Bram van Leer

Personal interests

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 was the formulation of the upwind numerical flux function for a hyperbolic system of conservation laws: F up = 1 2 ( F j + F j + 1 ) − 1 2 | A | j + 1 2 ( u j + 1 − u j ) .

Here the matrix | A | appears for the first time in CFD, defined as the matrix that has the same eigenvectors as the flux Jacobian A , but the corresponding eigenvalues are the moduli of those of A . The subscript j + 1 2 indicates a representative or average value on the interval ( 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) via scalar conservative non-oscillatory (part II) to conservative non-oscillatory Euler (part III). 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 (finite-volume scalar) and, finally, Part V (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.

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 and by V.P. Kolgan, a Russian researcher unknown in the West. In 2011 Van Leer devoted an article to Kolgan's contributions 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 and the development of the block-structured codes CFL2D and CFL3D which still are heavily used. Other seminal papers from these years are the review of upwind methods with Harten and Lax, the AMS workshop paper detailing the differences and resemblances between upwind fluxes and Jameson's flux formula, and the conference paper with Mulder 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,” 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. 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. 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. The same strategy for the Navier Stokes discretization was pursued by D. Lee.

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.

While the multi-grid project was going on, van Leer worked on two more subjects: multi-dimensional Riemann solvers, and time-dependent adaptive Cartesian grid. After conclusion of the multi-grid project, van Leer continued to work on local preconditioning of the Navier-Stokes equations together with C. Depcik. 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. 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. 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) 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. 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:

Three significant review papers by van Leer are:

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.

Education and training

Professional experience

Honors and awards

Recent publications

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