Until recently, I didn’t know much about the details of my grandfather’s career. I knew he was a mathematician, but not much else. His name was Eleazer Bromberg, although he usually went by Lazer.

Now, with a child of my own, I’ve felt the urge to excavate my family history. And in so doing, I’ve found some interesting bits and pieces.

Here, for example, is a photo of him (he’s in the unbuttoned jacket) during the delivery of the first-ever commercial mainframe computer: the UNIVAC, from Remington Rand. Pictured with him are James Rand, CEO of Remington Rand, and General Douglas MacArthur of WWII and Korean War fame, along with other notables including NYU president Henry Heald and General Leslie Groves, who had directed the Manhattan Project.

It turns out that he was around in the early days of commercial computing, and I wonder now if perhaps this explains my hobbies and interests in some ancestral sense.

I’ve come to discover that he was one of the first Secretaries of the still-thriving Association for Computing Machinery (ACM); he was head of the Mechanics branch of the Office of Naval Research; he helped with computations for Project Matterhorn (Wheeler & Spitzer’s Cold War-era fusion and hydrogen bomb research program at Princeton); and he worked at the National Bureau of Standards (now NIST) along with directing the Courant Computing Center at NYU.

He’s quoted in a 1965 New York Times story about the Courant Institute saying:

Dr. Eleazer Bromberg, assistant director, takes pride that “we accept all students who can benefit from the program.” He called the operation “an extremely advanced adult education program.” Some of the sons and daughters of women alumni are now beginning to enroll.

Dr. Bromberg stressed that, although many of the institute’s scholars do publish, nobody perishes for lack of it, and the general feeling is that a man should take his time rather than try to be prolific.

Every Saturday, under the direction of mathematician J. T. Schwartz, some 150 high school students get free computer lessons. Many, says Dr. Schwartz, come before 9 A.M. on weekday mornings to use the computers. The machines are off-limits to high school students between the hours of 9 and 5. No funds are available for this labor of love. “We put emphasis on doing things,” said Dr. Bromberg. “Whatever makes sense is done, and we hope the money follows afterwards.”

Lazer published several papers that I want to work my way through and try to understand. But so far, to my eyes, his most enduring mathematical and computational contribution seems to sit in an appendix to a 1957 technical report written at Los Alamos National Lab for the U.S. Atomic Energy Commission: The Particle-In-Cell Method for Hydrodynamic Calculations.

This report, written by Martha Evans and Francis Harlow, is arguably the true beginning of the field of computational fluid dynamics, or CFD. CFD is the science of using computers to predict how fluids like liquids and gases flow. It is used for everything from aerospace applications to computer cooling to HVAC and building design, gas turbines, windmills, nuclear reactors, blood flow modeling, and so much more.

Francis Harlow is one of the fathers of CFD, and in 2004 wrote an account of his time studying fluid dynamics at Los Alamos National Lab in the mid-20th century. (I can find little about Martha, although she published several papers with Harlow.) In Harlow’s account, he discusses how the paper came to be:

There were at that time two main ways to think about zoning space for resolving the behavior of a fluid. One of them we call the Lagrangian approach, which means having a mesh of computational cells that follow the motion of a fluid through its contortions and whatever processes that take place. … The other, the Eulerian viewpoint, has a fixed mesh of cells that stay in one place in the laboratory frame and the fluid flows through it.

Imagine you’re trying to model the behavior of two gases sloshing into each other in a container. Harlow is outlining the two ways they could do so at the time:

The first (the “Lagrangian approach”), roughly, is to computationally break the gases themselves up into “cells” and track the behavior of each of the cells. The cells each move and deform and interact physically with each other and the environment. After some sloshing, the “mesh” of cells is twisted, knotted, and unrecognizable from its initial state and hard to mathematically deal with (in some cases, impossible) — but each cell has retained its fundamental nature. No cells now contain a mixture of gases, because we’ve tracked and deformed the gas cells individually.

