AE/ME 339. K. M. Isaac. 8/31/2004 topic4: Implicit method, Stability, ADI method. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

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1 AE/ME 339 Comptatonal Fld Dynamcs (CFD) Comptatonal Fld Dynamcs (AE/ME 339) Implct form of dfference eqaton In the prevos explct method, the solton at tme level n,,n, depended only on the known vales of, -,n-,,n-, and +,n-, all whch are at tme level n-. n ote:,n and mean the same. We wll be sng these nterchangeably.

2 Comptatonal Fld Dynamcs (AE/ME 339) atre of solton n explct method can be llstrated graphcally as shown below. 3 Comptatonal Fld Dynamcs (AE/ME 339) In ths formlaton, solton at (,n),,n s affected only by the vales along and below bondary PQR (regon A) n the prevos fgre.vales n regon above PQR (regon B) do not nflence,n. Exact solton (x,y) at Q depends on the vales at all tmes earler than t n, a property of parabolc PDE. Is a lmtaton of the explct method. CFL crteron for stablty: 0 ( x) 4

3 Comptatonal Fld Dynamcs (AE/ME 339) Flly Implct Method The nodes sed n mplct method are llstrated n the fgre below 5 Comptatonal Fld Dynamcs (AE/ME 339) As before, crosses (x) denote grd ponts sed for ( t ) t for ( xx ) x The eqaton now becomes + ( x) n, + n,, n+ n, + +, n+ and crcles (o) 6 3

4 Comptatonal Fld Dynamcs (AE/ME 339) IC and BC are the same as before. ( ) 0, n+ g0 n + ( ) Mn, + g n +,0 f ( x) Eqatons smlar to the above shold be wrtten for each grd pont M ote: left bondary has 0 and rght bondary has M, ths total of (M+) grd ponts (also knows as nodes) are present. Ths we have (M-) lnear smltaneos eqatons wth (M-) nknowns. Explct solton s not possble. 7 Comptatonal Fld Dynamcs (AE/ME 339) Convergence of Implct form Can show sng Taylor seres λ λ + λ + z n, + n,, n+ n, + +, n+ n, Where zn, ( x) 4 tt + xxxx + O ( t) + O ( x) tt and xxxx are evalated at (, n+). 8 4

5 Comptatonal Fld Dynamcs (AE/ME 339) From the leadng n the above z n, O t+ ( x) It can be shown that mplct method converges to the exact solton of the PDE as 0 and x 0 for any vale of ( x) The dfference eqaton s now wrtten for M (+ λ) λ + λg ( t ), n+, n+, n 0 n+, n+ n, + +, n+ n, as follows. for for λ + (+ λ) λ M 9 Comptatonal Fld Dynamcs (AE/ME 339) Fnally, (show M, n fgre) λ + ( + ) g ( t ) M, n λ + + M, n+ M, n λ n+ for M 0 5

6 Comptatonal Fld Dynamcs (AE/ME 339) ote: In the matrx shown above, change n sbscrpt M- s adopted for smplcty. The RHS terms d,d, d are known qanttes. All matrx elements not shown are zero. The matrx above s called a trdagonal matrx.e., only sb-dagonal, dagonal, and sper dagonal terms are non zero. Solton can be obtaned by Gass elmnaton. Recrson solton γ c + Constants and γ are to be determned. Comptatonal Fld Dynamcs (AE/ME 339) Sbstttng nto the th eqaton of the set for - gves a γ c + b + c d + Rewrte as d aγ c + ac ac b b c γ + 6

7 Comptatonal Fld Dynamcs (AE/ME 339) ADI method Comparng the two eqatons we have recrson relatons for and γ ac b γ d aγ From the frst eqaton (see matrx n slde 0) d c b b b Where and γ d γ c + 3 Comptatonal Fld Dynamcs (AE/ME 339) From the last eqaton (see matrx n slde 0) d a b c γ + d c a γ b 4 7

