# Bram van Leer

Bram van Leer
Prof. van Leer at Aerospace Engineering building FXB at University of Michigan
Born Netherlands East-Indies
Fields CFD
Fluid dynamics
Numerical Analysis
Institutions University of Michigan
Alma mater Leiden State University
Doctoral advisor Hendrik C. van de Hulst
Known for MUSCL scheme

Bram van Leer is Arthur B. Modine Emeritus Professor of aerospace engineering at the University of Michigan, in Ann Arbor.[1] He specializes in Computational fluid dynamics (CFD), fluid dynamics, and numerical analysis, fields on which he has had a substantial influence.

An astrophysicist by education, van Leer made seminal contributions to CFD in his 5-part article series “Towards the Ultimate Conservative Difference Scheme (1972-1979),” where he extended Godunov’s finite-volume scheme to the second order (MUSCL), developed nonoscillatory interpolation using limiters, an approximate Riemann solver, and Discontinuous-Galerkin schemes for unsteady advection. Since joining the University of Michigan’s Aerospace Engineering Department (1986) he has worked on convergence acceleration by local preconditioning and multigrid relaxation for Euler and Navier-Stokes problems, unsteady adaptive grids, space-environment modeling, atmospheric flow modeling, extended hydrodynamics for rarefied flows, and Discontinuous Galerkin methods. He retired in 2012.

Throughout his career, van Leer has crossed interdisciplinary boundaries to export state-of-the-art CFD technology. Starting from astrophysics, he first made an impact on weapons research, followed by aeronautics, then space-weather modeling, atmospheric modeling, surface-water modeling and automotive engine modeling, to name the most important fields.

## Personal Interests

van Leer playing piano at Pierpont Commons, University of Michigan

van Leer is also an accomplished musician, playing the piano at the age of 5 and composing at 7. His musical education includes two years at the Royal Conservatory for Music of The Hague, NL. As a pianist he was featured in the Winter '96 issue of Michigan Engineering (Engineering and the Arts); as a carillonist he has played the carillon of the Central Campus Burton Tower on many football Saturdays.

He was the world's first and only cj (carillon-jockey), based on the North Campus Lurie Tower. In 1993 he gave a full-hour recital on the carillon of the City Hall in Leiden, the town where he studied for many years. van Leer prefers to improvise in the Dutch carillon-playing style; one of his improvisations is included on a 1998 CD featuring both UM carillons. His carillon composition "Lament" was published in the UM School of Music's carillon music series on the occasion of the Annual Congress of the Guild of Carilloneurs in North America, Ann Arbor, June 2002. A flute composition by van Leer was performed twice in 1997 by University of Michigan Professor Leone Buyse.

## Research work

Bram van Leer was a doctoral student in astrophysics at Leiden Observatory (1966–70) when he got interested in Computational Fluid Dynamics (CFD) for the sake of solving cosmic flow problems. His first major result in CFD[2] was the formulation of the upwind numerical flux function for a hyperbolic system of conservation laws: ${\displaystyle F^{\hbox{up}}={\frac {1}{2}}(F_{j}+F_{j+1})-{\frac {1}{2}}|A|_{j+{\frac {1}{2}}}(u_{j+1}-u_{j}).}$

Here the matrix ${\displaystyle |A|}$ appears for the first time in CFD, defined as the matrix that has the same eigenvectors as the flux Jacobian ${\displaystyle A}$, but the corresponding eigenvalues are the moduli of those of ${\displaystyle A}$. The subscript ${\displaystyle j+{\frac {1}{2}}}$ indicates a representative or average value on the interval ${\displaystyle (x_{j},x_{j+1})}$; it was no less than 10 years later before Philip L. Roe first presented his much used averaging formulas.

