“The scientific paper in its orthodox form does embody a totally mistaken conception, even a travesty, of the nature of scientific thought.” – Peter Medawar

A limiter is an intervention that steps into a numerical method, examines the data, and prevents the method from doing something unphysical, bad, or dangerous. The determination is based on focusing on the monotonicity of data to produce “safe” solutions. The first limiters were ad hoc, but they gained rigor over time. This is clearest with the TVD theory by Harten. A limiter acts when the solution the method would otherwise choose is going to cause harm, i.e., oscillations. In that moment it adaptively selects a less accurate method in exchange for a safe, robust solution.

Limiters are a major change in how numerics work. We had always solved nonlinear equations, but with limiters, we suddenly had nonlinear methods, and those methods were powerful. Before limiters, you had a choice between two poor options: accurate solutions that oscillated, or crude ways of imposing a physical principle. One such crude tool is artificial viscosity. Another is the safety of low-order dissipative methods like upwinding. Limiters are tightly tied to the entropy conditions that accompany both. An entropy condition is simply a way to select the physically realizable solution and separate it from the unphysical one. Artificial viscosity can achieve this as well. The power of a limiter is that it applies dissipation only where dissipation is necessary, and “limits” it everywhere else.

I asked five different AI engines, Claude, ChatGPT, Gemini, Llama, and Grok, about the history of limiters. The quality varied, but the answers were remarkably consistent. They were also consistently wrong: each returned the same incorrect, flattened version of reality, repeated five times over. That uniform wrongness is what makes this essay necessary. I have noticed an interesting side effect with AI. When I ask a question and the response is badly wrong, I feel like I should write a blog post about it. The information feels like it should be added to the corpus of knowledge. Right now, the corpus is missing key parts of this topic.

“No scientific discovery is named after its original discoverer.” – Stephen M. Stigler

What really stands out to me is what I should make of the other questions I ask these models. Especially the ones of equal or greater complexity than this one. If I already know the models are unreliable on expert nuance, how unreliable are they on everything else? How fast does reliability fall off as a topic becomes more complex? The decline seems significant. It is a thought, and a question, that frankly has been haunting me of late, and it is a warning that should be heeded broadly across most serious work. It also seems to reflect on the validity of the broader success claimed for “foundation” models.

When I queried these models, what I mostly learned was how wide and how deep the gaps in their knowledge are. They get the basic sketch right. The details, the truth of what actually happened, are missing. All they can hand back is what was processed through the published literature. This is a problem I have come to appreciate over a career: mathematicians, scientists generally, and engineers are terrible historians. The real story is not in the regular literature. Most people never see the details, so the details never get written down.

That is exactly why this is worth doing. I want the history of how things actually happened, as opposed to how they were later cleaned up. The real story has a great deal to teach us. We should know the truth about important discoveries like limiters. In learning how they were really made, we learn how discovery itself works: what the recurring patterns are, and how a rough, half-formed idea turns into state-of-the-art technology. How ideas are exchanged and bred for new ideas. That path is never as clean or as simple as the textbooks make it look. To sanitize it this soon after it happened is a real loss.

What I want to do here is put back the personal touches and the subtle narratives that get sanded off. The formal literature has little tolerance for these essential details, so we are left with a sanitized, formalized history that loses much of its humanity. These stories matter. A great deal has already been lost in the feedback loop between AI and Wikipedia: the depth and genuine understanding that come only from deeper research, and from having lived through some of it. This happens over and over in science, and limiters are another example.

“Everything of importance has been said before by somebody who did not discover it.” – Alfred North Whitehead

Now, back to the story of limiters.

The history of limiters is one of the most important developments in numerical methods. They were built specifically for hyperbolic PDEs, equations that are central to a vast range of applications. The methods we built to solve those equations are a large part of what sets modern computational technology apart. The limiter story is, at its core, the victory of constraints over simplicity. It is also a case of the same idea appearing independently and simultaneously. The conditions were right for inventing them. Both qualities make it worth understanding in depth.

If we step back to 1970, several things are happening at once. The previous twenty-five years had produced a set of competing methods for solving hyperbolic PDEs, and all of them were bad in some way. You had either oscillatory methods with ad hoc stabilization, or dissipative methods with terrible accuracy. Godunov had proved a barrier theorem that seemed to sentence us to one or the other unacceptable option: it stated that non-oscillatory linear methods are necessarily first-order accurate. At the same time, computing power was beginning to grow and to expand beyond the government and defense labs, and computational science was moving from a niche pursuit toward the center of science. As more varied people waded into computational science, ideas flowed. Things that had been accepted needed to be improved.

In this moment, two men set out to overcome the prohibition of Godunov’s theorem. One, Jay Boris, was at the Naval Research Laboratory in the United States. The other, Bram van Leer, was a student in the Netherlands studying astrophysics. At the time they did not know each other, yet they would devise strikingly similar ways of solving the same problem. It is clear that Boris knew of Godunov’s theorem at least in part, and that van Leer knew it as well. Boris understood its limitations. Van Leer did as well and also looked to modernize Godunov’s method.

It is also worth noting what else Boris was working on at the time: the technique that has become known as the Boris fix for Alfvén waves. In magnetics, one recurring difficulty is the appearance of extremely fast Alfvén waves as their speed approaches the speed of light, which severely constrains the time-step size in some MHD codes. Boris devised a method that artificially changes the speed of light to allow more efficient computation. Some of the basic concepts behind that fix are suspiciously similar to what you get with limiters.

In the Netherlands, Bram Van Leer was trying to figure out how to combine the high accuracy of a method like Lax-Wendroff’s with the guaranteed monotonicity of Godunov. I remember being at a conference in 2004 and asking Bram how he discovered limiters. He told me that he could just see that it could be done. He could overlay the solutions from the two methods and see that there was a way to blend them. He just needed to find the recipe. The key was that he knew it could be done. The key was the inspiration. After that it was just a matter of details.

