosequece of PDE(s).

5 Itroductio D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Dieretial Thomas Decompositio Deitio Let S = ad S be ite sets of dieretial polyomials such that S = cotais equatios ( s S = ) [s = 0] whereas S cotais iequatios ( s S ) [s 0]. The the pair ( S =, S ) of sets S = ad S is called dieretial system. Let Sol (S = /S ) deote the solutio set of the system ( S =, S ), i.e. the set of commo solutios of dieretial equatios { s = 0 s S = } that do ot aihilate elemets s S. Theorem Ay dieretial system ( S =, S ) is decomposable ito a ite set of ivolutive dieretial subsystems (S = i, S i ) with a disjoit set of solutios: (S = /S ) = i (S = i /S i ), Sol (S = /S ) = i Sol (S = i /S i ). (1)

6 Itroductio Navier-Stokes PDE system By completig to ivolutio the Navier-Stokes system of equatios for usteady two-dimesioal motio of icompressible viscous liquid of costat viscosity ca be writte i the followig form f 1 := u x + v y = 0, f 2 := u t + uu x + vu y + p x 1 Re F := (u xx + u yy ) = 0, f 3 := v t + uv x + vv y + p y 1 Re (v xx + v yy ) = 0, f 4 := ux 2 + 2v x u y + vy 2 + p xx + p yy = 0. Here f 1 - the cotiuity equatio, f 2, f 3 - the proper Navier-Stokes equatios, f 4 - the pressure Poisso equatio which is the itegrability coditio for {f 1, f 2, f 3 }, (u, v) - the velocity eld, p - the pressure, Re - the Reyolds umber. D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25

7 Itroductio D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Divergece form The ivolutive Navier-Stokes system admits coservatio law of the form P t I terms of {f 1, f 2, f 3, f 4 } this form reads Coservatio law form + Q x + R y = 0. f 1 : f 2 : f 3 : f 4 : x u + y v = 0, t u + ( x u 2 + p 1 Re u ) ( x + y vu 1 Re u y) = 0, t v + ( x uv 1 Re v ) ( x + y v 2 + p 1 Re v ) y = 0, x (uu x + vu y + p x ) + y (vv y + uv x + p y ) = 0.

8 Fiite Dierece Approximatios D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Computatioal mesh We use a orthogoal ad uiform computatioal grid as the set of poits (jh, kh, τ) R 3, τ > 0, h > 0, (j, k, ) Z 3. I a grid ode (jh, kh, τ) a solutio is approximated by the triple of grid fuctios {uj,k, vj,k, pj,k} := {u, v, p} x=jh,y=kh,t=τ. We itroduce diereces {σ x, σ y, σ t } actig o a grid fuctio φ(x, y, t) as σ x φ = φ(x + h, y, t), σ y φ = φ(x, y + h, t), σ t φ = φ(x, y, t + τ) ad deote by R the rig of dierece polyomials over K.

9 Fiite Dierece Approximatios D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Itegratio cotour To discretize NSS o the grid choose the itegratio cotour Γ i the (x, y) plae k + 2 k + 1 k j j + 1 j + 2

10 Fiite Dierece Approximatios D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 The Navie-Stokes system i itegral form Itegral coservatio law form vdx + udy = 0, Γ x j+2 x j x j+2 x j y k+2 y k y k+2 y k udxdy vdxdy t +1 t t +1 t t +1 t t +1 t ( Γ ( Γ ( vu 1 Re uy ) dx ( u 2 + p 1 Re ux ) dy ) dt = 0, ( v 2 + p 1 Re vy ) dx ( uv 1 Re vx ) dy ) dt = 0, ( ) ( ) (v 2 ) y + (uv) x + p y dx + (u 2 ) x + (vu) y + p x dy = 0. Γ