The second (the “Eulerian approach”) is to divide the space that the gases sit in into “cells” — imagine a 3d grid overlaid on the container. Each of these cells is perfectly sized and still, and instead of the cells moving, we monitor and calculate the properties of the cells as the gases pass through them. We get the benefit of no distortion — but now the gases are smeared across cells, and keeping track of which material is where becomes impossible.

You can see the difference in the animations below. The Lagrangian’s cells stay “pure,” but they get so bent out of shape as to become impossible physically. The Eulerian stays calculable, but you get “smeared” cells that are a combination of the two gases and thus lose track of the materials.

Lagrangian

Eulerian

Harlow goes on:

These problems with each of the computational viewpoints led us early in our work to think about some way to combine the two. Martha Evans and I started spending part of our time exploring ideas for accomplishing this amalgamation, and we developed a technique that turned out, despite a lot of initial skepticism, to be very useful. This technique we call the Particle-in-Cell (PIC) method.

PIC effectively takes the stance that there is both a Eulerian, fixed grid of cells and Lagrangian-style particles that move through those cells. They got the benefit of the core, fixed mesh that made calculations of pressures and gradients tractable, while also being able to take into account the bookkeeping of where the mass and materials actually are.

Here’s a way to visualize the PIC model.

Particle-in-Cell

This was revolutionary and highly effective. But Harlow goes on to say that the process can be viewed in two different ways (emphasis mine):

This process can be viewed in two distinctly different ways. One of them is based on the idea of time splitting, in which a computational cycle is broken into two parts, the first being an update of the quantities by all the processes except advection, while the second part moves the particles and accomplishes all of the advective fluxing.

The second way, as recognized by Eleazer Bromberg (a visitor to our Group from the Courant Institute at New York University), is to consider that the mesh of cells is actually Lagrangian, so that the first part of the calculations in a cycle advances all the variables, including the mesh coordinates and the particle positions, and the second part maps the mesh back into its original configuration, leaving the particles at their new locations.

My grandfather Lazer was visiting Los Alamos National Lab from NYU’s Courant Institute to help them with their nascent computing efforts. He is listed on the Evans & Harlow paper as one of the eight people who assisted with its “Machine Calculations.” But he is also called out as writing Appendix II in the paper, which is about exactly what Harlow highlights above.

I suspect what happened is that he was asked to help with the mechanical computations for Evans and Harlow’s approach, and while doing so, thought hard about the problem and came up with a different way of getting to the ultimate equations — hence the named-author appendix. The PIC technique is theirs, but the second way of looking at it is his.

In the core of the report, Evans and Harlow work through PIC as a sequence of phases. Each cycle begins by freezing the flow in place. On the stationary (Eulerian) grid, they let the pressure differences between neighboring cells “push” on the fluid, updating each cell’s velocity and energy, as if nothing were moving from one cell to the next. Once that’s done, they (computationally) let the fluid move, causing the particles to drift across cell boundaries. And because any boundary-crossing drags momentum and energy into a new cell, a final step reconciles the books.

This is, as Harlow described above, the breaking of the cycle into two parts: “the first being an update of the quantities by all the processes except advectionAdvection is the transport of a substance through bulk fluid motion — e.g. the movement of those mass-carrying particles in the Particle-in-Cell model., while the second part moves the particles and accomplishes all of the advective fluxing.” And in so doing, the grid doesn’t budge, so you don’t get the knotting of the Lagrangian failure mode. It is, however, not really how the fluid flows in real life, and doesn’t obviously guarantee conservation of mass, momentum, and energy.

This is one approach, and makes the method reasonably straightforward to program on those early computers.

But in Appendix II, Lazer avoids freezing anything. He lets the entire grid come “unstuck” and ride along with the fluid — for just one step, the cells themselves become Lagrangian, stretching, shearing, twisting. Everything advances at once with no forces or matter held back: velocities, energies, positions of cell walls, positions of the particles. This is closer to the real physical picture.

But then, before it becomes an issue, he “rezones” it all. He looks at where the distorted cells are, and maps them back onto the original grid, while pouring their contents (momentum, energy, mass) into those original fixed cells where they belong, conserving everything in the process. The particles are left exactly where the fluid carried them.

