Modified Mass Matrices and Positivity Preservation for Hyperbolic and Parabolic PDEs

Transcription

1 COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2000; 00:6 Prepared usng cnmauth.cls [Verson: 2000/03/22 v.0] Modfed Mass Matrces and Postvty Preservaton for Hyperbolc and Parabolc PDEs M. Berzns Λ Computatonal PDEs Unt, School of Computng, The Unversty of Leeds, Leeds LS2 9JT, UK. SUMMARY Modfcatons to the standard fnte element mass matrx are consdered wth the am of preservng the postvty of the dscrete soluton. The approach s used n connecton wth calculatng the ntal tme dervatve values for parabolc equatons and n connecton wth nonlnear Petrov-Galerkn schemes for hyperbolc equatons n one space dmenson. The extenson of the deas to unstructured meshes n two and three space dmensons s ndcated. Copyrght cfl 2000 John Wley & Sons, Ltd. KEY WORDS: Postvty Preservaton ; Mass Matrces, Fnte Element Methods. Introducton There are many stuatons n the numercal soluton of partal dfferental equatons n whch the computed soluton values should, on physcal grounds, reman non-negatve. One the smplest examples s that of the smple advecton equaton wth non-negatve ntal data whle other cases are those of concentratons of chemcal compounds n reactng flow calculatons. In the latter case preservng postvty s essental to avod the numercal calculaton becomng meanngless. Consder the soluton of the advecton equaton wth approprate ntal and boundary condton by usng the standard Galerkn method wth lnear bass (hat) functons φ (x) on a unformly spaced mesh x ; ;:::;N to get Z x+ x U t φ (x) dx Z x+ x U x φ (x) dx; ; :::;N; () where the approxmate soluton to ths p.d.e. as defned by U(x;t) N φ (x)u (t) where φ (x )δ. Evaluatng the ntegrals gves rse to the numercal scheme defned by Λ δx (U + U ) (2) Λ Correspondence to: M.Berzns, School of Computng, The Unversty of Leeds, Leeds LS2 9JT, UK. Receved 0 January 200 Copyrght cfl 2000 John Wley & Sons, Ltd. Revsed 22 March 200

2 2 M. BERZINS where δx s the unform mesh spacng n ths case and where du dt. Defnng the tme-dependent vector U by U [U ;:::;U N ] T allows ths system of equatons to be rewrtten n the form A (t) F(U(t)) (3) where the matrx A s referred to as the mass matrx. It s well-known that ths scheme s unsatsfactory n a very smlar way to that of lnear central dfference schemes, [8]. Many modfed Galerkn methods have been proposed to remedy ths stuaton. A survey of such methods s gven n [8] and ncludes Streamlne Upwnd Petrov-Galerkn (SUPG) methods [7] n whch the test functons are modfed to mprove the behavour of the method and Dscontnuous Galerkn (DG) methods [4, 7] n whch dscontnuous bass functons are used. There are many other approaches such as the modfed Petrov-Galerkn method of Cardle [3] n whch the test functon s modfed dfferently for the spatal and temporal terms. In ths case the numercal scheme that results s gven by + 6 ( β Λ ) δx (U + U α Λ )+ U 2U 2δx +U + (4) where β and α are the constants multplyng the Petrov-Galerkn addtonal polynomals n tme (cubc polynomal) and space (quadratc polynomal), see [3]. In the case of many of these methods t s clear that the magntude of unphyscal values s not as large as wth the standard Galerkn method and n the case of DG methods the mass matrx s the dentty matrx; ths makes t much easer to prove propertes such as postvty preservaton. The defnton used here for a postvty preservng scheme for the advecton equaton s one (see []) for whch the numercal soluton at tme t n+ may be wrtten n terms of the numercal soluton at tme t n n the form U (t n+ ) a U (t n ) where a ; and a 0 : (5) The key observaton wth regard to preservng postvty s due to Godunov [6] who proved that any scheme of better than frst order whch preserves postvty for the advecton equaton must be nonlnear. That s the coeffcents a n (5) above must depend on the numercal soluton to the p.d.e. Ths means that α and β n (4) must also depend on the soluton. In nvestgatng postvty preservng mass matrces and Galerkn fnte element methods for transent problems the startng pont wll be to rewrte the mass matrx as a postvty preservng matrx. Ths wll be appled to the soluton of a parabolc equaton. The same dea wll then be appled to hyperbolc equatons and lnked to the work of Cardle, [3] and to work on nonlnear fnte dfference schemes. Fnally the extenson of the approach to unstructured trangular and tetrahedral meshes wll be consdered. 2. Modfed Mass Matrces and the Intalsaton of Parabolc Equatons In tryng to solve parabolc equatons usng a Galerkn method-of-lnes approach Skeel and Berzns [9] showed that the ntal tme dervatves may have the wrong sgn. Ths s because the nverse of the mass matrx A may have have negatve entres. Suppose that the parabolc equaton s dscretsed n space to get a system of equatons of the form of (3) wth F 0. The ntal values of the tme dervatves are gven by solvng the equatons (3) for the ntal values of the tme dervatves (0). In computatonal experments tme dervatves wth the wrong sgn slow down the the ntegraton and Copyrght cfl 2000 John Wley & Sons, Ltd. Commun. Numer. Meth. Engng 2000; 00:6 Prepared usng cnmauth.cls

