Construction of Mimetic Numerical Methods

Transcription

1 Construction of Mimtic Numrical Mthods Blair Prot Thortical and Computational Fluid Dynamics Laboratory Dltars July 17, 013

2 Numrical Mthods Th Foundation on which CFD rsts.

3 Rvolution Math: Accuracy Stability Convrgnc Consistncy Physics: Consrvation Spurious Mods Wav propagation Maximum/minimum Constraints Mimtic mthods mimic th physics. 3

4 Rlationship Finit Diffrnc Mimtic Mthods Finit Elmnt SOM Edg/Fac Finit Volum Staggrd Natural Nighbors Mshlss All Numrical Mthods 4

5 Why it Mattrs Finding numrical issus in complx physics is hard. Adding nw physics is uncrtain. Masur twic, cut onc. 5

6 Mimtic Advction Jason Frank, CWI/Amstrdam 1D: Chang in msh siz (3x) T u T t Cntral Box Mthod Not upwinding, not tim stpping. 6

7 Mimtic Eignvalus Grritsma, Costabl/Daug, Arnold* v ( v) v m n 1,1,,4,4,5,5,8,.. *Finit Elmnt Extrior Calculus: From Hodg Thory to Numrical Stability Linar FE v nod Ndlc FE u vt dg 7

8 Mimtic Vctor Filds v( v) f Linar FE Ndlc FE 8

9 Mimtic Physics 9

10 Discrt Calculus How to always construct a mimtic mthod. Sparat Discrtization from Approximation PDE to LA LA to squar LA Do ALL discrtization xactly. This mans that th calculus and th physics rmain xact. Numrical approximation only in matrial laws Which ar nginring approximations alrady. Numrical approximation gos with physical approximation. 10

11 Exact PDE Discrtization Infinit numbr of quations Infinit numbr of unknowns How could you possibly mak this finit without loosing som information? Discrtization MUST b approximat! Answr: Collct thos infinit unknowns into a finit numbr of groups. Us intgral quantitis as th primary unknowns. 11

12 Exampl Shallow Watr Equations h t ( uh) 0 u uu g h u b u t 1

13 Basic Equations Vrtically avragd Navir-Stoks + Hydrostatic Consrvation of Mass h hu 0 t Physics Exact hu t Consrvation of Momntum ( uhu) ( g h ) ( hu) b( hu) Constant Dnsity No Msh Motion No Bottom Topography 13

14 Mimtic Discrtization Exact Discrtization of Physics n1 n hda hda dt hun ds c c f f 1 u l n n h d hu dl dt ( uq) dl ( pvdt pv 1dt) Exact discrt matrix systm H H DQ n 1 n c c f U U C( h ) ( h ) n 1 n u u G g v Numrically Exact But Not Solvabl 0 0 Clls and tim Dual dgs and tim 14

15 Solvabl Discrtization Exact discrt matrix systm H H DQ n 1 n c c f U U C( h ) ( h ) n 1 n u u G g n 0 Numrical Approximation (intrpolation in spac/tim) H p v c c hda U hudl Q dt hun ds f t n1 t n p dt v f n n1 c g 1 Hc H g ( A A ) hv pv c Q 1 U U S L L Flux to Momntum c Hydrostatic assumption n n1 f ( ) f Just on (vry simpl) possibl xampl 15

16 Dual Msh Viwpoint U U C( h ) ( h ) n 1 n u u G g n H H DQ n 1 n c c f 0 U hudl Q 1 U U S L L n n1 f ( ) f T n Q dt hun ds f f T n T n p v t n1 t n p dt v Dual Msh n n1 c g 1 Hc H g ( A A ) hv pv c c H Primary Msh c c hda 16

17 Vorticity Consrvation Momntum Eqn U U C( h ) ( h ) n 1 n u u G g n Transform th Gradint U U C( uhu) G( h ) ( h h ) G( gh ) n 1 n g 1 n n1 n n Sum ovr th dgs (Curl) n1 n1 U U h h h 1 C( uhu) G( gh ) 0 Discrt Dfinition of Vorticity is found 1 U da uda hu dl c c h h n 17

18 Mthod Choics Choic of th Dual Msh. Mdian Dual Choic of which msh to us. Nod cntrd prssur. Cll cntrd prssur. Voronoi Dual Choic of intrpolation. polynomial rconstruction in clls. rconstruction in dual clls wightd intrpolations (FE). 18

19 Rsults 19

20 Rfrncs Mattiussi, C., An Analysis of Finit Volum, Finit Elmnt, and Finit Diffrnc Mthods using som Concpts from Algbraic Topology, J. Computational Phys., 133, , Subramanian, V., and Prot, J. B. Highr Ordr Mimtic Mthods for Unstructurd Mshs, J. Computational Phys., 19, 68-85, 006. Prot, J. B., and Subramanian, V. Discrt Calculus Mthods for Diffusion, J. Computational Phys., 4 (1), 59-81, 007. J. B. Prot, Discrt Consrvation Proprtis of Unstructurd Msh Schms, Annual Rviws of Fluid Mchanics, 43, ,

21 Summary Numrical Mthods ar changing. Exact Discrtization and Approx Solution. Works on all typs of PDEs 1