SYNChronised numerical methods

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1 1 SYNChronsed numercal methods By Alan S Dawes AWE, Computatonal Physcs Group, Aldermaston, Berkshre, RG7 4PR, UK Abstract The dscretzaton of partal dfferental equatons wll produce naccuraces, and n partcular symmetry errors. Tradtonally, symmetry s ftted nto the numercal method[7] based on three approaches. The frst s to algn the mesh wth the underlyng symmetry and adapt the numercal method accordngly[4, 14]. A second approach s to use the natural coordnate frame and to use a coordnate transformaton from physcal to computatonal space. For ths type of strategy the partcular symmetry s contaned wthn the Jacoban for the transformaton and s embedded n a Geometrc Conservaton Law[17]. Fnally, a thrd approach s to modfy the numercal method to reduce the asymmetrc errors, nherent wthn the truncaton terms, by usng more nformaton from the wder computatonal mesh stencl[13]. Partal dfferental equatons reman unchanged when certan transformatons are performed; these symmetres leave the equatons nvarant after a gven transformaton. Le group theory has been shown to be a general way of determnng these symmetres[15, 3, 16, 12] and has been used to fnd analytc solutons to many partal dfferental equatons, and n partcular, hydrodynamc related equatons[5, 2]. However, n general the dscretzed equatons do not preserve these symmetres. By extendng Le groups to the dscrete system[8, 9, 6, 10, 11] we wll develop gudng prncples for SYNChronsed numercal methods where the SYmmetres of the partal dfferental equatons are Numercally Captured. By usng ths framework we wll show, through the concept of numercal symmetry breakng, why some numercal technques are more sutable than others for a gven applcaton. c Brtsh Crown Copyrght MOD/23 1. Fundamentals Le group theory s the marrage between co-ordnate geometry and group theory. Transformatons form a group and are contnuous based upon the varaton of a group parameter. Le group theory s based upon transformatons of both the ndependent and dependent varables. For m ndependent varables (x 1,...,x m ) and n dependent varables (u 1,...,u n ) these defne a m+n dmensonal phase space n whch all ponts are treated as ndependent. They are transformed n a contnuous manner based on the group parameter and the arbtrarly defned pontwse functons thus, x k = F k (x 1,...,x m,u 1,...,u n ;), ũ j = F j (x 1,...,x m,u 1,...,u n ;), (1.1) for 1 k m and m+1 j n+m and where the tlde represents the transformed varable. It s assumed that all of the functons are contnuous and contnuously dfferentable (C ) n all of the varables and. Note, there s an dentfy value = o that leaves the co-ordnates unchanged. Whence, x k = F k (x 1,...,x m,u 1,...,u n ; o ), u j = F j (x 1,...,x m,u 1,...,u n ; o ). (1.2) Mathematcs n Defence 23

2 SYNChronsed numercal methods 2 If we consder an nfntesmal transformatons from o o + then they can be wrtten as, x k = x k + ξ k (x 1,,u 1, ) +O( 2 ), ũ j = u j + η k (x 1,,u 1, ) +O( 2 ), (1.3) where ξ k and η j are respectvely the nfntesmal generators for the k-th ndependent and j-th dependent varables. The generator for the group can now be wrtten, X = ξ ξ m + η η n. (1.4) x 1 x m u 1 u n The computatonal scentst wll be famlar wth the use of transformatons mappng from physcal to computatonal space[17]. However, for these types of transformatons they are geometrc n nature. Furthermore, they assume that both the tme t and dependent varables (u 1,...,u n ) reman unchanged after the transformaton. When compared to the transformatons Eq.(1.1) they represent a subset of the Le group transformatons. For the general partal dfferental equaton, G(x k,u,u,m,u,mn, )=0, where u,m = u x m, u,mn = 2 u x m x n, (1.5) the transformed PDE s related to the orgnal PDE by the transformaton thus, G( x k,ũ,ũ, m,ũ, mn, ;)=G(x k,u,u,m,u,mn, ; o )+ (X G)+O( 2 ), (1.6) where 0 s the dentfy value that leaves the PDE unchanged, the tlde symbol represents the transformed PDE and the generator X has been prolonged[15] to nclude transformatons of the partal dervatves found wthn G( ) = 0; see next secton. For nvarance of the PDE X G( )=H( )G( ) for the arbtrary functonh( ) provded G( )=0. Whence, G( x k,ũ,ũ, m,ũ, mn, ;)=G(x k,u,u,m,u,mn, ; o ). (1.7) 2. Extenson to Numercal Methods For a gven numercal method (NM) expressed as, G NM (x n,tn,τn,h n +,hn,un 1,un,un +1,un+1, )=0, (2.1) t can be wrtten after the Le group transformaton as, G NM ( ;)=G NM ( ; o )+(X NM G NM ) +O( 2 ), (2.2) wherex NM s the group operator whch has been prolonged to nclude the transformatons of the numercal method. Ths operator must now nclude terms that represent the dscrete dependent varables and ndependent quanttes (mesh postons) wth respect to. For example, Fgure 1 llustrates one way that the ponts(x n,tn,un ) can vary after a transformaton. The group operator X NM wll leave the PDE nvarant and for numercal nvarance, X NM G NM ( )=H NM ( )G NM ( ), (2.3) for the arbtrary dscrete functonh NM (x n,tn,τn,h n +,hn,un 1,un,un +1,un+1, ) snce G NM ( )= 0. Whence, G NM ( ;)=G NM ( ; o ). (2.4) When ths s satsfed the numercal method wll be SYNChronsed wth the PDEs. Alternatvely, a numercal method satsfyng ths wll be known as a SYNC scheme; SYmmetres of the partal dfferental equatons are Numercally Captured. Mathematcs n Defence 23

