Bram van Leer

Bram van Leer is Arthur B. Modine Emeritus Professor of aerospace engineering at the University of Michigan, in Ann Arbor.

An appraisal of his early work has been given by C. Hirsch (1979)[1] An astrophysicist by education, van Leer made lasting contributions to CFD in his five-part article series “Towards the Ultimate Conservative Difference Scheme (1972-1979),” where he extended Godunov's finite-volume scheme to the second order (MUSCL).

Also in the series, he developed non-oscillatory 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 was the world's first and only CJ (carillon-jockey) based on the North Campus carillon, live streaming from the Lurie Tower.

A flute composition by van Leer was performed twice in 1997 by University of Michigan Professor Leone Buyse.

Bram van Leer was a doctoral student in astrophysics at Leiden Observatory (1966–1970) when he got interested in Computational Fluid Dynamics (CFD) for the sake of solving cosmic flow problems.

His first major result in CFD[3] was the formulation of the upwind numerical flux function for a hyperbolic system of conservation laws:

This paper[9] was reprinted in 1997 in the 30th anniversary issue of Journal Computational Physics with an introduction by Charles Hirsch.

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[10] and by V.P.

This led to van Leer's differentiable flux-vector splitting[12] and the development of the block-structured codes CFL2D and CFL3D [13][14] which still are heavily used.

In 1988, he embarked on a very large project, to achieve steady Euler solutions in O(N) operations by a purely explicit methodology.

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.

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 the University of Michigan at the time, to finish the project.

[23] While the multi-grid project was going on, van Leer worked on two more subjects: multi-dimensional Riemann solvers,[24][25] and time-dependent adaptive Cartesian grid.

[26] After conclusion of the multigrid project, van Leer continued to work on local preconditioning of the Navier-Stokes equations together with C.

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.

Starting in 2004, the recovery-based DG (RDG)[31] 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 non-linear diffusion equations that may or may not contain shear terms.

[32][33][34][35] On unstructured grids, the RDG was predicted to achieve the order of accuracy of 2p+2; this research unfortunately was not completed before van Leer retired.

Van Leer's early work, especially the series “Towards the ultimate conservative difference scheme” motivated by the needs of astrophysical modeling, has influenced a wide range of other disciplines; such interdisciplinary knowledge transfer is not self-evident.

The transition of ideas between disciplines through publications is a much slower process, as most researchers do not read journals based in fields other than their own expertise.

A case in point is the way Van Leer's ideas, contained in the series "Towards the Ultimate Conservative Difference Scheme," made their way into Atmospheric General Circulation Modeling (GCM).

Rood[38] from NASA's Goddard Space Flight Center published a comprehensive review of publications on advection schemes in 1987 that Van Leer's articles were unlocked to the GCM community.

Lin and Rood,[40] both at NASA Goddard, published a predictor-corrector version of the second-order Godunov method for use in atmospheric dynamics and implemented it in a shallow-water model.

On this occasion, van Leer presented a plenary lecture titled, “History of CFD Part II,” which covers the period from 1970 to 1995.

van Leer at the piano at Pierpont Commons, University of Michigan
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.