Next, van Leer succeeded in circumventing Godunov's barrier theorem (i.e., a monotonicity preserving advection scheme cannot be better than first-order accurate) by limiting the second-order term in the Lax-Wendroff scheme as a function of the non-smoothness of the numerical solution itself. This is a nonlinear technique even for a linear equation. Having discovered this basic principle, he planned a series of three articles titled "Towards the ultimate conservative difference scheme", which advanced from scalar nonconservative but non-oscillatory (part I[3]) via scalar conservative non-oscillatory (part II[4]) to conservative non-oscillatory Euler (part III[5]). The finite-difference schemes for the Euler Equations turned out to be unattractive because of their many terms; a switch to the finite-volume formulation completely cleared this up and led to Part IV[6] (finite-volume scalar) and, finally, Part V[7] (finite-volume Lagrange and Euler) titled, "A second-order sequel to Godunov's method", which is his most cited (approaching 5000 citations in 2015) article. This paper was reprinted in 1997 in the 30th anniversary issue of Journal Computational Physics with an introduction by Charles Hirsch.[8]

The series contains several original techniques that have found their way into the CFD community. In Part II two limiters are presented, later called by van Leer "double minmod" (after Osher's "minmod" limiter) and its smoothed version "harmonic"; the latter limiter is sometimes referred to in the literature as "van Leer's limiter." Part IV, "A new approach to numerical convection," describes a group of 6 second- and third-order schemes that includes two Discontinuous Galerkin schemes with exact time-integration. van Leer was not the only one to break Godunov's barrier using nonlinear limiting; similar techniques were developed independently around the same time by Boris[9] and by V.P. Kolgan, a Russian researcher unknown in the West. In 2011 Van Leer devoted an article to Kolgan's contributions [10] and had Kolgan's 1972 TsADI report reprinted in translation in the Journal of Computational Physics.

After the publication of the series (1972–79) van Leer spent two years at ICASE (NASA LaRC), where he was engaged by NASA engineers interested in his numerical expertise. This led to van Leer's differentiable flux-vector splittng[11] and the development of the block-structured codes CFL2D and CFL3D [12][13] which still are heavily used. Other seminal papers from these years are the review of upwind methods with Harten and Lax,[14] the AMS workshop paper [15] detailing the differences and resemblances between upwind fluxes and Jameson's flux formula, and the conference paper with Mulder[16] on upwind relaxation methods; the latter includes the concept of Switched Evolution-Relaxation (SER) for automatically choosing the time-step in an implicit marching scheme.

After permanently moving to the U.S., van Leer's first influential paper was “A comparison of numerical flux formulas for the Euler and Navier-Stokes equations,[17]” which analyzes numerical flux functions and their suitability for resolving boundary layers in Navier-Stoke's calculations. In 1988, he embarked on a very large project, to achieve steady Euler solutions in O(N) operations by a purely explicit methodology. There were three crucial components to this strategy: 1. Optimally smoothing multistage single-grid schemes for advections 2. Local preconditioning of the Euler equations 3. Semi-coarsened multigrid relaxation

The first subject was developed in collaboration with his doctoral student, C.H. Tai.[18] The second subject was needed to make the Euler equations look as much scalar as possible. The preconditioning was developed with doctoral student W. -T. Lee.[19] In order to apply this to the discrete scheme, crucial modification had to be made to the original discretization. It turned out that applying the preconditioning to an Euler discretization required a reformulation of the numerical flux function for the sake of preserving accuracy at low Mach numbers. Combining the optimal single grid schemes with the preconditioned Euler discretization was achieved by doctoral student J. F. Lynn.[20] The same strategy for the Navier Stokes discretization was pursued by D. Lee.[21]

The third component, semi-coarsened multigrid relaxation, was developed by van Leer's former student W. A. Mulder (Mulder 1989). This technique is needed to damp certain combinations of high- and low-frequency modes when the grid is aligned with the flow.

In 1994, van Leer teamed up with Darmofal, a post-doctoral fellow at University of Michigan at the time, to finish the project. The goal of the project was first reached by Darmofal and Siu (Darmofal, and Siu 1999), and later was done more efficiently by van Leer and Nishikawa.[22]

While the multi-grid project was going on, van Leer worked on two more subjects: multi-dimensional Riemann solvers,[23][24] and time-dependent adaptive Cartesian grid.[25] After conclusion of the multi-grid project, van Leer continued to work on local preconditioning of the Navier-Stokes equations together with C. Depcik.[26] A 1-D preconditioning was derived that is optimal for all Mach and Reynolds numbers. There is, however, a narrow domain in the (M, Re)-plane where the preconditioned equations admit a growing mode. In practice, such a mode, if it were to arise, should be damped by the time-marching scheme, e.g., an implicit scheme.