3 MODIFIED MASS MATRICES AND POSITIVITY PRESERVATION FOR PDES 3 may gve physcally msleadng soluton values. For these reasons Skeel and Berzns devsed a scheme that may be vewed as a lumped fnte element scheme n whch the mass matrx s replaced by the dentty matrx. The ssue of when a matrx may have an nverse consstng of postve entres s consdered n a large body of work on M matrces. See, for example, [2] who show that f A s a dagonally domnant M matrx wth negatve off dagonal entres then ts nverse A has only postve entres. The task s thus to modfy the mass matrx so that t has negatve off-dagonal entres. 2.. Dervaton of Modfed Mass Matrx For smplcty consder the case when lnear bass functons are used on a unform spatal mesh as n equatons (). The th row of the mass matrx s then gven by φ ;φ + + φ ;φ + + φ ;φ F : (6) Usng the dentty that on [x ;x + ] φ + φ + φ + gves: φ ; + + φ ;φ + + φ ;φ F : (7) + Defnng the rato s allows the th row of the mass matrx to be rewrtten as 2 3 φ ;φ φ ; φ ;φ 5 + F : (8) s Ths matrx s an M matrx f [:::] s postve (on a unform mesh ths requres 0 < s < ) as then the matrx s dagonally domnant wth negatve off-dagonal entres. In the case when s > on a unform mesh the th row s wrtten as φ ; + hφ ;φ + s φ ;φ F (9) whch agan s a row of an M matrx as [:::] s postve. In the case when s < 0 there appears h no alternatve but to dagonalse (lump) the matrx as φ ;φ + φ ;φ + φ ;φ +. The modfed matrx may also be wrtten as (wth approprate modfcatons f s 0) : (s + s ) h + 2s F δx : (0) In solvng the equatons (8) and (9) for the ntal values of the tme dervatves t s thus necessary to teratvely solve nonlnear equatons. Let m and s m be the values calculated at teraton m. The equatons solved n the case when s m > are then gven by m+ + γ m+ m In the case when 0 < s m < the teraton s defned by m+ + γ m+ + m F δx ; where γ F δx ; where γ» s m 6» s m 6 : () : (2) Copyrght cfl 2000 John Wley & Sons, Ltd. Commun. Numer. Meth. Engng 2000; 00:6 Prepared usng cnmauth.cls

4 4 M. BERZINS In order to llustrate that ths procedure produces ntal values of the tme dervatves wth the rght sgn the followng example s used. Skeel and Berzns [9] consder test examples such as the case when C the rght sde of equaton (3) s gven by F ß U t (x ;0) where C 0:fx < 0 and C :0 (Cx +:) otherwse. For a unform mesh of and 2 ponts across the nterval [;] Fgure shows that the.4 Mass matrx solve n.4 Mass matrx solve n Fgure. Mass Matrx Calculaton for Tme Dervatves - s true * s new, s orgnal method does result n tme dervatves of the correct sgn wthout the overshoots and undershoots of a standard Galerkn approach. The ssue of preservng postvty for the dffuson equaton has been consdered n very recent work by Farago and Horvath [5]. They show that for a d dmensonal problem usng the θ tme ntegraton method, t s necessary to restrct the choce of the parameter θ and the tmestep δt by 6θ d < δt δx 2 < 3d(θ ; where d ;2 Ther results also extends to three space dmensons. ) 3. Modfed Mass Matrces and Hyperbolc Equatons In general the tme dervatves may not have constant sgn and t s the non-negatvty of the soluton that must be consdered. In the case of the advecton equaton t s possble to rewrte equaton (4) as a smple explct method for hyperbolc equatons: AU (t n+ )AU (t n )+ δt F(U(t n )) (3) In usng the modfed mass matrx approach as part of a method for hyperbolc equatons t s nstructve to note other smlar approaches. The approxmaton of U x by a standard Galerkn method s dentcal to that of central fnte dfferences. A nonlnear central dfference method for hyperbolc equatons s gven by Swanson and Turkel [0] as : 2δx (U + U h )+ L 2δx + (U + U ) L (U U ) (4) where L ˆr +ˆr and ˆr U U 2 U U. The rght sde of ths may also be nterpreted as a nonlnear Petrov-Galerkn method n whch the test functon s ˆφ (x) where Λ dφ (x) ˆφ (x) φ (x)+δx L dx (5) Copyrght cfl 2000 John Wley & Sons, Ltd. Commun. Numer. Meth. Engng 2000; 00:6 Prepared usng cnmauth.cls

