On the application of explicit spatial filtering to the variables or fluxes of linear equations


 Laurence Taylor
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1 Journl of Computtionl Physics 225 (27) Short Note On the ppliction of explicit sptil filtering to the vriles or fluxes of liner equtions Christophe Bogey *, Christophe Billy Lortoire de Mécnique des Fluides et d Acoustique, UMR CNRS 559, Centre Acoustique, Ecole Centrle de Lyon, 36, Avenue Guy de Collongue, 6934 Ecully Cedex, Frnce Received 3 Decemer 26; received in revised form 2 April 27; ccepted April 27 Aville online 2 April 27 Keywords: Finite differences; Explicit filtering; Highorder; Wve eqution; Lrgeeddy simultion. Introduction The need for filtering highfrequency wves is recurrent issue in numericl simultions. These wves might indeed led to instility, nd they re in generl not clculted ccurtely y the discretiztion lgorithms. Selective filters hve therefore een designed in order to dmp highfrequency wves without ffecting significntly lowfrequency disturnces [ 6]. These filters re prticulrly used in computtionl erocoustics, ut they pper lso suitle for LrgeEddy Simultions (LES), in which only the scles lrger thn the grid size re computed, nd whose equtions re derived formlly y pplying filter opertor to the Nvier Stokes equtions [7]. Moreover LES sed specilly on explicit filtering hve een lso developed [8 ]. In prctice, the flow vriles re usully filtered explicitly fter ech time step. Consider for exmple the time integrtion of the following differentil eqution: ou of ðuþ þ ¼ ðþ ot ox where u is the unknown vrile nd the flux opertor F is function of u. The solution t (n + )th itertion t time t + Dt is otined from solution u n t nth itertion t time t, where Dt is the time step, in the following wy: the numericl integrtion of Eq. () provides u n+ t time t + Dt, which is then filtered, yielding the solution u nþ t (n + )th itertion (the r denotes the filtering). In this cse, the dissiptive effects of the filtering on the lrge scles depend on the filters pplied. In order to minimize undesirle dmping for ny filter, the explicit filtering of the flow fluxes hs een proposed insted of tht of the vriles [3]. The following eqution is then solved: ou of ðuþ þ ¼ ot ox ð2þ * Corresponding uthor. Fx: Emil ddresses: (C. Bogey), (C. Billy) /$  see front mtter Ó 27 Elsevier Inc. All rights reserved. doi:.6/j.jcp
2 22 C. Bogey, C. Billy / Journl of Computtionl Physics 225 (27) 2 27 The fluxes re thus filtered efore derivtion during the time integrtion. This method hs een used for LES [] nd for Direct Numericl Simultions [2], nd enles to void the cumultive dissiptive effects tht might result from the multiple filterings of the flow vriles fter ech time step. Unfortuntely other spurious negtive effects re likely to e produced. In this note, the influence of filtering the vriles or the fluxes is investigted for liner opertor F(u). Results re shown for stndrd explicit centered highorder schemes, nmely the thorder finite differences nd the 6th, 8th, th nd 2thorder filters, whose coefficients cn e found in the Appendix nd in Ref. [5] for instnce. The ccurcies in phse nd in mplitude re presented in the wve numer spce, nd they re illustrted y the solutions of test cse. 2. Effects of sptil filtering in the wve numer spce 2.. Filtering of vriles For simplicity, the onedimensionl wve eqution ou ot þ ou ox ¼ discretized on mesh with uniform spcing Dx is considered. The sptil derivtive is pproximted y centered, (2N + )point, finite differences, yielding X N ou ox ðxþ ¼ j uðx þ jdxþ ð4þ Dx j¼ N where the scheme coefficients re such s j = j, ensuring no dissiption. By pplying sptil Fourier trnsform to (4) s in [3] for instnce, the effective wve numer k fd of the scheme is otined k fd Dx ¼ 2 XN j sinðjkdxþ ð5þ j¼ The sptil derivtion thus leds to numericl wve numer k fd tht differs from the exct wve numer k. The phse errors E k =(kdx k fd Dx)/p otined for the thorder finite differences re represented in Fig. s function of the wve numer kdx. They re negligile for low wve numers, ut significnt for high wve numers. More quntittively, the ccurcy limit of the scheme, estimted from the ritrry criterium E k nd expressed in term of numer of points per wve length, is k k /Dx = In Fig., wves for kdx = p re shown not to e properly clculted. They cn e dmped y pplying centrl, (2N + )point filter to vrile u fter ech time step, providing uðxþ ¼uðxÞ r d ½D p ðuþšðxþ ð6þ ð3þ π/8 π/4 π/2 π 5 π/8 π/4 π/2 π Fig.. Representtion in logrithmic scles, s function of the wve numer kdx of: () phse error (kdx k fd Dx)/p otined for the stndrd thorder finite differences, () dmping functions D p ðkdxþ of the stndrd 6th, 8th, th nd 2thorder filters.