In the last decade of his career, van Leer occupied himself with extended hydrodynamics and discontinuous Galerkin method. The goal of the first project was to describe rarefied flow up to and including intermediate Knudsen numbers (Kn~1) by a hyperbolic-relaxation system. This works well for subsonic flows and weak shock waves, but stronger shock waves acquire the wrong internal structure.[27][28] For low speed flow, van Leer’s doctoral student H. L. Khieu tested the accuracy of the hyperbolic-relaxation formulation was tested by comparing simulations with the numerical results of a full-kinetic solver based on Boltzmann equation.[29] Recent research has demonstrated that a system of second order PDEs derived from the hyperbolic relaxation systems can be entirely successful; for details see Myong Over-reach 2014.

The second project was the development of discontinuous Galerkin (DG) methods for diffusion operators. It started with the discovery of the recovery method for representing the 1D diffusion operator.

Starting in 2004, the recovery-based DG (RDG)[30] has been shown an accuracy of the order 3p+1 or 3p+2 for even or odd polynomial-space degree p. This result holds for Cartesian grids in 1-, 2-, or 3-dimensions, for linear and nonlinear diffusion equations that may or may not contain shear terms.[31][32][33][34] On unstructured grids, the RDG was predicted to achieve the order of accuracy of 2p+2; this research unfortunately has not been completed before van Leer retired.

In addition to the above narrative, we list some subjects and papers related to van Leer’s interdisciplinary research efforts:

• Cosmic gas dynamics - van Albada, van Leer, and Roberts[35]
• Space Environment Modeling - Clauer et al.[36]
• Atmospheric Modeling - Ullrich, Jablonowski, van Leer[37]
• Automotive Engine modeling - Depsik, van Leer, Assanis[38]

Three significant review papers by van Leer are:

• The Development of Numerical Fluid Mechanics and Aerodynamics since the 1960s: US and Canada[39]
• Introduction to Computational Fluid Dynamics[40]
• B. van Leer, "Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution schemes," Communications in Computational Physics, Vol.1, pp. 192–205, 2006.

In 2010, van Leer received AIAA Fluid Dynamics award for his lifetime achievement. On this occasion, van Leer presented a plenary lecture titled, “History of CFD Part II,” which covers the period from 1970 to 1995. Below is the poster van Leer and his doctoral student Lo designed for this occasion.

The tableau is an allegory on the genesis of modern CFD in the period 1970-1985, specifically: the development of high-resolution methods (non-oscillatory methods of greater-than-first-order accuracy), and their ultimate adoption by the aerospace community. We see an exotic landscape dominated by a great pyramid. Three men are trying to reach its top by different means: Jay Boris (hammer en chisel), Bram van Leer (rope) and Vladimir Kolgan (ladder); the latter's untimely death in 1978 made him an unknown even in Russia. Note that the pyramid is also a giant capital Greek delta, symbol of the finite difference that pervades the equations of CFD. Keeper of the gate is John von Neumann, the father of CFD. From the prehistory of CFD are the busts, on the far left, of Richard Courant, Kurt Friedrichs and Hans Lewy, whose initials we know so well. On the far right we find, on beach chairs, Peter Lax and Sergei Godunov, giants of numerical analysis from the generation following Von Neumann. They relax while a younger generation struggles to raise the state of the art in CFD. In the foreground, going from left to right, we first encounter Bob MacCormack who, in the late 1960s, adapted the second-order Lax-Wendroff method to aeronautical uses, but was not able to tame its numerical oscillations. Next, Phil Roe, perhaps contemplating his approximate Riemann solver or Superbee limiter. Past the gate, Stan Osher and Ami Harten (died 1994), likely discussing TVD or ENO techniques. The latter three, together with van Leer, were most influential in getting high-resolution methods accepted in aerospace engineering; much of the technology transition took place at ICASE, NASA LaRC. Last but not least, in the airplane, Antony Jameson, who went his own way, developing a suite of highly efficient CFD codes for steady aeronautics.

