AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

Transcription

1 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 74 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS The Fnte Dfference Method These sldes are based on the recommended textbook: Culbert B. Laney. Computatonal Gas Dynamcs, CAMBRIDGE UNIVERSITY PRESS, ISBN / 74

2 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 74 Outlne 1 Conservatve Fnte Dfference Methods n One Dmenson 2 Forward, Backward, and Central Tme Methods 3 Doman of Dependence and CFL Condton 4 Lnear Stablty Analyss 5 Formal, Global, and Local Order of Accuracy 6 Upwnd Schemes n One Dmenson 7 Nonlnear Stablty Analyss 8 Multdmensonal Extensons 2 / 74

3 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 3 / 74 Note: The materal covered n ths chapter equally apples to scalar conservaton laws and to the Euler equatons, n one and multple dmensons. In order to keep thngs as smple as possble, t s presented n most cases for scalar conservaton laws: frst n one dmenson, then n multple dmensons. 3 / 74

4 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 4 / 74 Conservatve Fnte Dfference Methods n One Dmenson Recall that scalar conservaton laws are smple scalar models of the Euler equatons that can be wrtten n strong conservaton form as u t + f (u) x = 0 (1) Suppose that a 1D space s dvded nto grd ponts x and cells [x 1/2, x +1/2 ], where x +1/2 s called a cell edge Also suppose that tme s dvded nto tme-ntervals [t n, t n+1 ] The conservaton form of a fnte dfference method appled to the numercal soluton of equaton (1) s defned as follows ( )n u t = λ(ˆf n +1/2 t ˆf n 1/2 ) (2) where the subscrpt desgnates the pont x, the superscrpt n desgnates the tme t n, a hat desgnates a tme-approxmaton, and λ = t x, t = tn+1 t n, x = x +1/2 x 1/2 4 / 74

5 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 5 / 74 Conservatve Fnte Dfference Methods n One Dmenson One nterpretaton of the fnte dfference approach (2) and the conservaton form label s the approxmaton of the followng ntegral form of equaton (1) 1 x x+1/2 x 1/2 [u(x, t n+1 ) dx u(x, t n )] dx = 1 x t n+1 whch clearly descrbes a conservaton law t n [f ( u(x +1/2, t) ) f ( u(x 1/2, t) ) ] dt 5 / 74

6 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 6 / 74 Conservatve Fnte Dfference Methods n One Dmenson Not every fnte dfference method can be wrtten n conservaton form: Those whch can are called conservatve and ther assocated quanttes ˆf +1/2 n are called conservatve numercal fluxes fnte dfference methods derved from the conservaton form of the Euler equatons or scalar conservaton laws tend to be conservatve fnte dfference methods derved from other dfferental forms (for example, prmtve or characterstc forms) of the aforementoned equatons tend not to be conservatve conservatve fnte dfferencng mples correct shock and contact locatons 6 / 74

7 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 7 / 74 Conservatve Fnte Dfference Methods n One Dmenson Lke many approxmaton methods, conservatve fnte dfference methods can be dvded nto mplct and explct methods n a typcal mplct method ( u t )n ( ) u = (u K n t 1,..., u+k n 2 ; u n+1 L 1,..., u n+1 +L 2 ) ˆf n +1/2 = ˆf (u n K 1 +1,..., u n +K 2 ; u n+1 L 1 +1,..., u n+1 +L 2 ) (3) so that from (2) one has u n+1 = u(u n K 1,..., u n +K 2 ; u n+1 L 1,..., u n+1,..., u n+1 +L 2 ) (4) = the soluton of a system of equatons s requred at each tme-step Note: f u K n 1 +1 n (3) were wrtten as u K n 1, one would get the less convenent notaton u n+1 = u(u K n 1 1,..., u+k n 2 ; u n+1 L 1,..., u n+1 +L 2 ) nstead of (4) 7 / 74

8 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 8 / 74 Conservatve Fnte Dfference Methods n One Dmenson Lke many approxmaton methods, conservatve fnte dfference methods can be dvded nto mplct and explct methods (contnue) n a typcal explct method ( u t )n so that from (2) one has ( ) u = (u K n t 1,..., u+k n 2 ; u n+1 ) ˆf n +1/2 = ˆf (u n K 1 +1,..., u n +K 2 ) u n+1 = u(u n K 1,..., u n +K 2 ) = only functon evaluatons are ncurred at each tme-step (u K n 1,..., u+k n 2 ) and (u n+1 L 1,..., u n+1 +L 2 ) are called the stencl or drect numercal doman of dependence of u n+1 K 1 + K and L 1 + L are called the stencl wdths 8 / 74

9 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 9 / 74 Conservatve Fnte Dfference Methods n One Dmenson Summary: typcal stencl dagram 9 / 74

10 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 10 / 74 Conservatve Fnte Dfference Methods n One Dmenson Lke any proper numercal approxmaton, proper fnte dfference approxmaton becomes perfect n the lmt x 0 and t 0 an approxmate equaton s sad to be consstent f t equals the true equatons n the lmt x 0 and t 0 a soluton to an approxmate equaton s sad to be convergent f t equals the true soluton of the true equaton n the lmt x 0 and t 0 Hence, a conservatve approxmaton s consstent when ˆf (u,..., u) = f (u) = n ths case, the conservatve numercal flux ˆf s sad to be consstent wth the physcal flux A conservatve numercal method and therefore a conservatve fnte dfference method automatcally locates shocks correctly (however, t does not necessarly reproduce the shape of the shock correctly) A method that explctly enforces the Rankne-Hugonot relaton s called a shock-capturng method 10 / 74