3 INVARIANT TRUNCATION ERROR 3 t ( x n+1, t n+1,ũ n+1 ) (x n+1,t n+1,u n+1 ) (x+1 n+1,tn+1 +1,un+1 +1 ) ( x (x n,tn,un ) +1 n+1, tn+1 +1,ũn+1 +1 ) (x n ,tn +1,un +1 ) x ( x n, tn,ũn ) ( x n +1, tn +1,ũn +1 ) 3. Invarant Truncaton Error u FIGURE 1. Relatonshp between dscrete Le co-ordnate ponts. Usng Taylor s theorem a numercal method can be shown to be equal to a modfed equaton (ME) n whch the orgnal homogeneous PDE now equals a truncaton error (TE) source term thus, G ME ( )=G( )=TE( ). (3.1) By applyng the prolonged group operator to the modfed equaton t wll be nvarant, lke the PDE, when the truncaton error s also nvarant thus, X NM G ME = δ(te) δ = 0. (3.2) Ths s the condton for a SYNChronsed numercal method. 4. Numercal Symmetry Breakng The equatons of hydrodynamcs can be wrtten n conservaton law form as, u t + f(u) = 0, (4.1) x where the dependent varable u represents some conserved quantty, such as mass. Ths equaton can be wrtten numercally n explct Euleran conservaton law form usng a fxed mesh (please refer to the books by Hrsch[1] for general descrptons of types of numercal methods) as, G NM ( ; o )=u n+1 u n + τn h ( F(un p,...,un +q ;h,τn ) F(u n 1 p,...,un 1+q ;h,τn ))=0, (4.2) Mathematcs n Defence 23

4 SYNChronsed numercal methods 4 where h s the grd spacng (assumed fxed and unform), τ n the tme-step, and F( ) are the numercal flux functons. For smplcty we shall use the notaton F+1/2 n = F(un p,...,un +q ;h,τn ). The ntegers p and q defne the sze of the mesh stencl p+q+1 for the flux functons. The condton that ths wll be a SYNC scheme sx NM G NM =H NM G NM. That s, δu n+1 ( δ δun 1 δ + δτ n ) ( h δ τn δh δf h 2 (F n δ +1/2 F 1/2 n )+ τn n ) +1/2 δfn 1/2 =H NM G NM, (4.3) h δ δ We consder ths framework for the non-lnear vscous Burgers equaton, u t + u u x = u ν 2 x 2, (4.4) where ν s a parameter representng vscosty. The group structure for ths equaton s gven by Hydon, and conssts of fve groups representng scale nvarance, projectve nvarance, Gallean nvarance, temporal and spatal translaton. Wthout loss of generalty we shall consder a three pont stencl (p = 0 and q = 1) and by prolongng the fve group operators yelds, XNM 1 = xn x n + 2t n t u n n u n + h h + 2τn τ u n n 1 u n u n +1 1 u n +1 u n+1 XNM 2 = xn tn +(t x n ) 2 +(x n t n n un tn ) + ht u n n h +(tn +t n+1 )τ n + τ n u n+1 XNM 3 = tn x n + u n + u n + 1 u n +[1 τ n S + t (u x )] +1 u n+1 (4.5) X 4 NM = t n X 5 NM = x n where S t + s a shft operator. We do not provde the whole structure forxnm 2 due to ts complexty. From XNM 4 G NM andxnm 5 G NM t s mmedately obvous thath NM = 0 and that G NM cannot be a functon of x n and t n. ForXNM 1 G NM t can be shown (usng the method of characterstcs) that the flux can be wrtten n terms of dscrete nvarants, J 4 =G(J 1,J 2,J 3 ), (4.6) whereg s an arbtrary flux functon wth the dscrete nvarants gven by, J 1 = un +1 u n, J 2 = h2 τ n, J 3 = u n h, J 4 = τ n F n +1/2. (4.7) In practce the values of the dscrete nvarants wll be restrcted to satsfy numercal stablty and monotoncty. It can be shown that the nvarant flux functon Eq.(4.6) does not satsfy the Gallean and Projectve groups. Therefore, a numercal scheme of the type defned by Eq.(4.2) wll break the symmetry nherent n the PDE. The lack of nvarance les n the ntroducton of gradents wthn some of the prolonged group operators. The Gallean operator ncludes the gradent term [1 τ n S t + (u x )] and represents the relatve moton between the fxed and movng frames and s llustrated n Fgure 2. At t = t n both frames are concdent, but after a short perod τ n the Gallean frame has moved relatve to the fxed frame. Wthn the fxed frame x n = x n+1 and wthn the movng frame x n = x n+1, but the relatve postons between the dfferent frames changes. The value of ũ n+1 moves along a trajectory relatve to the fxed frame and ts value can be thought of as beng nterpolated from the values wthn the fxed frame. Mathematcs n Defence 23