11 Fiite Dierece Approximatios D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Additioal relatios Now we add itegral relatios betwee depedet variables ad derivatives Exact itegral relatios x j+1 (u 2 ) xdx = u(x j+1, y) 2 u(x j, y) 2, (v 2 ) ydy = v(x, y k+1 ) 2 v(x, y k ) 2, x j y k x j+1 (uv) xdx = u(x j+1, y)v(x j+1, y) u(x j, y)v(x j, y), x j y k+1 (uv) ydy = u(x, y k+1 )v(x, y k+1 ) u(x, y k )v(x, y k ), y k x j+1 u xdx = u(x j+1, y) u(x j, y), x j x j+1 v xdx = v(x j+1, y) u(x j, y), x j x j+1 p xdx = p(x j+1, y) u(x j, y), x j y k+1 y k+1 u ydy = u(x, y k+1 ) u(x, y k ), y k y k+1 v ydy = v(x, y k+1 ) u(x, y k ), y k y k+1 p ydy = p(x, y k+1 ) u(x, y k ). y k

12 Fiite Dierece Approximatios Fiite dierece approximatio 1 By usig the midpoit itegratio approximatio for the itegrals over x ad y ad the top-left corer approximatio for itegratio over t. The elimiatio of partial derivatives from the obtaied dierece system gives the followig FDA with a 5 5 stecil (Gerdt,Blikov'2009) FDA 1 = e 1 j,k := u j+1,k u j 1,k e 2j,k := u+1 jk u jk 1 Re τ e 3j,k := v +1 jk v jk 1 Re + v j,k+1 v j,k 1 = 0, + u 2 j+1,k u 2 j 1,k + v j,k+1 u j,k+1 v j,k 1 u j,k 1 ( u ) j+2,k 2u jk +u j 2,k + u j,k+2 2u jk +u j,k 2 = 0, 4h 2 4h 2 τ + u j+1,k v j+1,k u j 1,k v j 1,k v j,k+1 2 v 2 j,k 1 ( v ) j+2,k 2v jk +v j 2,k + v j,k+2 2v jk +v j,k 2 = 0, 4h 2 4h 2 e 4j,k := u 2 j+2,k 2u 2 j,k +u 2 j 2,k + v 2 j,k+2 2v j,k 4h 2 4h 2 2 +v 2 j,k 2 + p j+1,k p j 1,k + p j,k+1 p j,k u j+1,k+1 v j+1,k+1 u j+1,k 1 v j+1,k 1 u j 1,k+1 v j 1,k+1 +u j 1,k 1 v j 1,k 1 4h 2 + p j,k+2 2p jk +p j,k 2 4h 2 = 0. + p j+2,k 2p jk +p j 2,k 4h 2 D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25

13 Fiite Dierece Approximatios D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Fiite dierece approximatio 2 If oe applies the trapezoidal approximatio to the itegral relatios for u x, u y, v x, v y, u 2 ) x, (v 2 ) y ad p istead of the midpoit approximatio, the it produces FDA with a 3 3 stecil (Gerdt,Blikov'2009) FDA 2 = e 1 j,k := u j+1,k u j 1,k + v j,k+1 v j,k 1 = 0, e 2j,k := u+1 jk u jk τ + ujk u j+1,k u j 1,k + vjk u j,k+1 u j,k 1 + p j+1,k p j 1,k ( 1 u ) j+1,k 2u jk +u j 1,k Re + u j,k+1 2u jk +u j,k 1 = 0, h 2 h 2 e 3j,k := v +1 jk v jk e 4 j,k := 1 Re ( u + ujk v j+1,k v j 1,k + vjk v τ ( v j+1,k 2v jk +v j 1,k h 2 j+1,k u j 1,k + p j+1,k 2p jk +p j 1,k h 2 j,k+1 v j,k 1 + v j,k+1 2v jk +v j,k 1 h 2 ) = 0, ) 2 v + 2 j+1,k v j 1,k u j,k+1 u j,k p j,k+1 2p jk +p j,k 1 h 2 = 0 + p j,k+1 p j,k 1 ( v j,k+1 v ) 2 j,k 1