11 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 11 / 74 Forward, Backward, and Central Tme Methods Forward Tme Methods Forward Tme (FT) conservatve fnte dfference methods correspond to the choces ( u )n t = u n+1 u n and ˆf +1/2 n t = ˆf (u K n 1+1,..., u+k n 2 ) wth Forward Space (FS) approxmaton of the term u x (x, t n ), ths leads to the FTFS scheme u n+1 = u n λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = f (u n +1) wth Backward Space (BS) approxmaton of the term u x (x, t n ), ths leads to the FTBS scheme u n+1 = u n λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = f (u n ) wth Central Space (CS) approxmaton of the term u x (x, t n ), ths leads to the FTCS scheme u n+1 = u n λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = 1 2 (f (un +1) + f (u n )) 11 / 74

12 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 12 / 74 Forward, Backward, and Central Tme Methods Backward Tme Methods Backward Tme (BT) conservatve fnte dfference methods correspond to the choces ( u )n t = u n+1 u n and ˆf +1/2 n t = ˆf (u n+1 K 1+1,..., un+1 +K 2 ) wth Forward Space (FS) approxmaton of the term u x (x, t n ), ths leads to the BTFS scheme u n+1 = u n λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = f (u n+1 +1 ) wth Backward Space (BS) approxmaton of the term u x (x, t n ), ths leads to the BTBS scheme u n+1 = u n λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = f (u n+1 ) wth Central Space (CS) approxmaton of the term u x (x, t n ), ths leads to the BTCS scheme u n+1 = u n λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = 1 2 ( ) f (u n+1 +1 ) + f (u n+1 ) 12 / 74

13 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 13 / 74 Forward, Backward, and Central Tme Methods Central Tme Methods Central Tme (CT) conservatve fnte dfference methods correspond to the choces ( u )n t = 1 t 2 (un+1 u n 1 ) and ˆf +1/2 n = ˆf (u K n 1+1,..., u+k n 2 ) wth Forward Space (FS) approxmaton of the term u x (x, t n ), ths leads to the CTFS scheme u n+1 = u n 1 2λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = f (u n +1) wth Backward Space (BS) approxmaton of the term u x (x, t n ), ths leads to the CTBS scheme u n+1 = u n 1 2λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = f (u n ) wth Central Space (CS) approxmaton of the term u x (x, t n ), ths leads to the CTCS scheme u n+1 = u n 1 2λ(ˆf n +1/2 ˆf n 1/2), wth ˆf n +1/2 = 1 2 (f (un +1) + f (u n )) 13 / 74

14 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 14 / 74 Doman of Dependence and CFL Condton Numercal and Physcal Domans of Dependence Recall the theory of characterstcs: A pont n the x t plane s nfluenced only by ponts n a fnte doman of dependence and nfluences only ponts n a fnte range of nfluence Hence, the physcal doman of dependence and physcal range of nfluence are bounded on the rght and left by the waves wth the hghest and lowest speeds In a well-posed problem, the range of nfluence of the ntal and boundary condtons should exactly encompass the entre flow n the x t plane 14 / 74

15 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 15 / 74 Doman of Dependence and CFL Condton Numercal and Physcal Domans of Dependence The drect numercal doman of dependence of a fnte dfference method s ts stencl: For example, f the soluton approxmated by an mplct fnte dfference method can be wrtten as u n+1 = u(u n K 1,..., u n +K 2 ; u n+1 L 1,..., u n+1 +L 2 ) ts drect numercal doman of dependence s the regon of the x t plane covered by the ponts (u K n 1,..., u+k n 2 ; u n+1 L 1,..., u n+1 +L 2 ) Smlarly, f the soluton approxmated by an explct fnte dfference method can be wrtten as u n+1 = u(u n K 1,..., u n +K 2 ) ts drect numercal doman of dependence s the regon of the x t plane covered by the ponts (u n K 1,..., u n +K 2 ) The full (or complete) numercal doman of dependence of a fnte dfference method conssts of the unon of ts drect numercal doman of dependence and the doman covered by the ponts of the x t plane upon whch the numercal values n the drect numercal doman of dependence depend upon 15 / 74

16 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 16 / 74 Doman of Dependence and CFL Condton Numercal and Physcal Domans of Dependence The Courant-Fredrchs-Lewy or (CFL) condton The full numercal doman of dependence must contan the physcal doman of dependence Any numercal method that volates the CFL condton msses nformaton affectng the exact soluton and may blow up to nfnty: For ths reason, the CFL condton s necessary but not suffcent for numercal stablty 16 / 74

17 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 17 / 74 Doman of Dependence and CFL Condton Scalar Conservaton Laws Consder frst the lnear advecton problem u t + a u = 0 x { 1 f x < 0 u(x, 0) = 0 f x 0 Assume that a > 0: The exact soluton s { 1 f x at < 0 u(x, t) = u(x at, 0) = 0 f x at 0 The FTFS approxmaton wth x = cst s u n+1 where as before, λ = t x = (1 + λa)u n λau+1 n { u 0 1 f 1 = u( x, 0) = 0 f 0 17 / 74

18 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 18 / 74 Doman of Dependence and CFL Condton Scalar Conservaton Laws Then u 1 = 1 f λa f = 1 0 f 0 u 2 = 1 f 3 (1 + λa)(1 λa) f = 2 (1 + λa)(1 + λa) f = 1 0 f 0 and so forth The frst two tme-steps reveal that FTFS moves the jump n the wrong drecton (left rather than rght!) and produces spurous oscllatons and overshoots Furthermore, the exact soluton yelds u(0, t) = 1, but FTFS yelds u 1 0 = 0! 18 / 74