3 C. Bogey, C. Billy / Journl of Computtionl Physics 225 (27) with ½D p ðuþšðxþ ¼ XN j¼ N d j uðx þ jdxþ ð7þ where the filter coefficients re such s d j = d j, ensuring no phse error, nd r d is constnt etween nd. The ppliction of Fourier trnsform to (7) gives the dmping function of the filter D p ðkdxþ ¼d þ 2 XN j¼ d j cosðjkdxþ ð8þ in the wve numer spce. The dmping functions otined for the 6th, 8th, th nd 2thorder filters re presented in Fig.. Gridtogrid oscilltions re found to e removed y the filters, wheres low wve numers re wekly ffected. To determine n ccurcy limit for ech filter, the criterium r d D p is used. The vriles eing filtered fter ech itertion, the effects of filtering re cumultive, nd it is not necessry to set r d =. A vlue of r d =.2 is therefore chosen here. The limits otined for the filters re reported in Tle in terms of numer of points per wve length. They rnge from k d /Dx = 8.33 for the 6thorder filter down to k d /Dx = 4.82 for the 2thorder filter. When centered schemes re used nd tht filtering is pplied explicitly to the flow vriles, the sptil discretiztion thus genertes phse errors due to the derivtion nd dmping due to the filtering, which cn e evluted seprtely s in Tle Filtering of fluxes In the cse of flux filtering, the flow fluxes re filtered efore sptil derivtion in order to remove gridtogrid oscilltions, so tht the frequency content of the solution is controlled. The following eqution is then solved: ou ot þ o ox ½u D pðuþš ¼ ð9þ The ppliction of sptil Fourier trnsform, denoted y ht, yields o^u ot þ ikfd ½ D p ðkdxþš^u ¼ ðþ where the expressions of k fd nd D p re given y (5) nd (8). The numericl wve numer ssocited with the sptil discretiztion is now k H ¼ k fd ½ D p ðkdxþš, which suggests tht the explicit filtering of the fluxes does not introduce dditionl dissiption of the vrile, ut modifies the wve numer clculted y the finite differences. These effects of flux filtering re shown in Fig. 2 nd, where, respectively, the pproximted wve numer k w Dx nd the phse errors E k =(kd x k w Dx)/p, otined for thorder finite differences used in comintion with 6th, 8th, th nd 2thorder filters, re presented s functions of the exct wve numer kdx. Filtering the fluxes is clerly found to increse the phse errors, oth for low nd high wve numers. The deteriortion in phse ccurcy is however less importnt when higher order filters re used. These oserv Tle Accurcy limits in phse nd in mplitude, k k nd k d, of the lgorithm comining the th order finite differences with the 6th, 8th, th nd 2th filtering of the vriles nd of the fluxes Vriles (r d =.2) Fluxes k k /Dx k d /Dx k k /Dx k d /Dx 6th order th order th order th order Note tht k d /Dx = 5.6 when 2th filtering of the vriles is used with r d =.