5 CONCLUSIONS 5 x n+1 τ n x n+1 t n+1 = t n + τ n x n h xn x n + h t = t n FIGURE 2. Relatonshp between fxed (black) and movng frames (red). Ths s the source of the gradent term. Note, ths type of analyss can be appled to the projectve operator. A numercal scheme based upon a fxed stencl can break certan symmetres. A scheme that s SYNChronsed wth all of the PDEs symmetres s one where the mesh moves to ts natural postons. In ths paper we wll not consder how ths can be done due to space lmtatons. However, the reader should consder numercal methods based on dscrete nvarants[19, 10, 11] and those based on Movng Frames[18, 8]. 5. Conclusons Usng Le group theory we have presented a concse mathematcal defnton for the nvarance of partal dfferental equatons under a gven transformaton. In ths paper ths framework was extended to ts representatve numercal method and t was shown that t can reman nvarant only f ts truncaton error remans nvarant. Numercal schemes of ths type are SYNChronsed wth ther respectve PDEs symmetres. In ths paper we have shown how numercal nvarance errors are due to numercal symmetry breakng. A scheme that s SYNChronsed s one where the mesh moves to ts natural postons. REFERENCES [1] C. Hrsch. Numercal Computaton of Internal and External Flows, Volumes 1 and 2. John Wley & Sons. [2] S. Atzen and J. Meyer-Ter-Vehn. The Physcs of Inertal Fuson. Clarendon Press, Oxford, [3] G.W. Bluman and J.D. Cole. Smlarty Methods for Dfferental Equatons. Sprnger-Verlag, New York, [4] E. Caramana and P. Whalen. Numercal Preservaton of Symmetry Propertes of Contnuum Problems. J. Comp. Phys. 141, pages , [5] S.V. Coggeshall and R.A. Axford. Le Group Invarance Propertes of Radaton Hydrodynamcs Equatons and Ther Assocated Smlarty Solutons. J. Phys. Fluds, pages , [6] A.S. Dawes. Invarant Numercal Methods. ICFD Conference on Numercal Methods n Fluds, Readng Unversty, UK, March, [7] A.S. Dawes. Sources of Cartesan Mesh Induced Asymmetres Based Upon the Lagrangan + Remap Method. Numercal Methods for Mult-Materal Flud Flows, Praque, CZ, September 2007, [8] A.S. Dawes. Invarant Numercal Methods. Int. J. Num. Meth. Fluds, 56, pages , [9] A.S. Dawes. Invarant Numercal Methods. Symmetry Preservng Methods n CFD and Beyond, ICFD Workshop, Unversty of Readng, Mathematcs n Defence 23

6 SYNChronsed numercal methods 6 [10] V.A. Dorodntsyn. Fnte dfference methods entrely nhertng the symmetry of the orgnal equatons. n Modern Group Analyss: Advanced analytcal and computatonal methods n mathematcal physcs, edted by N. Ibragmov (CRC Press, New York), [11] V.A. Dorodntsyn. Symmetry of Fnte Dfference Equatons. n CRC Handbook of Le Group Analyss of Dfferental Equatons, Volume 1: Symmetres, Exact Solutons and Conservaton Laws, edted by N. Ibragmov (CRC Press, New York), [12] P.E. Hydon. Symmetry Methods for Dfferental Equatons. Cambrdge Unversty Press, [13] A. Kumar. Isotropc Fnte-Dfference. J. Comp. Phys. 2, pages , [14] Margoln and M. Shashkov. Usng a Curvlnear Grd to Construct Symmetry Preservng Dscretsatons for Lagrangan Gas Dynamcs. J. Comp. Phys. 149, pages , [15] P. Olver. Applcaton of Le Groups to Dfferental Equatons. Sprnger, [16] H. Stephan. Dfferental Equatons. Ther soluton usng symmetres. Cambrdge Unversty Press, [17] P.D. Thomas and C.K. Lombard. Geometrc Conservaton Law and ts Applcatons to Flow Computatons on Movng Grds. AIAA Journal, Vol. 17, No. 10, pages , [18] M.Chhay et al. Comparson of some Le-symmetry-based ntegrators. J. Comp. Phys., Volume 230 Issue 5, pages , 21. [19] Bakrova and Dorodntsyn and Kozlov. Symmetry preservng dscrete schemes for the heat transfer equatons. J. Phys. A: Math. Gen., 30, pages , 1997 Mathematcs n Defence 23