19 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 19 / 74 Doman of Dependence and CFL Condton Scalar Conservaton Laws Ths s because FTFS volates the CFL condton u n+1 = (1 + λa)u n λau n / 74

20 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 20 / 74 Doman of Dependence and CFL Condton Scalar Conservaton Laws FTCS satsfes the CFL condton for 1 λa 1 However, t almost always blow up (as wll be seen n a homework): Ths llustrates the fact that the CFL condton s a necessary but not suffcent condton for numercal stablty You can also check that when appled to the soluton of any scalar conservaton law, the BTCS method always satfes the CFL condton 20 / 74

21 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 21 / 74 Doman of Dependence and CFL Condton Scalar Conservaton Laws For scalar conservaton laws, the CFL condton translates nto a smple nequalty restrctng the wave speed lnear advecton equaton and explct forward-tme method wth u n+1 = u(u K n 1,..., u+k n 2 ) n the x t plane, the physcal doman of dependence s the lne of slope 1/a n the x t plane, the full numercal doman of dependence of u n+1 s bounded t on the left by a lne of slope K = λ and on the rght by a lne of slope 1 x K 1 t K = λ 2 x K 2 hence, the CFL condton s K2 λ a K1 K2 λa K1 λ whch requres that waves travel no more than K 1 ponts to the rght or K 2 ponts to the left durng a sngle tme-step 21 / 74

22 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 22 / 74 Doman of Dependence and CFL Condton Scalar Conservaton Laws For scalar conservaton laws, the CFL condton translates nto a smple nequalty restrctng the wave speed (contnue) lnear advecton equaton and explct forward-tme method wth = u(u K n 1,..., u+k n 2 ) (contnue) u n+1 f K 1 = K 2 = K, the prevous CFL condton becomes λ a K (5) for ths reason, λa s called the CFL number or the Courant number keep n mnd however that n general, a = a(u) and therefore the CFL number depends n general on the soluton s range 22 / 74

23 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 23 / 74 Doman of Dependence and CFL Condton Scalar Conservaton Laws For scalar conservaton laws, the CFL condton translates nto a smple nequalty restrctng the wave speed (contnue) lnear advecton equaton and mplct backward-tme method wth u n+1 = u(u K n 1,..., u+k n 2 ; u n+1 L 1,..., u n+1 +L 2 ) f L 1 > 0 and L 2 = 0, the full numercal doman of dependence of u n+1 ncludes everythng to the left of x = x and beneath t = t n+1 n the x t plane f L 1 = 0 and L 2 > 0, the full numercal doman of dependence of u n+1 ncludes everythng to the rght of x = x and beneath t = t n+1 n the x t plane f L 1 > 0 and L 2 > 0, the full numercal doman of dependence of u n+1 ncludes everythng n the entre x t plane beneath t = t n+1 concluson: as long as ther stencl ncludes one pont to the left and one to the rght, mplct methods avod CFL restrctons by usng the entre computatonal doman (hence, ths ncludes BTCS but not BTFS or BTBS) 23 / 74

24 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 24 / 74 Doman of Dependence and CFL Condton The Euler Equatons In 1D, the Euler equatons have three famles of waves that defne the physcal doman of dependence For each famly of waves, a CFL condton of a gven numercal method can be establshed as n the case of a scalar conservaton law: Then, the overall CFL condton s the most restrctve of all establshed CFL condtons For example, f K 1 = K 2 = K, A s the Jacoban matrx of the conservatve flux vector, and ρ(a) denotes ts spectral radus ( ρ(a) = max ( vx a, v x, v x + a ), the CFL condton of an explct forward-tme method becomes ( recall (5) ) λρ(a) K λρ(a) s called the CFL number or the Courant number 24 / 74

25 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 25 / 74 Doman of Dependence and CFL Condton The Euler Equatons λρ(a) K For supersonc flows, all waves travel n the same drecton, ether left or rght the mnmum stencl allowed by the CFL condton contans ether W 1 n and W n for rght-runnng supersonc flow, or W n and W+1 n for left-runnng supersonc flow For subsonc flows, waves travel n both drectons, and the mnmum stencl should always contan W 1 n, W n, and W+1 n Hence, a smart or adaptve stencl can be useful for the case of the Euler equatons! 25 / 74

26 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 26 / 74 Lnear Stablty Analyss Unstable solutons exhbt sgnfcant spurous oscllatons and/or overshoots Unstable solutons of lnear problems exhbt unbounded spurous oscllatons: Ther errors grow to nfnty as t Hence the concept of nstablty dscussed here for the soluton of lnear problems s that of ubounded growth Snce unstable solutons typcally oscllate, t makes sense to descrbe the soluton of a lnear problem such as a lnear advecton problem as a Fourer seres (sum of oscllatory trgonometrc functons) 26 / 74

27 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 27 / 74 Lnear Stablty Analyss von Neumann Analyss The Fourer seres for the contnuous (n space) soluton u(x, t n ) on any spatal doman [a, b] s u(x, t n ) = a n 0 + m=1 ( am n cos 2πm x a ) b a + m=1 ( bm n sn 2πm x a ) b a (6) For the dscrete soluton u n u(x, t n ), the Fourer seres s obtaned by samplng (6) as follows ( u n = a0 n + am n cos 2πm x ) a ( + bm n sn 2πm x ) a (7) b a b a m=1 m=1 Assume x +1 x = x = cst, x 0 = a, and x N = b x a = x and b a = N x: Ths transforms (7) nto ( ( u n = a0 n + am n cos 2πm ) ( + bm n sn 2πm )) (8) N N m=1 27 / 74