## References

1. ^ "van Leer at University of Michigan". Retrieved 2009-04-04.
2. ^ van Leer, B. (1970). A Choice of Difference Schemes for Ideal Compressible Flow (Ph.D.). Sterrewacht, Leiden, The Netherlands.
3. ^ B. van Leer. Towards the ultimate conservative difference scheme I. The quest of mono- tonicity. In Lecture Notes in Physics. Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, pages 163–168. Springer, 1973.
4. ^ B. van Leer. Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. Journal of Computational Physics, 14(4):361–370, 1974. doi:10.1016/0021-9991(74)90019-9.
5. ^ B. van Leer. Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow. Journal of Computational Physics, 23(3):263–275, 1977. doi:10.1016/0021-9991(77)90094-8.
6. ^ B. van Leer. Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection. Journal of Computational Physics, 23(3):276–299, 1977. doi: 10.1016/0021-9991(77)90095-X.
7. ^ B. van Leer. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. Journal of Computational Physics, 32(1):101–136, 1979. doi:10.1016/0021-9991(79)90145-1.
8. ^ B. van Leer. Towards the ultimate conservative difference scheme. Journal of Computa- tional Physics, 135(2):229–248, 1997. doi:10.1006/jcph.1997.5704.
9. ^ Boris, Jay P.; Book, David L. (1973), "Flux-corrected transport. I. SHASTA, A fluid transport algorithm that works", Journal of computational physics, 11.1: 38–69, Bibcode:1973JCoPh..11...38B, doi:10.1016/0021-9991(73)90147-2
10. ^ van Leer, B. (2011), "A historical oversight: Vladimir P. Kolgan and his high-resolution scheme", Journal of Computational Physics, 230.7: 2378–2383, Bibcode:2011JCoPh.230.2378V, doi:10.1016/j.jcp.2010.12.032
11. ^ van leer, B. (1982), "Flux-vector Splitting for the Euler Equations", Lecture Notes in Physics, International Conference on Numerical Methods in Fluid dynamics, Springer, Berlin, 170: 507–512
12. ^ Anderson, W.K.; Thomas, J.L.; van Leer, B. (1985), "A comparison of finite-volume flux-vector splittings for the Euler equations", AIAA Paper
13. ^ Thomas, J.L.; Walters, R.W.; Van Leer, B.; Anderson, W.K. (1985), "Implicit flux-split schemes for the Euler-equations", AIAA Paper, 85: 1680
14. ^ Harten, A.; Lax, P.D.; van Leer, B. (1983), "Upstream Differencing and Godunov-type Schemes for Hyperbolic Conservation Laws", SIAM Rev., 25: 35–61, doi:10.1137/1025002
15. ^ van Leer, Bram (1985). "Upwind-difference methods for aerodynamic problems governed by the Euler equations". In Engquist, Bjorn E.; Osher, Stanley; Somerville, Richard C. J. Large-Scale Computations in Fluid Mechanics, Part 2 (Lectures in Applied Mathematics). pp. 327–336.
16. ^ Mulder, W.A.; van Leer, B. (1985), "Experiments with Implicit Upwind Methods for the Euler Equations", J. Comp. Phys., 59: 232–246, Bibcode:1985JCoPh..59..232M, doi:10.1016/0021-9991(85)90144-5
17. ^ van Leer, B.; Thomas, J. L.; Roe, P. L.; Newsome, R. W. (1987), "A comparison of numerical flux formulas for the Euler and Navier-Stokes equations", AIAA Paper CP-874: 36–41
18. ^ van Leer, B.; Tai, C.-H.; Powell, K. G. (1989), "Design of Optimally Smoothing Multi-Stage Schemes for the Euler Equations", AIAA Paper 89-1933-CP
19. ^ van Leer, B.; Lee, W. T.; Roe, P. L. (1991), "Characteristic Time-Stepping or Local Preconditioning for the Euler Equations", AIAA 10th Computational Fluid Dynamics Conference, AIAA Paper CP-91-1552: 260–282
20. ^ van Leer, B.; Lynn, J. (1995), "A semi-coarsened multi-grid-solver for the Euler equations with local preconditioning", 12th AIAA Computational Fluid Dynamics Conference, AIAA Paper 95-1667-CP: 242–252