4 24 C. Bogey, C. Billy / Journl of Computtionl Physics 225 (27) 2 27 π 3π/4 π/2 2 3 π/4 4 π/4 π/2 3π/4 π 5 π/8 π/4 π/2 π Fig. 2. Representtion s function of the wve numer kdx of: () pproximted wve numer k w Dx, in liner scles, nd () phse error E k =(kdx k w Dx)/p, in logrithmic scles, otined using the stndrd thorder finite differences, lone, nd in comintion with explicit 6th, 8th, th nd 2thorder filtering of the fluxes. tions re supported y the ccurcy limits reported in Tle, which re estimted from E k s previously. The limits otined using flux filtering re indeed for instnce k k /Dx = 8.57 for the 6thorder filter nd k k /Dx = 5.54 for the 2thorder filter, which is higher thn the limit k k /Dx = 5.25 otined without filtering. In summry, the numericl errors due to the filtering of the flow fluxes re dded to those of the sptil derivtion, which decreses the ccurcy in phse. 3. Test cse 3.. Definition nd reference solutions A onedimensionl prolem is considered to illustrte the influence of sptil filtering. The wve eqution (3) is solved on mesh with uniform spcing Dx. The disturnces t t = re defined s uðxþ ¼sin 2px x 2 exp lnð2þ 8Dx 3Dx In the wve numer spce, s shown in [5], they re chrcterized y dominnt component for kdx = p/4, corresponding to wve length of 8Dx, nd lso y significnt components for < kdx < p/2. To emphsize the errors resulting from the sptil discretiztion, the perturtions re propgted over 8Dx, nd Eq. (3) is integrted in time y lowstorge 6stge Runge Kutt lgorithm [5], using the smll time step Dt =.2Dx. The sptil derivtion is tken into ccount y thorder explicit finite differences. The solution clculted without filtering is displyed in Fig. 3. Compred to the exct solution, it is slightly distorted y the finite differences. The error rte evluted etween the exct nd the clculted solutions s e num ¼ X ðu cl u exct Þ 2 = X =2 u 2 exct ðþ is in this cse e num = Fig. 3. Test cse: exct solution t t = 8, numericl solution otined without filtering.
5 3.2. Solutions otined using filterings C. Bogey, C. Billy / Journl of Computtionl Physics 225 (27) The solutions otined using filtering of the vriles fter ech time step with r d =.2 re presented in Fig. 4, nd compred with the solution clculted without filtering. The dissiptive effects of the filtering re visile, especilly with the filters of 6th nd 8th order in Fig. 4 nd, ut they re significntly reduced with filters of higher order. The error rtes with respect to the exct solution re thus oserved in Tle 2 to decrese from.84 with the 6thorder filter down to.245 for the 2thorder filter. The error otined with the 2thorder filter is moreover smller thn tht without filtering, which indictes higher numericl ccurcy. In this cse, the filtering ffects only wve numers tht re not properly clculted y the finite differences, s shown in Tle. The solutions otined using flux filtering re displyed in Fig. 5. They re clculted y solving Eq. (9). Flux filtering is therefore pplied t ech stge of the Runge Kutt lgorithm. With respect to the solution without filtering, the solutions show lrger dispersion of the disturnces, with the strengthening of the til of the wve pcket. This is oserved with the 6thorder filter in Fig. 5, s well s with the 2thorder filter in Fig. 5d. The dditionl errors in phse due to flux filtering however decrese using filters of higher order. The error rtes etween the numericl nd the exct solutions in Tle 2 re thus etween.234 nd.44, which is still lrger thn the error of.37 otined without filtering. As the order of the filter increses, one cn indeed c d Fig. 4. Test cse: numericl solution otined without filtering, solutions using explicit filtering of the vriles (r d =.2). Filters of: () 6th order, () 8th order, (c) th order, (d) 2th order. Tle 2 Test cse: errors e num etween the exct solution nd the numericl solutions otined using explicit 6th, 8th, th nd 2thorder filtering of the vriles (r d =.2) nd of the fluxes; without filtering, e num =.37 Vriles Fluxes 6th order th order th order th order
