Philip Roe University of Michigan Searching for Truthiness in Computational Fluid Dynamics 2015 AIAA Fluid Dynamics Lecture Philip Roe University of Michigan

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

The order of difficulty of a problem The smallest number of good ideas sufficient to solve the problem CFD is a high-order problem 6/22/2015 Philip Roe, Searching for truthiness

Philip Roe, Searching for truthiness “My first cfd paper” 6/22/2015 Philip Roe, Searching for truthiness

“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

Flow over a waisted axisymmetric body From an unpublished manuscript in which the shock relationships were used as characteristic equations, and thought of as characteristic equations.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. 6/22/2015 Philip Roe, Searching for truthiness

7,800 citations cant be wrong! 6/22/2015 Philip Roe, Searching for truthiness

Going two-dimensional C. C. (Lytton) Sells, RAE Tech Report, 1979. 6/22/2015 Philip Roe, Searching for truthiness

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 6/22/2015 Philip Roe, Searching for truthiness

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. 6/22/2015 Philip Roe, Searching for truthiness

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) 6/22/2015 Philip Roe, Searching for truthiness

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……. 6/22/2015 Philip Roe, Searching for truthiness

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. 6/22/2015 Philip Roe, Searching for truthiness

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 6/22/2015 Philip Roe, Searching for truthiness

Low speed flow past a cylinder 2nd –order TVD Matrix PSI Elliptic/hyperbolic Mach number contours; M. Rad, thesis Michigan 2001 6/22/2015 Philip Roe, Searching for truthiness

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 6/22/2015 Philip Roe, Searching for truthiness

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 6/22/2015 Philip Roe, Searching for truthiness

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

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. 6/22/2015 Philip Roe, Searching for truthiness

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 6/22/2015 Philip Roe, Searching for truthiness

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). 6/22/2015 Philip Roe, Searching for truthiness

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. 6/22/2015 Philip Roe, Searching for truthiness

Philip Roe, Searching for truthiness Typical results 6/22/2015 Philip Roe, Searching for truthiness

Standard accuracy tests “Scheme V” is about twice as accurate as DG1 and about five times less expensive. 6/22/2015 Philip Roe, Searching for truthiness

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 6/22/2015 Philip Roe, Searching for truthiness

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. 6/22/2015 Philip Roe, Searching for truthiness

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. 6/22/2015 Philip Roe, Searching for truthiness

Philip Roe, Searching for truthiness 6/22/2015 Philip Roe, Searching for truthiness

Acoustic solution for an initial pressure hill Pressure absolute velocity 6/22/2015 Philip Roe, Searching for truthiness

Closeup of advancing shock 6/22/2015 Philip Roe, Searching for truthiness

Clarifying the extremum 6/22/2015 Philip Roe, Searching for truthiness

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 6/22/2015 Philip Roe, Searching for truthiness

Translation of a vortex through a 10x 10 box Initial density at O=0.8796 After one circuit =0.8925 6/22/2015 Philip Roe, Searching for truthiness

We have third-order convergence! 6/22/2015 Philip Roe, Searching for truthiness

Philip Roe, Searching for truthiness The Well of Truthiness Power Originality Fashion Truth Hope Simplicity Orthodoxy Tradition Popularity 6/22/2015 Philip Roe, Searching for truthiness

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