21. ^ Lee, D.; van Leer, B.; Lynn, J. (1997), "A Local Navier-Stokes Preconditioner for all Mach and Cell Reynolds Numbers", 13th AIAA CFD Conference, AIAA-97-2024, Snowmass, CO
22. ^ Nishikawa, H.; van Leer, B. (2003), "Optimal Multigrid Convergence by Hyperbolic/Elliptic Splitting", Journal of Comput. Phys., 190: 52–63, doi:10.1016/s0021-9991(03)00253-5
23. ^ Levy, D. W.; Powell, K. G.; van Leer, B. (1993), "Use of a Rotated Riemann Solver for the Two-Dimensional Euler Equations", Journal of Comput. Phys., 106: 201–214, doi:10.1016/s0021-9991(83)71103-4,
24. ^ Rumsey, C. L.; van Leer, B.; Roe, P. L. (1993), "A multidimensional flux function with applications to the Euler and Navier-Stokes equations", Journal of Comput. Phys., 105 (2): 306–323, doi:10.1006/jcph.1993.1077
25. ^ Chiang, Y.-L.; van Leer, B. (1992), "Simulation of Unsteady Inviscid Flow on an Adaptively Refined Cartesian Grid", AIAA Paper 92-0443
26. ^ Depcik, C.; van Leer, B. (2003), "In Search of an Optimal Local Navier-Stokes Preconditioner", 16th AIAA Conference on Computational Fluid Dynamics, AIAA 2003-3703, Orlando, FL
27. ^ Suzuki, Y.; van Leer, B. (2005), "Application of the 10-Moment Model to MEMS Flows", AIAA Paper 2005-1398
28. ^ Suzuki, Y.; Khieu, H. L.; van Leer, B. (June 2009), "CFD by First-Order PDE's", Continuum Mechanics and Thermodynamics, 21: 445–465, doi:10.1007/s00161-009-0124-2
29. ^ Khieu, L.; van Leer, B. (2011), "Solid-boundary treatment for moment equations", 20th AIAA Computational Fluid Dynamics Conference, 3
30. ^ van Leer, B.; Nomura, S. (2005), "Discontinuous Galerkin for Diffusion", AlAA Paper 2005-5108
31. ^ van Leer, B.; Lo, M.; van Raalte, M. (2007), "A Discontinuous Galerkin Method for Diffusion-Based on Recovery", 18th AlAA Computational Fluid Dynamics Conference, AIAA Paper 2007-4083, Miami, FL.
32. ^ van Leer, B.; Lo., M. (2009), "Unification of discontinuous Galerkin methods for advection and diffusion", 19th AIAA Computational Fluid Dynamics Conference, AIAA-2009-0400, Orlando, FL
33. ^ Lo, M.; van Leer, B. (2009), "Analysis and Implementation of the Recovery-based Discontinuous Galerkin Method for Diffusion", AIAA Paper Nr. 2009-3786, San Antonio, TX
34. ^ Lo, M.; van Leer, B. (2011), "Recovery-Based Discontinuous Galerkin for Navier Stokes Viscous Terms", AIAA Paper 2011-3406, Honolulu, HI
35. ^ van Albada, G.D.; van Leer, B.; Roberts, W.W. Jr. (1982), "A Comparative Study of Computational Methods in Cosmic Gas Dynamics", Astronomy and Astrophysics, 108: 76–84
36. ^ Clauer, C.R.; Gombosi, T.I.; Dezeeuw, D.L.; Ridley, A.J.; Powell, K.G.; van Leer, B.; Stout, Q.F.; Groth, C.P.T.; Holzer, T.E. (2000), "High Performance Computer Methods Applied to Predictive Space Weather Simulations", IEEE Transactions on Plasma Science, 28: 1931–1937, doi:10.1109/27.902221
37. ^ Ullrich, P.A.; Jablonowski, C.; van Leer, B. (2010), "High-order finite-volume methods for the shallow-water equations on the sphere", J Computational Phys.
38. ^ Depcik, C.; van Leer, B.; Assanis, D. (2005), "The Numerical Simulation of Variable-Property Reacting-Gas Dynamics: New Insights and Validation", Numerical Heat Transfer . Part A: Applications, 47: 27–56, doi:10.1080/10407780490520823
39. ^ van Leer, Bram (1985). "The Development of Numerical Fluid Mechanics and Aerodynamics since the 1960's: US and Canada". In Hirschel, Ernst Heinrich; Karuse, Egon. 100 Volumes of ‘Notes on Numerical Fluid Mechanics. Springer. pp. 159–185.
40. ^ van Leer, Bram (2010). "Part 7: Introduction to Computational Fluid Dynamics". In Richard, Blockley; Shyy, Wei. Encyclopedia of Aerospace Engineering. 2. Wiley. pp. 1–14.