14 Fiite Dierece Approximatios D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Fiite dierece approximatio 3 The third approximatio with 3 3 stecil is obtaied from NSS by the covetioal discretizatio what cosists of replacig the temporal derivatives with the forward diereces ad the spatial derivatives with the cetral diereces. FDA 3 = e 1 j,k := u j+1,k u j 1,k + v j,k+1 v j,k 1 = 0, e 2j,k := u+1 jk u jk τ + ujk u j+1,k u j 1,k + vjk u j,k+1 u j,k 1 + p j+1,k p j 1,k ( 1 u ) j+1,k 2u jk +u j 1,k Re + u j,k+1 2u jk +u j,k 1 = 0, h 2 h 2 e 3j,k := v +1 jk v jk e 4 j,k := 1 Re ( u + ujk v j+1,k v j 1,k + vjk v τ ( v j+1,k 2v jk +v j 1,k h 2 j+1,k u j 1,k + p j+1,k 2p jk +p j 1,k h 2 j,k+1 v j,k 1 + v j,k+1 2v jk +v j,k 1 h 2 ) = 0, ) 2 v + 2 j+1,k v j 1,k u j,k+1 u j,k p j,k+1 2p jk +p j,k 1 h 2 = 0 + p j,k+1 p j,k 1 ( v j,k+1 v ) 2 j,k 1

15 Cosistecy Aalysis D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Dieretial ad dierece cosequeces A perfect dierece ideal F geerated by F R is the smallest dierece ideal cotaiig F ad such that for ay f R ad k 1, k 2, k 3 N 0 (σ x f ) k 1 (σ y f ) k 2 (σ t f ) k 3 F = f F. I dierece algebra, perfect ideals play the same role as radical ideals i commutative ad dieretial algebra. Set F R (NSS) geerates radical dieretial ideal F. Let a ite set of dierece polyomials f1 = = f p = 0, F := { f1,... f p } R be a FDA to F. Dieretial ad dierece cosequeces A dieretial (resp. dierece) polyomial f R (resp. f R) is dieretial-algebraic (resp. dierece-algebraic) cosequece of F (resp. F) if f F (resp. f F ).

16 Cosistecy Aalysis D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Covetioal (weak) cosistecy of FDA We shall say that a dierece equatio f = 0 implies (i the cotiuous limit) the dieretial equatio f = 0 ad write f f if f does ot cotai the grid spacigs h, τ ad the Taylor expasio about a grid poit (uj,k, v j,k, p j,k ) trasforms equatio f = 0 ito f + O(h, τ) = 0 where O(h, τ) deotes expressio which vaishes whe h ad τ go to zero. Deitio The dierece approximatio F is (weakly or w-)cosistet with F if p = 4 ad ( f F ) ( f F ) [ f f ]. The requiremet of w-cosistecy which has bee uiversally accepted i the literature, is ot satisfactory by the followig two reasos: 1 The cardiality of FDA to a system of dieretial equatios may be dieret from that i the system. 2 A w-cosistet FDA may ot be good i view of iheritace of properties of the uderlyig dieretial equatio(s) at the discrete level.

17 Cosistecy Aalysis D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Strog cosistecy Deitio A FDA to PDE(s) is strogly cosistet or s-cosistet if ( f F ) ( f [F ] ) [ f f ]. The algorithmic approach (Gerdt'12) to vericatio of s-cosistecy is based o the followig statemet. Theorem A dierece approximatio F R to F R is s-cosistet i a (reduced) stadard basis G of the dierece ideal [ F] satises ( g G ) ( f [F] ) [ g f ]. Give a dieretial polyomial f R, oe ca algorithmically check its membership i F by performig the ivolutive Jaet reductio.

18 Cosistecy Aalysis D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 S-cosistecy aalysis of FDA 1,2 ad 3 All three FDAs are w-cosistet. This ca be easily veried by the Taylor expasio of the ite diereces i the set F := {e 1 j,k, e 2 j,k, e 3 j,k, e 4 j,k } about the grid poit {hj, hk, τ} whe the grid spacigs h ad τ go to zero. Propositio Amog weakly cosistet FDAs 1,2, ad 3 oly FDA 1 is strogly cosistet. Corollary A stadard basis G of the dierece ideal geerated by the set of polyomials i FDA 1 satises the coditio ( g G ) ( f [F] ) [ g f ].

