1. Philip Roe University of Michigan
Searching for Truthiness in Computational Fluid Dynamics AIAA Fluid Dynamics Lecture
Philip Roe
University of Michigan

2. Philip Roe, Searching for truthiness
Two definitions
truthiness n. informal, the quality of seeming or being felt to be true, even if not necessarily true. physicsicity n. coinage, the quality of appearing to be based on physical principles, even if those principles are false or inappropriate.
6/22/2015
Philip Roe, Searching for truthiness

3. The order of difficulty of a problem
The smallest number of good ideas sufficient to solve the problem
CFD is a high-order problem
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Philip Roe, Searching for truthiness

4. Philip Roe, Searching for truthiness
“My first cfd paper”
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Philip Roe, Searching for truthiness

5. “Uniformly valid” characteristic equations
Suppose that points P1, P2 either lie
on the same C+ characteristic in a
smooth flow, or on different sides
of a C- shock. P2 P1

6. Flow over a waisted axisymmetric body
From an unpublished manuscript in which the shock relationships were used as characteristic equations, and thought of as characteristic equations. What happens between consec- -utive grid points is unknown, but it might be a shock and the method must not break down.
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Philip Roe, Searching for truthiness

7. 7,800 citations cant be wrong!
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Philip Roe, Searching for truthiness

8. Going two-dimensional
C. C. (Lytton) Sells, RAE Tech Report, 1979.
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Philip Roe, Searching for truthiness

9. The problem with the ICASE equation
We gained many incredible insights from the analytical study of numerical methods for the linear advection equation,
but we were misled by those successes into thinking that all hyperbolic problems are alike. In one dimension, acoustic waves and advection “waves” behave much alike, but in two/three dimensions they behave quite differently.
advection acoustics
linear domain of dependence isotropic domain of dependence
bounded variation unbounded variation
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Philip Roe, Searching for truthiness

10. Do we need discontinuous representations?
Pro
Discontinuities are introduced by geometrical limiting processes.
Discontinuous Galerkin methods have mass matrices that are local, in accordance with the nature of hyperbolic flow.
Con
Discontinuities aligned with cell faces introduce locally one-dimensional physics.
The scaling of pressure disturbances at low Mach number is badly wrong.
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Philip Roe, Searching for truthiness

11. Philip Roe, Searching for truthiness
Low speed flow
Bernouillis equation p+½ρq2 = p0 shows that for almost incompressible flow δp/δq = O(M2). For the one-dimensional flow assumed by Riemann solvers δp/δq =O(1). Pressure changes are actually proportional to ∇⋅q, which is O(M2)
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Philip Roe, Searching for truthiness

12. Residual distribution schemes (1)
Data is stored at vertices of triangles,. For each element T, compute ∅𝑇= 𝑇𝑭𝑥+𝑮𝒚 𝑑𝑠 = 𝜕𝑇(𝑭𝑑𝑦−𝑮𝑑𝑥) And then distribute 𝜙= 𝜙 𝑇 1 + 𝜙 𝑇 2 + 𝜙 𝑇 3
I wrote a small note in 1987, and subsequently worked with students Tim Tomaich, Lisa Mesaraos, Mani Rad.
Extensive contributions from Hermann Deconinck, Remi Abgrall, Mario Richiutto…….
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Philip Roe, Searching for truthiness

13. Residual distribution (2)
For scalar problems , both linear and nonlinear,
there are good simple schemes that recognize the
flow direction as special even when it is not aligned
with an edge. For systems, there are “matrix versions”
of these schemes.
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Philip Roe, Searching for truthiness

14. Potential flow past a nonlifting ellipse
In elliptic-hyperbolic splitting, the residual is divided into two parts based on the real and complex eigen-vectors of the steady flow. The real part is handled by the scalar method, the complex part by least squares
It never proved possible to extend these results to
unsteady or three-dimensional flows
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Philip Roe, Searching for truthiness

15. Low speed flow past a cylinder
2nd –order TVD Matrix PSI Elliptic/hyperbolic
Mach number contours; M. Rad, thesis Michigan 2001
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Philip Roe, Searching for truthiness

16. Philip Roe, Searching for truthiness
Operator Splitting
For a system of equations 𝒖 𝑡 +𝐴𝒖+𝐵𝒖=0 we can perform the sequence 𝒖 ∗ = 𝒖 𝑛 +𝐴 𝑢 𝑛 𝒖 𝑛+1 = 𝒖 ∗ +𝐵 𝒖 ∗ with no error at the PDE level if and only if 𝐴𝐵−𝐵𝐴=0 so that 𝐴, 𝐵 must have the same eigenvectors. The commuting property failed in unsteady or three-dimensional flow
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Philip Roe, Searching for truthiness