28 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 28 / 74 Lnear Stablty Analyss von Neumann Analyss Recall that samples can only support wavelengths of 2 x or longer (the Nyqust samplng theorem states that samples spaced apart by x perfectly represent functons whose shortest wavelengths are 4 x): Hence (8) s truncated as follows N/2 u n a0 n + m=1 ( a n m cos ( 2πm N ) ( 2πm + bm n sn N )) An equvalent expresson n the complex plane usng I as the notaton for the pure magnary number (I 2 = 1) s u n N/2 m= N/2 C n me I 2πm N = N/2 m= N/2 u n m (9) From (8), (9), and Euler s formula e I θ = cos θ + I sn θ t follows that C n 0 = a n 0, Cm n = an m Ibm n, C m n = an m + Ibm n / 74

29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 29 / 74 Lnear Stablty Analyss von Neumann Analyss Hence, each term of the Fourer seres can be wrtten as u n m = C n me 2πm (I N ) = C n m e I φm where φ m = 2πm and m = N/2,, N/2 N Because of lnearty the amplfcaton factor G m = C m n+1 Cm n = G m (λ) does not depend on n: However, t depends on λ (snce u n+1 and u n are produced by the numercal scheme beng analyzed) whch tself depends on t Hence, each term of the Fourer seres can be expressed as u n m = C m n Cm n 1 C m 2 Cm 1 Cm 1 Cm 0 Cme 0 I φm = G m G m Cme 0 I φm = GmC n me 0 I φm where G n m = G n m(λ) = (G m (λ)) n 29 / 74

30 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 30 / 74 Lnear Stablty Analyss von Neumann Analyss Fnally, assume that C 0 m = 1 (for example): Ths leads to u n m = G n m(λ)e I φm Conclusons the lnear approxmaton s lnearly stable f G m(λ) < 1 for all m t s neutrally lnearly stable f G m(λ) 1 for all m and G m(λ) = 1 for some m t s lnearly unstable f G m(λ) > 1 for some m Each of the above concluson can be re-wrtten n terms of λ = t x Applcaton (n class): apply the von Neumann analyss to determne the stablty of the FTFS scheme for the lnear advecton equaton 30 / 74

31 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 31 / 74 Lnear Stablty Analyss Matrx Method Shortcomngs of the von Neumann stablty analyss method requres the soluton to be perodc (u n 0 = u n N) requres constant spacng x does not account for the boundary condtons Alternatve method: so-called Matrx (egenvalue analyss) Method based on the fact that for a lnear problem and a lnear approxmaton method, one has u n+1 = M(λ)u n, where u n = (u n o u n 1 u n N) T and M s an amplfcaton matrx whch depends on the approxmaton scheme and on λ Ths mples u n = M n (λ)u 0 31 / 74

32 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 32 / 74 Lnear Stablty Analyss Matrx Method u n = M n (λ)u 0 Suppose that M s dagonalzable M(λ) = Q 1 Λ(λ)Q, Λ = dag (λ 1 (λ),, λ N (λ)) Then M n (λ) = Q 1 Λ n Q (Qu) n = Λ n (λ)(qu) 0 Conclusons the lnear approxmaton s lnearly stable f ρ (M(λ)) < 1 t s neutrally lnearly stable f ρ (M(λ)) = 1 t s lnearly unstable f ρ (M(λ)) > 1 32 / 74

33 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 33 / 74 Lnear Stablty Analyss Matrx Method Advantages of the Matrx Method for (lnear) stablty analyss does not requre the soluton to be perodc does not requre constant grd spacng ncorporates the effects of the boundary condtons Shortcomng: n general, the computaton of ρ(m) s not trval However, the above shortcomng s not an ssue when the objectve s to prove the uncondtonal stablty of an (mplct) scheme re-wrte the lnear verson of equaton (2) before tme-dscretzaton n matrx form as du + B( x)u = 0 (10) dt suppose that B s dagonalzable and transform equaton (10) nto the set of ndependent scalar equatons dv m dt + µ m( x)v m = 0, µ m > 0, m = 1,, N focus on one of the above equatons and dscretze t n tme apply the scalar form of the Matrx Method for stablty analyss: f the concluson turns out to be ndependent of µ m, then the aforementoned shortcomng s not an ssue example (n class): apply the Matrx Method to determne the stablty of a BT scheme for the lnear advecton equaton 33 / 74

34 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 34 / 74 Formal, Global, and Local Order of Accuracy Formal order of accuracy measures the order of accuracy of the ndvdual space and tme approxmatons Taylor seres expansons modfed lnear equatons However due to nstablty, formal order of accuracy may not be ndcatve of the actual performance of a method: For example, recall that a stablty condton s λρ(a) K tρ(a) K x and observe that such a stablty condton prevents, for example, fxng t and studyng the order of accuracy of the ndvdual space approxmaton when x 0 34 / 74

35 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 35 / 74 Formal, Global, and Local Order of Accuracy Besdes formal order of accuracy, one way to measure the order of accuracy s to reduce x and t smultaneously whle mantanng λ = t constant and fxng the ntal and boundary condtons x In ths case, a method s sad to have global p-th order of accuracy (n space and tme) f e C x x p = C t t p, e = u(x, t n ) u n for some constant C x and the related constant C t = C x λ p Other error measures can be obtaned by usng the 1-norm, 2-norm, or any vector norm, or f the error s measured pontwse 35 / 74