8 Comptatonal Fld Dynamcs (AE/ME 339) Rearrangng yelds d a γ ac b γ c γ + 5 Comptatonal Fld Dynamcs (AE/ME 339) Algorthm smmary: Recrson formlas for and γ b, d γ ac b,, 3,, γ d aγ,,, 3,, 6 8

9 Comptatonal Fld Dynamcs (AE/ME 339) Once the coeffcents have been calclated, the solton vector can be calclated startng wth and gong backwards, as follows: γ c + γ, -, -,, ote: n denotes tme level and denotes the last bt one node n the -drecton. 7 Comptatonal Fld Dynamcs (AE/ME 339) Recrson formlas for b, d γ ac b γ and γ,, 3,, d aγ,,, 3,, Gass elmnaton can case large rond-off errors. Implct schemes sally reqre more comptatonal steps, bt the rato /( x) has no restrctons, s a defnte advantage. 8 9

10 Comptatonal Fld Dynamcs (AE/ME 339) Stablty (7.0) A fnte dfference form s convergent f the solton tends to the exact solton as (, x) 0 (n the absence of rond off error). Stablty refers to amplfcaton of nformaton present n IC, BC or ntrodced by errors n the nmercal procedre sch as rond off error. Von-emann s stablty analyss: Stablty mples only bondedness, not the magntde of devaton from the tre solton. Key featres of stablty analyss: Assme that: ) At any stage, t0 here, a Forer expanson can be made of some ntal fncton f(x), and a typcal term n the expanson can be wrtten as e jx where s a postve constant and j 9 Comptatonal Fld Dynamcs (AE/ME 339) ) Separaton of tme and space dependence can be made. j x At tme t, the term becomes ψ ( t) e By sbstttng n the dfference eqaton, the form of ψ(t) can be determned and stablty crteron establshed. Example Explct fnte dfference form + ( x) n, + n,, n n, +, n Sbsttte for each. 0 0

11 Comptatonal Fld Dynamcs (AE/ME 339) ψ( t+) e jx ψ( t) e jx ψ( t) j ( x x) j x j ( x x) t ( ) e e e + + x Cancel exp (jx) throghot ψ( t + t) ψ( t) λ( e j x e j x + + ) e θ + e θ Trg. dentty: Cos ( θ ) can be sed to get ψ ( t + t) ψ ( t) + λ( + cos( x)) Comptatonal Fld Dynamcs (AE/ME 339) Snce cosθ sn ( θ / ) ( ) ( ) sn x ψ t + t ψ t λ ψ ( t t) ψ( t) 4λsn x + If we choose ψ(0), ths has the solton ( ) ( / x ) ψ () t 4λsn( /) t t And can be proven by sbsttton (see next slde).

12 Comptatonal Fld Dynamcs (AE/ME 339) ( ) ( )/ ψ ( t + t) 4λsn ( x/) t+ ψ()4 t λsn ( x/) For stablty, ψ(t) mst be bonded as (, x) 0 Ths reqres 4λsn ( x /) An amplfcaton factor ξ s sally defned as follows: 3 Comptatonal Fld Dynamcs (AE/ME 339) ξ 4λ sn ( x / ) Whch shows ξ for stablty. In the Forer expanson we consdered only one term correspondng to vale of. When all possble vales of are consdered sn( x/) cold become. Therefore, the stablty condton becomes λ ( x) λ 0.5, ξ λ.0, ξ 3 (nstable) Inttvely t mples that,n affects,n+ n a non-negatve manner. 4

13 Comptatonal Fld Dynamcs (AE/ME 339) A smlar analyss for the mplct method wold gve ξ + 4λsn ( x /) Snce ξ for all λ, the procedre s ncondtonally stable. Consstency means that the procedre may n fact approxmate the solton of the PDE nder stdy and not the solton of some other PDE. 5 Program Completed Unversty of Mssor-Rolla Copyrght 00 Crators of Unversty of Mssor 6 3