Higher-Order Discrete Calculus Methods

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1 Highr-Ordr Discrt Calculus Mthods J. Blair Prot V. Subramanian

2 Ralistic Practical, Cost-ctiv, Physically Accurat Paralll, Moving Msh, Complx Gomtry, Slid 1

3 Contxt Discrt Calculus Mthods Finit Dirnc Mimtic SOM Finit Elmnt Edg/Fac Staggrd Finit Volum Natural Nighbors Mshlss DC Mthods ar a subst o many othr classical approachs Slid 2

4 Exact Discrtization Discrtization: Continuous PDE => Finit Dimnsional Matrix Problm Solution: This Can B Don Exactly Rquirs Approximations/Error But all approximation rrors occur in th constitutiv quations (in matrial proprtis). All numrical rrors appar with th modling rrors. Discrtization Approximation Slid 3

5 Exampl Physical Equation (Hat Equation) ( ρct ) = k T t Componnts o th Physical Equation i q = 0 t Consrvation o Enrgy g = T Dinition o Gradint Physics Math Matrial Approximation q = kg Fourir s Law i = ρct Prctly Caloric Matrial Th Physical (Continuous) Systm Slid 4

6 Exampl Exact Discrtization o Physics and Calculus + + = n 1 n idv idv dt da 0 c c q n g d l = T T n2 n1 Numrical Approximations Q = M g 1 A Q = k L g I + I + DQ = n 1 n c c g = GT n Numrically Exact 0 I c = M T 2 n I = ρcv T c c n Numrically Approximat Th Numrical (Discrt) Systm Slid 5

7 Highr Ordr Discrt Calculus Highr Ordr Exact Gradint g dl = T T dg dg ac n2 n1 n2 n2 n1 n1 Applicabl to Any Polyhdron n2 ( t x) g dl = t ( x T x T ) Tdl n gda = t dgs Tdl n1 T n Td d g l dg [2] ( t x) g dl= G dg da n g ac Sam Msh Mor Unknowns Tn Tdl Slid 6

8 Highr Ordr Discrt Calculus Highr Ordr Exact Divrgnc Nd on quation or ach dg. Intgrat ovr th vry thin CVs surrounding th dual acs. Tak th thin limit carully so thin lmnts align. dg clls ( q n ) n da + n c qdl = or dg acs qda + [ q] n dl = 0 dg acs l 0 c dx Slid 7

9 Rconstruction Hav gdl xg dl n gda Nd q n da ( q n ) n da q dl Assum q=-kg q g is linar on primary msh clls. Basis Functions not Rquird Slid 8

10 Rsults Good Cost/Accuracy or Smooth Solutions Slid 9

11 Conclusions Exact discrtization o PDEs is possibl and strongly ncouragd. Excllnt numrical/mathmatical proprtis ar NOT rstrictd to FE mthods. Applicabl to ANY polygon msh (including mshlss), no xplicit basis unctions, no nd to build matrics. Slid 10

12 JCP Publications Subramanian, V., and Prot, J. B. Highr Ordr Mimtic Mthods or Unstructurd Mshs, J. Comput.. Phys., 219,, 68-85, 85, 2006 Prot, J. B., and Subramanian, V. Discrt Calculus Mthods or Diusion, J. Comput.. Phys., 224 (1), 59-81, Subramanian, V., and Prot, J. B. A Discrt Calculus Analysis o th Kllr-Box Schm and a Gnralization o th Mthod to Arbitrary Mshs, Accptd to J. Comput.. Phys.,, Slid 11

14 Discrt Calculus Oprators Gradint Oprator G ( T T GT ) n 2 n 1 n Divrgnc Oprator D Q DQ Curl Oprator C s Cs Rotation Oprator R U RU G C = D = R T T φ = 0 φ = constant ( ) = 0 GT = 0 { T } = { c} DC = 0 n n ( ) = 0 CG = 0 Discrt Calculus Oprators mimic Continuous Oprators Slid 6

15 Th Discrt Calculus Approach Physics i + q = t g = T 0 Physically Exact Numrics I I Q n + 1 n c c + D = 0 g = GT n Numrically Exact q = kg i = ρct Physically Approximat g Q = k A L Ic = ρcvctn Numrically Approximat ( ρct ) t = k T ( ρcv I T ) c n A = D k GT t L n Continuous vs. Discrt Systm Slid 7

16 Discrt Calculus Mthods Nod-Basd Mthod T n Cll-Basd Mthod (Mixd) Tn, Q Fac-Basd Mthod (KB) T, Q Highr Ordr Mthod T, T n DC Approach is a mthodology not a particular mthod! Slid 9

5 < x < 1 T xact 1+ x + y 0 x 0.5 = 4x + y 0.

17 Physical Accuracy Discontinuous Diusion: Hat Flow at an Angl k 4 0 < x < 0.5 = < x < 1 T xact 1+ x + y 0 x 0.5 = 4x + y x 1 Linarly Complt as wll as Physically Ralistic Slid 10

18 Cost o DC Mthods For any accuracy lvl, DC is mor cost-ctiv than Finit Volum Mor Cost-Ectiv than Traditional Mthods Slid 12

19 Implications Exact Discrtization: Discrt Oprators (Div, Grad, Curl) bhav just lik th continuous oprators. Mimtic. Discrt d Rham Complx (algbraic topology). ( ) = 0, tc Physics is always capturd (consrvation, wav propagation, max principal, ). No spurious mods. No surpriss. Mthods that captur physics wll Slid 3

20 Incomprssibl Navir-Stoks Tsts

21 Total cost or a dsird accuracy lvl Fac-basd DC convrgs mor slowly than Finit Volum High Matrix Condition Numbrs Advrsly Impacts th Cost Slid 8 / 16

22 Discrt Calculus Oprators Discrt Gradint Oprator n g 1 = Tn 2 Tn bc g 2 = Tn 2 + T bc g 3 = Tn 2 + T G = bc g 4 = Tn 1 + T bc g = T + T 1 0 Discrt Gradint Oprator ( ) 5 n 1 5 Discrt Divrgnc Oprator ( c) DQ = Q + Q + Q c DQ = Q + Q + Q c D = G = D T

23 Mor Discrt Calculus Oprators Discrt Curl Oprator ( ) C = Discrt Rotation Oprator ( ) C = C = C T

Construction of Mimetic Numerical Methods

Construction of Mimetic Numerical Methods Construction of Mimtic Numrical Mthods Blair Prot Thortical and Computational Fluid Dynamics Laboratory Dltars July 17, 013 Numrical Mthods Th Foundation on which CFD rsts. Rvolution Math: Accuracy Stability