36 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 36 / 74 Formal, Global, and Local Order of Accuracy Determnng analytcally the global order of accuracy defned above can be challengng: For ths reason, t s usually predcted by comparng two dfferent numercal solutons obtaned usng the same numercal method but two dfferent values of x p = log( e 2 / e 1 ) log( x 2 / x 1 ) = log( e 2 / e 1 ) log( t 2 / t 1 ) where e l = u(x, t n ) u n s the absolute error for x l and t l = λ x l, l = 1, 2 36 / 74

37 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 37 / 74 Formal, Global, and Local Order of Accuracy Another way to measure the order of accuracy s to assume that the soluton s perfect at tme t n that s, u n = u(x, t n ), whch s usually true for n = 0 and measure the local (n tme) truncaton error nduced by a sngle tme-step ē = u(x, t n+1 ) u n+1 t Now, let t 0 and x 0 whle mantanng λ = t, and the x ntal and boundary condtons fxed: Then, a method s sad to have local p-th order of accuracy (n space and tme) f ē C x x p = C t t p for some constant C x and the related constant C t = C x λ p Unlke the global order of accuracy, the local order of accuracy s relatvely easy to determne analytcally Example (n class): determne analytcally the local order of accuracy of the FTFS scheme for the lnear advecton equaton 37 / 74

38 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 38 / 74 Upwnd Schemes n One Dmenson Consder a nonlnear scalar conservaton law In 1D, there are rght-runnng waves and left-runnng waves: For rght-runnng waves, rght s the downwnd drecton and left s the upwnd drecton Smlarly for left-runnng waves, left s the downwnd drecton and rght s the upwnd drecton Then every numercal approxmaton to a scalar conservaton law can be descrbed as Centered: f ts stencl contans equal numbers of ponts n both drectons Upwnd: f ts stencl contans more ponts n the upwnd drecton Downwnd: f ts stencl contans more ponts n the downwnd drecton Upwnd and downwnd stencls are adjustable or adaptve stencls: Upwnd and downwnd methods test for wnd drecton and then, based on the outcome of the tests, select ether a rght- or a left-based stencl 38 / 74

39 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 39 / 74 Upwnd Schemes n One Dmenson 39 / 74

40 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 40 / 74 Upwnd Schemes n One Dmenson 40 / 74

41 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 41 / 74 Upwnd Schemes n One Dmenson Upwndng ensures shock avodance f the shock reverses the wnd, whereas central dfferencng does not 41 / 74

42 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 42 / 74 Upwnd Schemes n One Dmenson Upwndng does not ensure shock avodance f the shock does not reverse the wnd downwndng on the rght above avods the shock but volates the CFL condton and thus would create larger errors than crossng the shock would 42 / 74

43 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 43 / 74 Upwnd Schemes n One Dmenson General remarks upwnd methods are popular because of ther excellent shock capturng ablty among smple FT or BT methods, upwnd methods outdo centered methods: However, hgher-order upwnd methods often have no specal advantages over hgher-order centered methods Sample technques for desgnng methods wth upwnd and adaptve stencls flux averagng methods flux splttng methods wave speed splttng methods 43 / 74

44 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 44 / 74 Upwnd Schemes n One Dmenson Introducton to Flux Splttng Flux splttng s defned as f (u) = f + (u) + f (u) df + du 0, df du 0 Hence, f + (u) s assocated wth a rght-runnng wave and f (u) s assocated wth a left-runnng wave Usng flux splttng, the governng conservaton law becomes u t + f + x + f x = 0 whch s called the flux splt form Then, f + can be dscretzed conservatvely usng at least one pont x to the left, and f can be dscretzed conservatvely usng at least x one pont to the rght, thus obtanng conservaton and satsfacton of the CFL condton 44 / 74

45 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 45 / 74 Upwnd Schemes n One Dmenson Introducton to Flux Splttng Unfortunately, because n general f + ( ) + f u and u f u ( ) f u flux splttng cannot descrbe the true connecton between fluxes and waves, unless all waves run n the same drecton f all waves are rght-runnng, the unque physcal flux splttng s f + = f and f = 0 f all waves are left-runnng, the unque physcal flux splttng s f = f and f + = 0 Ths s the case only for (nonlnear) scalar conservaton laws away from sonc ponts, and for the Euler equatons n the supersonc regme 45 / 74

46 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 46 / 74 Upwnd Schemes n One Dmenson Introducton to Flux Splttng Assume that f + s dscretzed wth a leftward bas so that the x approxmaton at x = x s centered or based towards x = x 1/2 ( ) f + x ˆf + 1/2 x for some ˆf + 1/2 Assume that f s dscretzed wth a rghtward bas so that the x approxmaton at x = x s centered or based towards x = x +1/2 ( ) f x ˆf +1/2 x for some ˆf +1/2 Usng forward Euler to perform the tme-dscretzaton leads to u n+1 = u n λ( ˆf +n n 1/2 + ˆf +1/2 ) whch s called the flux splt form of the numercal approxmaton 46 / 74

