Mimetic Methods and why they are better

Transcription

1 Mimetic Methods and why they are better Blair Perot Theoretical and Computational Fluid Dynamics Laboratory February 25, 2015

2 Numerical Methods: Are more than Mathematics Math: Accuracy Stability Convergence Consistency Physics: Conservation Spurious Modes Wave propagation Maximum/minimum Constraints Mimetic methods mimic the physics. 2

3 Relationship Finite Difference Mimetic Methods Finite Element SOM Edge/Face Finite Volume Staggered Natural Neighbors Meshless All Numerical Methods 3

4 Mimetic Advection 1D Advection: Change in mesh size (3x more mesh on the right side) T u T t Central Spurious Wave Reflection Box Method 4

5 Eigenvalues: Theory Vector Laplacian v ( v) v m 2 n 2 1,1,2,4,4,5,5,8,.. Linear FE v node Nedelec FE u vt edge 5

6 Eigenvectors: Practice v( v) f Arnold Linear FE Nedelec FE 6

7 Mimetic Surface Tension 7

8 Overview Use Exact Discretization Separate Discretization from Approximation PDE to LA LA to square LA Do ALL discretization exactly. This means that the calculus and the physics remain exact. Numerical approximation in material laws. Which are engineering approximations already. Numerical approximation goes with physical approximation. 8

9 Discrete Calculus: Part 1 Exact Discretization a b 0 t a A B r b Infinite Dimensional Partial Differential Eqn. Finite Dimensional Matrix Problem Basic unknowns are integral quantities. Collect infinite data into finite groups. 9

10 Discrete Calculus: Part 2 Solution requires Approximation a A Ba r A B r b C Db 0 Underdetermined Unique Square Relate discrete unknowns to each other. This relation is a material law. Also related to interpolation. Also related to discrete inner products 10

11 Example: Heat Eqn ct t kt Components of the Physical Equation i q 0 t g T Conservation of Energy Definition of Gradient Physics Math Material Approximation q i kg ct Fourier s Law Perfectly Caloric Material 11

12 Exact Discretization Perfect representation of Physics and Calculus n 1 n idv idv dt qnda 0 c c f f f e gdl Tn Tn 2 1 I I DQ n 1 n c c f 0 g e GT n Exact 12

13 Solution Numerical Approximation of Constitutive Eqns. Qf M g 1 e I c M T 2 n A Q k f g f Le I cv T c c n e Approximate Dependent on Mesh 13

14 Dual Mesh Viewpoint I I DQ n 1 n c c f 0 g e GT n I idv n1 n1 c c I cv T c c n T n T n T n T n Q dt qn da f f f gdl g e e A Dual Mesh f Q k f L g e e T n T n Primary Mesh 14

15 Properties Conservation of Energy Entropy Production Maximum Principal Any continuous principle for the PDE All errors appear as imperfect material properties. 15

16 Variations Choice of the Dual Mesh. Median Dual Dual or Primary Node centered pressure. Cell centered pressure. Voronoi Dual Choice of interpolation. polynomial reconstruction in cells. reconstruction in dual cells weighted interpolations (FE). 16

17 Results Mimetic Regular CV Log scale 17

18 References Mattiussi, C., An Analysis of Finite Volume, Finite Element, and Finite Difference Methods using some Concepts from Algebraic Topology, J. Computational Phys., 133, , Perot, J. B., and Subramanian, V. Discrete Calculus Methods for Diffusion, J. Computational Phys., 224 (1), 59-81, J. B. Perot, Discrete Conservation Properties of Unstructured Mesh Schemes, Annual Reviews of Fluid Mechanics, 43, , E. Tonti, Why starting from differential equations for computational physics? J. Computational Phys., 257, ,

19 Summary Numerical Methods are changing. Exact Discretization Approx Solution. Works on all types of PDEs 19

20 Questions 20

21 Navier-Stokes Results 21