5 MODIFIED MASS MATRICES AND POSITIVITY PRESERVATION FOR PDES 5 where L Λ L f x < x < x + and L Λ L + otherwse. Although ths scheme s postvty preservng t s qute dffuse. A less dffuse scheme s one n whch L Λ s defned by L Λ V (r )r r +r +r f x + > x > x and L Λ V(r )r f x > x > x where V (r ) Routne manpulaton shows that ths defnton gves the well-known van Leer scheme, e.g []: " and r U + U U U. U (t) δx β (U (t) U (t)); where β V (r ) V (r ) + : (6) 2 2r An alternatve vew of ths scheme s thus as a nonlnear Petrov-Galerkn method. Requrng postvty for forward Euler tmesteppng requres the CFL type restrcton 0» β < δx δt. The nonlnear extenson of the type of Petrov-Galerkn method gven by (3) s thus gven by equaton (0) wth F defned by the rght sde of (6). An outlne proof that, when combned wth forward Euler tmesteppng, ths s postvty preservng follows from a modfed verson of (). (The proof for the case n (2) beng smlar). The teraton from () may be wrtten as: m+ (t) + γ» γ m (t)+ F δx ; m 0;;::: where γ s defned as n equaton () and so depends on s m and where F δx s defned by the rghthand sde of equaton (6). Hence ths equaton may also be wrtten as m+ (t) + γ " γ m (t) β δx (U (t) U (t)) # # ; m 0;;::: (7) The predcted values, 0 (t n+ ), are gven by equaton(6). An outlne of the approach used to defne soluton postvty n terms of equaton (5) may now be gven. The ntal guesses for the tme drvatves are gven by equaton (6): 0 (t h n+ ) U δx (t n ) U 2 (t n ) β (8) where β s also defned as n (6). Substtutng for 0 n (7) and applyng forward Euler tmesteppng gves: U (t n+ ) U (t δt n) δx( + γ) h β (U (t n ) U (t n )) + γβ (U (t n ) U 2 (t n )) where U (t n+ ) s the frst teraton estmate for U (t n+ ). Ths may be rewrtten as U (t n+ ) ( β E)U (t n)+(β γβ )EU (t n )+(γβ )EU 2 (t n ) (20) where E δt δx ( + γ) : In provng postvty of ths consder the worst case n equaton (9) and suppose that β (U (t n ) U (t n )) has a dfferent sgn to β (U (t n ) U 2 (t n )). It then follows that r s negatve and hence that β h + V (r ) 2 and β» V (r 2 ) 2r 2. Hence V (r ) β γβ + γ + γ V (r 2 ) 2 2r 2 (9) Copyrght cfl 2000 John Wley & Sons, Ltd. Commun. Numer. Meth. Engng 2000; 00:6 Prepared usng cnmauth.cls