47 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 47 / 74 Upwnd Schemes n One Dmenson Introducton to Flux Splttng A method n flux splt form s conservatve f and only f Proof ˆf +n n +1/2 + ˆf +1/2 = g n +1 g n for some g n (11) u n+1 = u n λ( ˆf +n 1/2 + g n g n + ˆf n +1/2 ) compare wth the conservaton form u n+1 = u n λ(ˆf n +1/2 ˆf n 1/2) = ˆf n +1/2 = ˆf n +1/2 + g n, n ˆf 1/2 = ˆf +n 1/2 + g n = ˆf n +1/2 = ˆf n +1/2 g n, ˆf +n 1/2 = ˆf n 1/2 + g n requre now that ˆf n +1/2 = ˆf n (+1) 1/2 ˆf n +1/2 + g n = ˆf +n +1/2 + g +1 n = ˆf +n n +1/2 + ˆf +1/2 = g +1 n g n Snce there are no restrctons on g n, every conservatve method has nfntely many flux splt forms that are useful for nonlnear stablty analyss 47 / 74

48 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 48 / 74 Upwnd Schemes n One Dmenson Introducton to Flux Splttng Example: Desgn a frst-order upwnd method for Burgers equaton usng flux splttng then re-wrte t n conservaton form for Burgers equaton, the unque physcal flux splttng s f (u) = u2 2 = max(0, u) u 2 } {{ } f + (u) + mn(0, u) u 2 }{{} f (u) 48 / 74

49 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 49 / 74 Upwnd Schemes n One Dmenson Introducton to Flux Splttng Example: Desgn a frst-order upwnd method for Burgers equaton usng flux splttng then re-wrte t n conservaton form (contnue) a flux splt form of Burgers equaton s u t x (max(0, u)u) + 1 (mn(0, u)u) = 0 2 x a backward-space approxmaton of f + x gves ( ) n (max(0, u)u) max(0, un )u n max(0, u 1)u n 1 n x x a forward-space approxmaton of f x gves ( ) n (mn(0, u)u) mn(0, un +1)u+1 n mn(0, u n )u n x x combnng these wth an FT approxmaton yelds u n+1 = u n λ 2 (max(0, un )u n max(0, u n 1)u n 1) λ 2 (mn(0, un +1)u n +1 mn(0, u n )u n ) 49 / 74

50 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 50 / 74 Upwnd Schemes n One Dmenson Introducton to Flux Splttng Example: Desgn a frst-order upwnd method for Burgers equaton usng flux splttng then re-wrte t n conservaton form (contnue) the reader can check that the frst-order upwnd method descrbed n the prevous page can be re-wrtten n conservaton form usng g n = 1 2 (un ) 2 50 / 74

51 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 51 / 74 Upwnd Schemes n One Dmenson Introducton to Wave Speed Splttng In contrast wth flux splttng, wave speed splttng uses the governng equatons n non conservaton form and tends to yeld non conservatve approxmatons Hence n most cases, flux splttng s preferred over wave speed splttng... except when the flux functon has the property f (u) = df du u = a(u)u whch means that f (u) s a homogeneous functon of degree 1 (recall Euler s theorem whch states that a dfferentable functon f (u) s a homogeneous functon of degree p f and only f (df /du) u = pf (u)): Ths property makes flux splttng and wave speed splttng closely related 51 / 74

52 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 52 / 74 Upwnd Schemes n One Dmenson Introducton to Wave Speed Splttng For scalar conservaton laws, wave speed splttng can be wrtten as a(u) = a + (u) + a (u) a + (u) 0, a (u) 0 Then, the scalar conservaton law can be wrtten as u t u u + a+ + a x x = 0 whch s called the wave speed splt form Then, a + u can be dscretzed conservatvely usng at least one x pont to the left, and a u can be dscretzed conservatvely usng x at least one pont to the rght, thus obtanng satsfacton of the CFL condton 52 / 74

53 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 53 / 74 Upwnd Schemes n One Dmenson Introducton to Wave Speed Splttng Next, consder vector conservaton laws such as the Euler equatons Splt the Jacoban matrx as follows A(u) = A + (u) + A (u) where the egenvalues of A + are non negatve and those of A are non postve A + 0, A 0 Recall that A + and A are obtaned by computng and splttng the egenvalues of A The wave speed splt form of the Euler equatons can then be wrtten as u t u u + A+ + A x x = 0 Agan, A + u can then be dscretzed conservatvely usng at least x one pont to the left, and A u usng at least one pont to the x rght, thus obtanng satsfacton of the CFL condton 53 / 74

54 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 54 / 74 Upwnd Schemes n One Dmenson Introducton to Wave Speed Splttng If f (u) s a homogeneous functon of degree 1, then from Euler s theorem t follows that f (u) = a(u)u f ± (u) = a ± (u)u However, the above flux vector splttng may or may not satsfy df + df 0 and du du 0 54 / 74

55 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 55 / 74 Upwnd Schemes n One Dmenson Introducton to Wave Speed Splttng Assume that a + u s dscretzed wth a leftward bas so that the x approxmaton at x = x s centered or based towards x = x 1/2 ( a + u ) n u a +n n u 1 n x 1/2 x Assume that a u s dscretzed wth a rghtward bas so that the x approxmaton at x = x s centered or based towards x = x +1/2 ( a u ) n u a n +1 n un x +1/2 x Usng forward Euler to perform the tme-dscretzaton leads to u n+1 = u n λa n +1/2 (un +1 u n ) λa +n 1/2 (un u n 1) whch s called the wave speed splt form of the numercal approxmaton 55 / 74