Simplifying in another way: the Evans & Harlow approach is to keep the imagined, fixed grid in place, compute the flow of the properties (e.g. energy) between its cells, then move the particles, then re-account for that particle movement. The Bromberg approach is to forget about the grid, let the fluid flow like it really does, then re-instantiate the grid over the new state and calculate the new values for those cells (leaving the particles wherever they landed).

Below you can see a visualization of this. Harlow & Evans calculate the forces, then move the particles; Bromberg treats the whole thing as a fluid in one step, and then rezones the grid. They arrive at the same place — it is the same method, but just a different mathematical approach to deriving the result.

Harlow & Evans

Bromberg

This arrives at the same ultimate answer, but it is in some sense more theoretically correct, useful, and intuitive. Lazer’s method inherently wires in conservation of mass, momentum, and energy by focusing on boundaries instead of points. Today we would call this a “finite-volume method” for fluid dynamics (a term coined in the 1970s).

The approach isn’t just an aesthetic preference. In the body of the report, Evans and Harlow’s multi-step update process carries a small energy bookkeeping error (δD\delta D) which monotonically increases the calculated energy above the physically intended value. It’s an artifact of the discretization — if you shrink the timestep, you can shrink the error arbitrarily — but it’s there in the calculations regardless. Lazer’s derivation sidesteps it by starting from the integral conservation laws — looking at total mass, momentum, and energy inside each cell, balanced by what crosses the cell’s faces.

In his retrospective, directly after Harlow mentions Lazer’s alternative PIC view, he notes (editors’ note mine):

Some years later, Tony Hirt and Dan Butler of our Group discovered that instead of a complete mapping back to the original mesh each cycle [what Lazer did], the mapping could be partial, leading to a technique that is called an Arbitrary–Lagrangian–Eulerian (ALE) method, which limits to either of the two viewpoints.

What I find striking is how cleanly Lazer’s Appendix II prefigures the ALE idea. His move was to let the mesh behave like a Lagrangian for the physics step, then remap back to the original Eulerian grid before the distortion becomes fatal to the math.

ALE takes that same idea and relaxes the last part. Instead of always mapping all the way back to the original grid, you are allowed to map to some other convenient grid (maybe closer to the material motion, or the laboratory frame, or maybe adaptively to keep the cells well-shaped). That freedom is the “Arbitrary” in ALE’s “Arbitrary Lagrangian-Eulerian.” So perhaps Lazer came up with just plain ol’ “LE” in his work.

That is why the appendix feels less like a footnote than a seed crystal. It has the essential ALE rhythm: move with the material and rezone. Hirt and Butler’s later contribution was to see that the rezone did not have to be total.

There are still new ALE formulations being created and used across topics like gas modeling, novel flying object design, pavement responsiveness, pure numerical-methods research and more (all of those just in the last couple years!).

ALE methods remain important because many of our hardest simulations live in the middle ground of too much deformation for a Lagrangian mesh, but also too much interface detail or moving-boundary geometry for an Eulerian one.

So in this sense, Lazer’s 1957 approach of doing Lagrangian updates and then Eulerian rezones — separating the Lagrangian physics from the computational remap, spurred by his presumed search for a more elegant formulation of Evans & Harlow’s groundbreaking fluid dynamics work — remains scientifically relevant here, nearly seven decades later.

Note that I am not claiming that all of that work is built on his appendix — I don’t know if those later scientists ever read it. But the ideas are on the same foundation regardless (and Harlow, the original progenitor, seems to think they are connected).

If I were writing a history of CFD, this would all read very differently, more prominently highlighting contributions by these many scientists and drawing harder academic lineage linkages. But this isn’t a history of CFD; it’s a blog post about my grandfather.

And with that, one last photo I discovered: my grandfather Lazer is on the left, with James Stoker (an excellent mathematician and one of the founders of Courant) and Louis Nirenberg (one of the best mathematicians of the 20th century, co-awarded the 2015 Abel Prize with Nobel Laureate John Nash). I wonder what they’re looking at.

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