6 26 C. Bogey, C. Billy / Journl of Computtionl Physics 225 (27) c d Fig. 5. Test cse: numericl solution otined without filtering, solutions using explicit filtering of the fluxes. Filters of: () 6th order, () 8th order, (c) th order, (d) 2th order. expect the solution to progressively collpse the solution otined without filtering, nd consequently the error rte would tend to.37 while remining lrger. 4. Concluding remrks The pplictions of explicit sptil filtering to the flow vriles nd to the fluxes in simultions might oth generte numericl rtifcts, which re not of the sme nture. For liner equtions, using centered schemes, the filtering of the vriles might led to dditionl dissiption of the solution, wheres filtering the fluxes might decrese the ccurcy in phse. These unwnted effects cn e minimized y using filters of highorder, nmely t lest of the order of the sptil differentition. Filtering the vriles my however pper more relevnt thn filtering the fluxes ecuse it cn enle to remove highfrequency wves tht re not properly clculted, nd consequently to improve the numericl solutions. The computtionl cost of filtering the vriles is lso smller, ecuse the vriles cn e filtered every time step, or even every nth time step, wheres flux filtering must e pplied t ech stge of the time integrtion lgorithm. Finlly, there might e prolem for numericl stility using flux filtering, ecuse this method does not directly remove gridtogrid oscilltions, ut only prevents their genertion during the simultion. Appendix The coefficients of the 6thorder centered explicit filter re d =5/6,d = 5/64, d 2 = 3/32 nd d 3 = /64, nd d j = d j. The coefficients of the other schemes used in this note cn e found in Ref. [5] for instnce. References [] S.K. Lele, Compct finite difference schemes with spectrllike resolution, J. Comput. Phys. 3 () (992) [2] C.K.W. Tm, H. Shen, Direct computtion of nonliner coustic pulses using high order finite difference schemes, AIAA Pper , 993.
7 C. Bogey, C. Billy / Journl of Computtionl Physics 225 (27) [3] O.V. Vsilyev, T.S. Lund, P. Moin, A generl clss of commuttive filters for LES in complex geometries, J. Comput. Phys. 46 (998) [4] M.R. Visl, D.V. Gitonde, Highorderccurte methods for complex unstedy susonic flows, AIAA J. 37 () (999) [5] C. Bogey, C. Billy, A fmily of low dispersive nd low dissiptive explicit schemes for flow nd noise computtions, J. Comput. Phys. 94 () (24) [6] J. Berlnd, C. Bogey, O. Mrsden, C. Billy, Highorder, low dispersive nd low dissiptive explicit schemes for multiplescle nd oundry prolems, J. Comput. Phys. 224 (2) (27) [7] P. Sgut, Lrgeeddy Simultion for Incompressile Flows An Introduction, Springer, Berlin, 2. [8] D.P. Rizzett, M.R. Visl, G.A. Blisdell, A timeimplicit highorder compct differencing nd filtering scheme for lrgeeddy simultion, Int. J. Num. Meth. Fluids 42 (23) [9] J. Mthew, R. Lechner, H. Foysi, J. Sesterhenn, R. Friedrich, An explicit filtering method for lrge eddy simultion of compressile flows, Phys. Fluids 5 (8) (23) [] C. Bogey, C. Billy, Lrge eddy simultions of trnsitionl round jets: influence of the Reynolds numer on flow development nd energy dissiption, Phys. Fluids 8 (6) (26) 65. [] T.S. Lund, The use of explicit filters in lrge eddy simultion, Comput. Mth. Appl. 46 (23) [2] E. Lmllis, J.H. Silvestrini, Direct numericl simultion of interctions etween mixing lyer nd wke round cylinder, J. Turulence 3 (22) 28. [3] C.K.W. Tm, J.C. We, Dispersionreltionpreserving finite difference schemes for computtionl coustics, J. Comput. Phys. 7 (993)
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