56 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 56 / 74 Upwnd Schemes n One Dmenson Introducton to Wave Speed Splttng The flux splt form and wave speed form are connected va ˆf ±n +1/2 = a±n +1/2 (un +1 u n ) From the above relaton and equaton (11), t follows that (a +n +1/2 + a n +1/2 )(un +1 u n ) = g n +1 g n for some flux functon g n (12) Hence, the transformaton from conservaton form to wave speed form and vce versa s ˆf n +1/2 = a n +1/2 (un +1 u n ) + g n, ˆf 1/2 n = a+n 1/2 (un u 1) n + g n (13) 56 / 74

57 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 57 / 74 Upwnd Schemes n One Dmenson Introducton to Wave Speed Splttng u n+1 = u n λa n +1/2 (un +1 u n ) λa +n 1/2 (un u n 1) The above notaton for the wave speed splt form s the standard notaton when wave speed splttng s used to derve new approxmaton methods Wave speed splt form s also often used as a prelmnary step n nonlnear stablty analyss, n whch case the standard notaton s Hence u n+1 = u n + C +n +1/2 (un +1 u n ) C n 1/2 (un u n 1) C +n +1/2 = λa n +1/2, C n 1/2 = λa+n 1/2 C n +1/2 = λa+n +1/2 (14) 57 / 74

58 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 58 / 74 Upwnd Schemes n One Dmenson Introducton to Wave Speed Splttng From (14), t follows that f a method s derved usng wave speed splttng and not just wrtten n wave speed splt form, the splttng underlyng (12) can also be wrtten as λa(u) = C (u) C + (u), C + (u) 0, C (u) 0 Then, the conservaton condton (12) becomes (C n +1/2 C +n +1/2 )(un +1 u n ) = λ(g n +1 g n ) And equatons (13) become λˆf +1/2 n +n = C +1/2 (un +1 u n )+λg n, λˆf 1/2 n n = C 1/2 (un u 1)+λg n n 58 / 74

59 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 59 / 74 Nonlnear Stablty Analyss Focus s set here on explct FT dfference approxmatons Recall that unstable solutons exhbt sgnfcant spurous oscllatons and/or overshoots Recall also that lnear stablty analyss focuses on these oscllatons and reles on the Fourer seres representaton of the numercal soluton: It requres only that ths soluton should not blow up, or more specfcally, that each component n ts Fourer seres representaton should not ncrease to nfnty because of lnearty, ths s equvalent to requrng that each component n the Fourer seres should shrnk by the same amount or stay constant at each tme-step 59 / 74

60 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 60 / 74 Nonlnear Stablty Analyss Smlarly, nonlnear stablty analyss focuses on the spurous oscllatons of the numercal soluton, but wthout representng t by a Fourer seres t can requre that the overall amount of oscllaton remans bounded, whch s known as the Total Varaton Bounded (TVB) condton t can also requre that the overall amount of oscllaton, as measured by the total varaton, ether shrnks or remans constant at each tme-step ( ths s known as the Total Varaton Dmnshng (TVD) condton ) however, whereas not blowng up and shrnkng are equvalent notons for lnear equatons, these are dfferent notons for nonlnear equatons: In partcular, TVD mples TVB but TVB does not necessarly mply TVD 60 / 74

61 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 61 / 74 Nonlnear Stablty Analyss Monotoncty Preservaton The soluton of a scalar conservaton law on an nfnte spatal doman s monotoncty preservng: If the ntal condton s monotone ncreasng (decreasng), the soluton s monotone ncreasng (decreasng) at all tmes Suppose that a numercal approxmaton nherts ths monotoncty preservaton property: Then, f the ntal condton s monotone, the numercal soluton cannot exhbt a spurous oscllaton Monotonocty preservaton was frst suggested by the Russan scentst Godunov n 1959: It s a nonlnear stablty condton, but not a great one for the followng reasons: t does not address the case of nonmonotone solutons t s a too strong condton: t does not allow even an nsgnfcant spurous oscllaton that does not threaten numercal stablty attemptng to purge all oscllatory errors, even the small ones, may cause much larger nonoscllatory errors Godunov s theorem: For lnear methods (NOT to be confused wth lnear problems), monotoncty preservaton leads to frst-order accuracy at best 61 / 74

62 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 62 / 74 Nonlnear Stablty Analyss Total Varaton Dmnshng TVD was frst proposed by the amercan appled mathematcan Amram Harten n 1983 as a nonlnear stablty condton The total varaton of the exact soluton may be defned as follows TV (u(, t)) = sup all possble sets of samples x = u(x +1, t) u(x, t) Laney and Caughey (1991): the total varaton of a functon on an nfnte doman s a sum of extrema maxma counted postvely and mnma counted negatvely wth the two nfnte boundares always treated as extrema and countng each once, and every other extrema countng twce 62 / 74

63 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 63 / 74 Nonlnear Stablty Analyss Total Varaton Dmnshng 63 / 74

64 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 64 / 74 Nonlnear Stablty Analyss Total Varaton Dmnshng Numercal effects that can cause the total varaton to ncrease 64 / 74

65 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 65 / 74 Nonlnear Stablty Analyss Total Varaton Dmnshng The exact soluton of a scalar conservaton law s TVD TV (u(, t 2 )) TV (u(, t 1 )), t 2 t 1 What about the numercal soluton of a scalar conservaton law? The total varaton of a numercal approxmaton at tme t n may be equally defned as TV (u n ) = = u n +1 u n t s the sum of extrema maxma counted postvely and mnma counted negatvely wth the two nfnte boundares always treated as extrema and countng each once, and every other extrema countng twce Now, a numercal approxmaton nherts the TVD property f n, TV ( u n+1) TV (u n ) 65 / 74

