Acceleration Methods for Numerical Solution of the Boltzmann Equation

1. Acceleration Methods for Numerical Solution of the Boltzmann Equation Husain Al-Mohssen

2. Motivation & Introduction Problem Statement Proposed Approach Important Implementation Details Examples Discussion Future Work Outline

3. Motivation • Nano-Micro devices have been developed recently with very small dimensions: • DLP (Length) • HD read/write head (Gap Length) • At STP an air molecule travels an average distance between collisions • As may be expected the Navier-Stokes (NS) description of the flow starts to break down as system length becomes comparable to l • Accurate engineering models are essential for the understanding and design of such systems

5. Motivation (cnt) • The Knudsen number is defined as the ratio of the mean free path to a characteristic dimension (Kn= l/L). Kn is a measure of the degree of departure from the NS description • Kn Regimes: • Recent applications are at low Ma number

6. Introduction

7. Introduction (cnt) The Boltzmann Equation (BE) in normalized form: • Follows from the dilute gas assumption • Valid for all Kn • 7D(1time+3Space+3Velocity) nonlinear Integro-differential equation

8. Introduction (cnt) Numerical Methods of Solving the BE: • Particle based: DSMC • Collisionless advection step + collision steps are successively applied. • Can be shown to simulate BE exactly in the limit of large numbers [Wagner 1992]. • Chronic sampling problems at low speeds [Hadjiconstantinou et al, 2003]. • Low Ma lmit particularly troublesome • Approximations of the BE • Linearized (has many advantages espcially when Ma<<1; still requires numcerical solution) • BGK CI Replaced with • Numerical solutions of the BE • Recently Baker and Hadjiconstantinou (B&H) proposed a method to solve the BE at low Ma in a relatively efficient manner.

9. Introduction (cnt)

10. Problem Statement

11. Proposed Solution Methodology F(ui) and F’(ui) F(u) x ui+1 ui

12. Proposed Solution Methodology (cnt)

13. Simplified Flow Chart of Method Start Find Estimate Integrate BE to find Use Broyden to find from and Find Converged? No Yes End

14. Important Implementation Details(for Broyden Portions)

15. 1D Graphical Analog F[u] u

16. Important Implementation Details (BE Portions) Shift f to target mean Integrate BE 1 2 3

17. Step1: Equilibrate f Step2: Sample Calculation to find Flow Chart of Method Start Find Estimate Integrate BE Use Broyden to find from and Find Converged? No Yes End

18. U 0.04 Exaggerated Kn Layer 0.02 100 200 300 400 500 Node # -0.02 -0.04 Exaggerated Kn Layer Examples

19. kn=0.1 0.0015 0.001 0.0005 20 40 60 80 100 120 -0.0005 -0.001 -0.0015 Examples (cnt) Knudsen Layer Broyden Solution Exact layer Convergence History 512 nodes, kn =0.1

20. Discussion

21. Future Work

22. The End Questions?

23. DSMC Performance Scaling

24. B&H Performance Scaling

25. Plot of Convergence Rates of Different Methods • Plot of error for Direct integration, Broyden and Baker Implicit code. Kn=0.025 # of nodes 128. (log[Error] vs. log[CI evaluations])

26. Error of Broyden vs. noise of F • Show how sig=sig/N_inf in multidimensions

27. Broyden Step • Broden formula • Formula constraints • Broyden Formula derivation

28. Backup slides+notes • [[check conv. History 4 high kn and 512]] • “proper” kndsen layer with 100^3 and lower noise kn=0.1 and at least 128 nodes. Replace one already in presentation • Change Conv. History plto to 512 and kn0.025 and 30^3 cells • N_inf vs. Kn for our pb’s to show our rough break point….

29. DSMC Performance Scaling (Backup) Direct Integration Cost: Broyden Cost: Slope Sampling Scaling is key: Analysis assumes sampling a small portion of run =>

30. dt .01 , g .1 Red dt .1 g .1 Green dt .1 g90 Blue dt .1 , .01 g1 Orange = = = = 0.0001 0.00001 - 6 1. 10 ´ - 7 1. 10 ´ - 8 1. 10 ´ 10 100 1000 B&H Noise for Different Paramters(Backup) For little extra computational Effort you get a dramatic decrease in measurement error. compare for example pt. A, B and C. A Kn=? If only interested in eng. Accuracy N_inf=10^-4/sig_sample Cost A=Cost B Cost C=10 Cost A B C

31. Distribution Function initilization (Backup) • Plot of norm f vs. step [[Possibly for multiple kn [[what kn? What state of F?]]

32. Scaling Arguments (Backup) • Why is it always O(10)? Well possibly because of this: • As per Kelly Newton’s is q-Quadratic and secent is Q-superlinear; Broyden is somewhere in between. • The other plot is the MMA result using [a] x/nnn + noise • Kelly says eps=K eps^2 not exp[-2t] MMA Model Problem in Multi-D with Noise

33. Can u answer these Questions • Is it possible that O(10) will increase with less noise Requrement • If u reduce Dt sample to decrease noise, don’t u increase N_inf??!!! • [[Re-initializing a Run after it reaches its minimum noise level with less noise as a method of Confirming convergance or reducing noise (NB: since we are somehow finding the null space of the Jacobian aren’t we somehow garanteed to have a sick matrix when we stall?)]]

34. Can u Explain B&H? • What is importance sampling? & how is it applied to CI? Write the appt. version of CI. • What is control variate M/C interation? • How is the finite volume Spliting method implemented? What are the various Stability conditions?

35. Integration Stability Codnition • CI step • Convection Step • Implicit step?

36. -7 -6 -5 -4 Log10[Input Noise] -1 -2 Log10[error] 32 -3 16 512 -4 128 -5 64