17. This cannot be done by Flux Vector Splitting
Many codes attempt a “convective/acoustic”, or “wave/particle” splitting such as
Advection plus dilation
Although the eigenvalues of these fluxes are
correct, the eigenvectors are not, and the
operators do not commute
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Philip Roe, Searching for truthiness

18. Advection-Acoustic Splitting of the Euler Equations
transport
dilation
pressure
Advection Acoustics

19. Primitive variables make a clean split
advection acoustics
Nonconservative forms of the equations can be split into two commuting subsystems. The split is not unique, Any other thermodynamic variable can be substituted for pressure.
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Philip Roe, Searching for truthiness

20. The objective of the project
A robust third-order scheme
Continuous data representation
No one-dimensional physics
Insensitive to mesh quality
Not too complicated
Fully discrete
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Philip Roe, Searching for truthiness

21. Van Leers “new approach to numerical advection”
Reconstruct in some basis,
Evolve exactly,
Project back into basis
Independently rediscovered by Popov and Ustyagov (2007), Akoh, Li and Xiao (2010), Berthon et al (2013), Zeng (2013).
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Philip Roe, Searching for truthiness

22. Philip Roe, Searching for truthiness
Noteworthy features
Exact for any quadratic data,
Therefore third-order on any grid,
reconstruction,
Upwind, but no Riemann solver,
Physical argument needed only to second order,
Final step needs no physics except conservation,
Fully discrete,
Stable up to by construction,
Low storage (2DOF/cell, same as DG1).
In 3d, storage is only 2.20 DOF/cell, compared with 4.0 for DG1 or 10.0 for DG2.
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Philip Roe, Searching for truthiness

23. Philip Roe, Searching for truthiness
Typical results
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Philip Roe, Searching for truthiness

24. Standard accuracy tests
“Scheme V” is about twice as accurate as DG1
and about five times less expensive.
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Philip Roe, Searching for truthiness

25. Philip Roe, Searching for truthiness
Poisson’s integral
The genuinely, truly, fully multidimensional upwind method (really!)
Given some function u(x,y,z) its spherical means are defined by dS ct
And the initial-value-problem for the linear acoustic system
is solved by
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Philip Roe, Searching for truthiness

26. Reduction to one dimension
If the function 𝑢(𝑥,𝑦,𝑧) depends only on 𝑥 , then
𝑴 𝑹 𝒖(𝒙) = 𝟏 𝟐𝑹 −𝑹 𝑹 𝒖 𝒙 𝒅𝒙
Inserting this into the
Poisson integral gives the standard one-dimensional characteristic equations.
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Philip Roe, Searching for truthiness

27. Sketch of Active Flux method in two dimensions
In the first stage, boundary values are updated, using particle tracing for the advective terms, or Poisson integrals for the acoustic terms. Note that this stage uses primitive variables. These two operations are sequential (operator splitting)
In the second stage, fluxes are calculated from the updated boundary values, and new cell averages are calculated. A “conservative fix-up” is needed at this stage.
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Philip Roe, Searching for truthiness

28. Philip Roe, Searching for truthiness
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Philip Roe, Searching for truthiness

29. Acoustic solution for an initial pressure hill
Pressure absolute velocity
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Philip Roe, Searching for truthiness

30. Closeup of advancing shock
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Philip Roe, Searching for truthiness

31. Clarifying the extremum
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Philip Roe, Searching for truthiness

32. The pressureless Euler equations
represent a cloud of particles travelling independently.
The “momentum” equations have the implicit solution
The particle that will be at x after time Δt is currently at x* where
and the new values will be
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Philip Roe, Searching for truthiness

33. Translation of a vortex through a 10x 10 box
Initial density at O=0.8796
After one circuit =0.8925
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Philip Roe, Searching for truthiness

34. We have third-order convergence!
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Philip Roe, Searching for truthiness

35. Philip Roe, Searching for truthiness
The Well of Truthiness
Power
Originality
Fashion
Truth
Hope
Simplicity
Orthodoxy
Tradition
Popularity
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Philip Roe, Searching for truthiness

36. Acknowledgements
Financial support from Grant NNX12AJ70A administered by Dr Mujeeb Malik under the NASA Revolutionary Computational Aeronautics Progrem.
Past Students
Lisa Mesaros, Mani Rad, Hiroaki Nishikawa, Tim Eymann
Current students Doreen Fan Brad Maeng
6/22/2015
Philip Roe, Searching for truthiness