6 6 M. BERZINS and n order to guarantee a postve soluton at the end of the frst teraton we need to mpose the condton γ < :0. Thus, from equatons () and (2), the teratve method s only appled f 7 < sm < 7 The same approach can then be used nductvely to prove postvty for second and subsequent teratons. Fgure 2 shows the numercal results obtaned when the method s appled to the advecton of a square pulse functon from x 0:2 to0:7 usng a spatal mesh of 5 equally spaced ponts. The fgure compares the soluton obtaned wth the van Leer method wth the new approach and shows the effect of usng a mass matrx s to gve a stll postve but more skewed profle than that obtaned from the van Leer method. Advecton cfl 50 Advecton cfl Solutons at t Solutons at t x 0. 0 x * Exact, + van Leer FV, new FE Fgure 2. Numercal soluton of advected square wave 4. Extenson to Trangular and Tetrahedral Meshes In the case of trangular mesh examples t s possble to use the same dea as n one space dmenson. In ths case, for example, the mass matrx for a mesh fragment consstng of three trangles wth node n common and a permeter consstng of nodes, k and l s gven by: (φ ;)+ φ ;φ + k φk ;φ + l φl ;φ (2) where φ ;φ 2 ha k + A l, φ ;φ 2 ha k + A l + A lk and where A k s the area of trangle wth nodes, and k. As the contrbuton of element k to the mass matrx s A k 2 h k : (22) Copyrght cfl 2000 John Wley & Sons, Ltd. Commun. Numer. Meth. Engng 2000; 00:6 Prepared usng cnmauth.cls

7 MODIFIED MASS MATRICES AND POSITIVITY PRESERVATION FOR PDES 7 The same deas as n one space dmenson may be used to rewrte 2 + k by usng the same approach as n Secton 2.. Consder the ; ;k trangle and let ( +k)2 be mdpont soluton value on k edge. The decomposton gven by h h 2 + k 2 + ( +k)2) k + 2 ( +k)2) (23) allows the terms on the rght sde of ths equaton to be vewed as second order approxmatons to second and frst space dervatves. Hence n dscardng these terms we ntroduce second order errors as n one space dmenson. The same dea extends to tetrahedral mesh examples. Consder a sngle tetrahedron wth ts four nodes labelled as ; ;k;l and assocated lnear bass functons φ ;φ ;φ k and φ l. The mass matrx ntegral assocated wth ths tetrahedron s h V kl (φ ;φ )+ φ ;φ + k φk ;φ + l φl ;φ : (24) Evaluatng the volume ntegral gves: V kl 20 h k + l whch may be rewrtten as three terms of the form V kl 20 appled as n one and two space dmensons. (25) h k and same deas 5. Summary In ths paper a novel approach to preservng postvty has been taken. The approach reles on usng a nonlnear form of the mass matrx n conuncton wth nonlnear Petrov-Galerkn type terms. The approach has applcatons n one,two and three space dmensonal cases, but further work s clearly needed to assess ts usefulness. REFERENCES. Berzns M, Ware JM. Postve Cell Centered Fnte Volume Dscretsaton Methods for Hyperbolc Equatons on Irregular Meshes, Appled Numercal Mathematcs 995; 6: Berman A, Plemmons RJ. Nonnegatve Matrces n the Mathematcal Scences. Acadedmc Press, Cardle JA. A modfcaton of the Petrov-Galerkn method for the transent convecton-dffuson equaton. Internatonal Journal for Numercal Methods n Engneerng 995; 38: Cockburn B, Karnadaks GE, Shu C-W. (eds). Dscontnuous Galerkn Methods, Theory Computatons and Applcatons. Lecture Notes n Computataonal Scence and Engneerng Sprnger Berln Hedelberg, 2000; Farago I, Horvath R. On the nonnegatvty conservaton of fnte element soluton of parabolc problems. In 3D Fnte Element Conference Proceedngs Krzek M., et al (eds). Gakkotosho Co., Tokyo Godunov SK. Fnte dfference method for the numercal computaton of dscontnuous solutons of the equatons of flud dynamcs. Math Sbornk 959; 47: Johnson C. Numercal Soluton of Partal Dfferental Equatons by the Fnte Element Method Cambrdge Unversty Press, Segal A. Fnte element methods for advecton-dffuson equatons. In Numercal Methods for Advecton Dffuson Problems. Notes on Numercal Flud Mechancs, Volume 45 Vreugdenhl CB, Koren B (eds). Veweg: Braunscweg/Wesbaden, 993; Skeel RD, Berzns M. A Method for the Spatal Dscretsaton of Parabolc Equatons, SIAM Journal on Scentfc Computng 990; (): Swanson RC, Turkel E. On central-dfference and upwnd schemes. Journal of Computatonal Physcs 992; 0: Copyrght cfl 2000 John Wley & Sons, Ltd. Commun. Numer. Meth. Engng 2000; 00:6 Prepared usng cnmauth.cls