66 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 66 / 74 Nonlnear Stablty Analyss Total Varaton Dmnshng Important result: TVD mples monotoncty preservaton and therefore mples nonlnear stablty Proof: Suppose that the ntal condton s monotone the TV of the ntal condton s u u f t s monotone ncreasng and u u f t s monotone decreasng f the numercal soluton remans monotone, TV = cst; otherwse, t develops new maxma and mnma causng the TV to ncrease f the approxmaton method s TVD, ths cannot happen and therefore the numercal soluton remans monotone TVD can be a stronger nonlnear stablty condton than the monotoncty preservng condton 66 / 74

67 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 67 / 74 Nonlnear Stablty Analyss Total Varaton Dmnshng Drawback: Clppng phenomenon (llustrated wth the lnear advecton of a trangle-shaped ntal condton) The TV should ncrease by x between tme-steps but a TVD scheme wll not allow ths clppng error ( here ths error s O( x) because t happens at a nonsmooth maxmum, but for most smooth extrema t s O( x 2 ) ) 67 / 74

68 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 68 / 74 Nonlnear Stablty Analyss Total Varaton Dmnshng Summary of what should be known about TVD n practce, most attempts at constructng a TVD scheme end up enforcng stronger nonlnearty stablty condtons such as the postvty condton dscussed next TVD mples monotoncty preservaton: Ths s desrable when monotoncty preservaton s too weak but less desrable when monotoncty preservaton s too strong gven that TVD can be stronger TVD tends to cause clppng errors at extrema: In theory, clppng does not need to occur at every extrema snce, for example, the local maxmum could ncrease provded that a local maxmum decreased or a local mnmum ncreased or a local maxmum-mnmum par dsappeared somewhere else and may be only moderate when t occurs: However, n practce, most TVD schemes clp all extrema to between frst- and second-order accuracy n theory, TVD may allow large spurous oscllatons but n practce t rarely does n any case, t does not allow the unbounded growth type of nstablty 68 / 74

69 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 69 / 74 Nonlnear Stablty Analyss Postvty Recall that the wave speed splt form of a FT scheme s gven by u n+1 = u n + C +n +1/2 (un +1 u n ) C n 1/2 (un C +n +1/2 0 and C n +1/2 0 u n 1) Suppose that a gven FT numercal scheme can be wrtten n wave speed splt form wth C +n +1/2 0, C n +n +1/2 0 and C +1/2 + C n +1/2 1 (15) Condton (15) above s called the postvty condton (also proposed frst by Harten n 1983) What s the connecton between the postvty condton and the nonlnear stablty of a scheme? 69 / 74

70 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 70 / 74 Nonlnear Stablty Analyss Postvty The answer s: The postvty condton mples TVD Example: FTFS appled to the nonlnear advecton equaton u t + a(u) u x = 0 s postve f 1 λan +1/2 0 Proof: FTFS can be wrtten n wave speed splt form for the purpose of nonlnear stablty analyss as follows u n+1 = u n + C +n +1/2 (un +1 u n ) C n 1/2 (un u n 1) where C +n +1/2 = λan +1/2 and C n 1/2 = 0 λa n +1/2 0 C +n +1/2 = λan +1/2 0 C +n +1/2 + C n +1/2 = λan +1/2 and therefore the condton (15) becomes n ths case 1 λa n +1/2 0 also, note that the postvty condton s n ths case equvalent to the CFL condton 70 / 74

71 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 71 / 74 Multdmensonal Extensons The extenson to multple dmensons of the computatonal part of the materal covered n ths chapter may be tedous n some cases but s straghtforward (except perhaps for the characterstc theory) The expressons of the Euler equatons n 2D and 3D can be obtaned from Chapter 2 (as partcular cases of the expresson of the Naver-Stokes equatons n 3D) For smplcty, the focus s set here on the 2D Euler equatons W t + F x x (W ) + F y y (W ) = 0 71 / 74

72 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 72 / 74 Multdmensonal Extensons 2D structured grd 72 / 74

73 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 73 / 74 Multdmensonal Extensons W t + F x x (W ) + F y y (W ) = 0 For the above 2D Euler equatons, the equvalent of equaton (2) on a 2D structured grd s ( W t t where )n,j = λ x ( F n x +1/2,j F n x 1/2,j ) λ y ( F n y,j+1/2 F n y,j 1/2 ) λ x = t, λ y = t x y j x = x +1/2,j x 1/2,j j, y j = y,j+1/2 y,j 1/2 and F x+1/2,j and F y,j+1/2 are constructed exactly lke ˆf +1/2 n 1D 73 / 74

74 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 74 / 74 Multdmensonal Extensons For example, a 2D verson of FTCS has the followng conservatve numercal fluxes F x n +1/2,j = 1 ( Fx (W n 2 +1,j) + F x (W n = 1 2,j) ) (ρv x ) +1,j + (ρv x ),j (ρv 2 x ) +1,j + (ρv 2 x ),j + p +1,j + p,j (ρv x v y ) +1,j + (ρv x v y ),j (Ev x ) +1,j + (Ev x ),j + (pv x ) +1,j + (pv x ),j F y n,j+1/2 = 1 ( Fy (W n 2,j+1) + F y (W n = 1 2,j) ) (ρv y ),j+1 + (ρv y ),j (ρv x v y ),j+1 + (ρv x v y ),j (ρv 2 y ),j+1 + (ρv 2 y ),j + p,j+1 + p,j (Ev y ),j+1 + (Ev y ),j + (pv y ),j+1 + (pv y ),j 74 / 74