## Education and Training

• 1963 - Candidate Astronomy, Leiden State University
• 1966 - Doctorandus Astrophysics, Leiden State University
• 1970 - Ph.D. Astrophysics, Leiden State University, 1970
• 1970-72 - Miller Fellow Astrophysics, University of California Berkeley

## Professional Experience

• 2012–Present - Arthur B. Modine Professor Emeritus, University of Michigan
• 2007-2012 - Arthur B. Modine Professor of Engineering, University of Michigan
• 1986-2007 - Professor of Aerospace Engineering, University of Michigan
• 1982-86 - Research Leader, Delft University of Technology
• 1979-81 - Visiting Scientist, NASA Langley (ICASE)
• 1978-82 - Research Leader, Leiden Observatory
• 1970-72 - Miller Fellow Astrophysics, University of California Berkeley
• 1966-77 - Research Associate, Leiden Observatory

## Honors and Awards

• 2010 - AIAA Fluid dynamics Award
• 2007 - Arthur B. Modine Professor of Aerospace Engineering
• 2005-2009 - Senior Fellow of the University of Michigan
• 2005 - Dept. of Aerospace Engineering Service Award, Univ. of Michigan
• 2003 - Computational Mechanics Award, Japan Society of Mechanical Engineers
• 1996 - College of Engineering Research Excellence Award, Univ. of Michigan
• 1995 - AIAA Fellow
• 1992 - Public Service Group Achievement Award, NASA Langley
• 1992 - Dept. of Aerospace Engineering Research Award, Univ. of Michigan
• 1990 - Group Achievement Award, NASA Langley
• 1990 - Honorary Doctorate, Free University Brussels
• 1978 - C. J. Kok Prize, Leiden University

## Recent Publications

The following articles all relate to the discontinuous Galerkin method for diffusion equations:

• B. van Leer and S. Nomura, "Discontinuous Galerkin for diffusion," AIAA Paper 2005-5108, 2005.
• B. van Leer, M. Lo and M. van Raalte, "A Discontinuous Galerkin Method for diffusion based on recovery," AIAA paper 2007-4083, 2007.
• M. van Raalte and B. van Leer, "Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion," Communications in Computational Physics Vol. 5, pp. 683–693, 2009.
• B. van Leer and M. Lo, "Unification of Discontinuous Galerkin methods for advection and diffusion," AIAA paper 2009-0400, 2009.
• M. Lo and B. van Leer, "Analysis and implementation of the Recovery-based Discontinuous Galerkin method for diffusion," AIAA Paper 2009-3786, 2009.
• Lo, M.; van Leer, B., "Recovery-Based Discontinuous Galerkin for Navier Stokes Viscous Terms", AIAA Paper 2011-3406, 2011.