19 Numerical Tests D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Exact Solutio Suppose that the NSS is deed for t 0 i the square domai Ω = [0, π] [0, π] ad provide iitial coditios for t = 0 ad boudary coditios for t > 0 ad (x, y) Ω accordig to the exact solutio (Pearso'64) u := e 2t/Re cos(x) si(y), v := e 2t/Re si(x) cos(y), p := e 4t/Re (cos(2x) + cos(2y))/4. We compute the error by meas of formula: e g = max j,k gj,k N g(x j, y k, t f ). 1 + g(x j, y k, t f )

20 Numerical Tests Relative error i u, v ad p with FDA 1 for Re = x 10 5 x x y 0 0 x y 0 0 x 50 y 0 0 x Computed error with FDA 1 (u, v ad p, respectively): N = 40, tf = 1, Re = 102 ad m = 100 D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25

21 Numerical Tests D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Numerical problem We simulate a Karma vortex street by solvig the Navier-Stokes system umerically over time usig the three above preseted FDAs. The relative error of the coguratio vector orm (p, u, v) is measured over time. The superior behavior of the s-cosistet FDAs compared to the s-icosistet FDA ca clearly be observed. Whereas FDA 2,3 performs slightly better tha FDA 1 for small t < 2 s, FDA 1 outperforms FDA 2,3 i the log term. As expected, stability ca be improved by icreasig spatial resolutio m. Sice i our experimets we are essetially iterested i comparig dieret discretizatios of u, v, ad p o the space domai, the value of the time step was always chose i order to provide stability. Usig Re = 220 we ca observed the characteristic repeatig patter of the swirlig vortices.

22 Numerical Tests D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Numerical problem We simulate a Karma vortex street by solvig the Navier-Stokes system umerically over time usig the three above preseted FDAs. The relative error of the coguratio vector orm (p, u, v) is measured over time. The superior behavior of the s-cosistet FDAs compared to the s-icosistet FDA ca clearly be observed. Whereas FDA 2,3 performs slightly better tha FDA 1 for small t < 2 s, FDA 1 outperforms FDA 2,3 i the log term. As expected, stability ca be improved by icreasig spatial resolutio m. Sice i our experimets we are essetially iterested i comparig dieret discretizatios of u, v, ad p o the space domai, the value of the time step was always chose i order to provide stability. Usig Re = 220 we ca observed the characteristic repeatig patter of the swirlig vortices.

23 Numerical Tests D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Numerical problem We simulate a Karma vortex street by solvig the Navier-Stokes system umerically over time usig the three above preseted FDAs. The relative error of the coguratio vector orm (p, u, v) is measured over time. The superior behavior of the s-cosistet FDAs compared to the s-icosistet FDA ca clearly be observed. Whereas FDA 2,3 performs slightly better tha FDA 1 for small t < 2 s, FDA 1 outperforms FDA 2,3 i the log term. As expected, stability ca be improved by icreasig spatial resolutio m. Sice i our experimets we are essetially iterested i comparig dieret discretizatios of u, v, ad p o the space domai, the value of the time step was always chose i order to provide stability. Usig Re = 220 we ca observed the characteristic repeatig patter of the swirlig vortices.

24 Numerical Tests D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25 Simulatio of the K arm a vortex street

25 Numerical Tests Simulatio of the K arm a vortex street 0-2 log(error) Time / s Ðèñ.: Temporal evolutio of the relative error of the K arm a vortex street simulatio usig dieret FDAs: FDA 1 (red curves), FDA 2 (gree curves), ad FDA 3 (blue curve). Moreover, dieret spatial resolutios are used: m = 250 (dotted curves), m = 500 (dashed curves), ad m = (solid curves). D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25

26 Coclusios Coclusios Mai results obtaied We ivestigated s-cosistecy of three ite dierece approximatios to the Navier-Stokes equatios for usteady two-dimesioal motio of icompressible viscous liquid of costat viscosity. By usig algorithmic methods of dieretial ad dierece algebra we show that oe of the approximatios which is characterized by a 5 5 stecil is s-cosistet whereas the other two with a 3 3 stecil are ot. This result is at variace with uiversally accepted opiio that discretizatio with a more compact stecil is umerically favoured. Our computer experimetatio revealed much better umerical behavior of the s-cosistet approximatio i compariso with the cosidered s-icosistet oes. D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25

27 Coclusios Ackowledgmets D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April / 25