This recipe was what he discovered over the course of a series of papers and has come to be known as limiters. The most common of these are the Van Leer limiters, along with a host of numerical methods for the advection equation and ultimately a higher-order extension of the Godunov method for the Euler equations. His limiters were similar to what Boris created but different in approach. The success of Van Leer’s methodology is attributable to two things: his knowledge, inspiration, and brilliance, and some fortuitous circumstances. I’ll get into this later, but Bram had help in extending his methods to a broader community. He also made better choices for outlets.

The target audience for Van Leer’s work was astrophysics and aerodynamics. Both were essential to the success of the methodology, particularly aerodynamics, where the mathematical theories of Lax and the influence of the Courant Institute were substantial. Astrophysics was important for personal connections and for tying into the national laboratories. Jay Boris worked within the plasma physics community, which was not as fertile a place for expanding his ideas to a broader set of humans. To some extent, this is simply the luck of the draw.

What both men created would become known as limiters, and it produced the Limiter War. They waged pitched battles over credit for years, and the fact is that they both deserve it. They both did the work, and the invention was nearly simultaneous because the conditions were right. Those conditions were the ones laid out at the start of the 1970s: the expansion of computational science and the powerful computing that let it be pursued. What is less well known is that there were two further inventions of limiters, again independent, in nearly the same period.

“In science the credit goes to the man who convinces the world, not to the man to whom the idea first occurs.” – Francis Darwin

The first came from an aerodynamics engineer named Vladimir Kolgan, who worked in the Soviet Union. He developed a numerical scheme that looks very much like what we would today call a second-order ENO scheme. It used a limiter that returns the slope with the smallest magnitude. Kolgan was also producing what we would now call the first high-order Godunov method. He took the scheme Godunov introduced in the very paper that contained the barrier theorem and extended it to second order. The development and popularization of the high-order Godunov method is most often credited to Bram van Leer as one of his chief contributions, yet Kolgan got there first.

“Kolgan succumbed to lung cancer in 1978, at the age of 37; at that time the final papers by Boris’ group and by Van Leer had yet to appear.”– Bram van Leer,

Kolgan’s limiter function is quite similar to the minmod function, which is now used ubiquitously in the design of limiters. Jay Boris invented minmod, and it was so well designed that it has been reused over and over across the field. For the uninitiated, minmod is short for “minimum modulus.” It returns the argument with the smallest magnitude, provided all the arguments share the same sign. If they differ in sign, it returns zero.

The fourth inventor of limiters is Ami Harten, a student of Peter Lax. Harten developed a class of methods known as the artificial compression method. It looks at the magnitude of gradients and removes dissipation from low-order methods such as first-order schemes. The effect is to blend a second-order method with a first-order one, removing the dissipation inherent in the first-order method. Harten is best known for his papers on TVD methods, a mathematically rigorous version of high-order methods. TVD was a formalization of van Leer’s ideas, and it supplied a host of theoretical properties. It rests on four very simple linear test equations, and it carries significant limitations on the achievable order of accuracy. Nonetheless, it provided a mathematical rigor that powered the acceptance of limiters broadly.

“Discovery consists of seeing what everybody has seen and thinking what nobody has thought.” – Albert Szent-Györgyi,

As I argued in the case of the Lax equivalence theorem, the limited but rigorous theory behind TVD methods provides a firm foundation for them. The theory is narrow, but it is exactly that rigor that drives acceptance, development, and the eventual expansion of the methods to a broad range of applications. Peter Lax put much of this into order in his less well-known work on high-resolution methods, where he described how combining accuracy with nonlinear limiters yields what we now call high-resolution methods. That is the most accurate single description of the broad class of methods discussed here.

These applications are strongest in aerodynamics, but they also include a major shift in how the nuclear weapons labs chose to solve their problems. That leads to a couple of closing threads to pull.

In the late 1970s, van Leer was visited by a young scientist from Lawrence Livermore who was on sabbatical. His name was Paul Woodward. A genuine synthesis of ideas followed. Van Leer’s 1979 paper on the MUSCL method was substantially influenced by Woodward’s visit. It included a version of Godunov’s method that used a number of implementations and tricks common in the Lawrence Livermore codes, among them building an Eulerian method through a Lagrangian remap, along with the time-integration ideas Woodward had promoted.

Woodward, in turn, took van Leer’s ideas and worked them into a fine art. In collaboration with Phil Colella, he developed the piecewise parabolic method (PPM), in a sense an extension of van Leer’s Scheme V from his 1979 paper. PPM removed some of the extra variables that earlier scheme required and reconstructed them from finite-volume values. To this day, PPM remains one of the most powerful methods for solving hyperbolic PDEs.

The methods adopted most directly by the weapons labs were the ones van Leer championed, and they took hold most completely in numerical advection and remap algorithms. Van Leer’s high-order methods with limiters for the advection equation revolutionized remap and changed the codes forever. Over the span of a few years every code that did advection or remap adopted Van Leer’s ideas. His methods were so much better that adoption was immediate.

They also had an enormous impact on astrophysics, enabling a phase change in the calculations that could be attempted. This was seen most acutely in accretion disks, which cannot form naturally if you simulate with first-order methods. You need the combination of accuracy with mononticity to compute the phenomena. Once limiters were introduced, you could suddenly compute them. The degree of improvement in capability was quantum in nature.

The explanation is relatively simple. A first-order method effectively adds a linear viscosity, and at almost any reasonable resolution the flow you compute is effectively laminar. You never get the transition to high-Reynolds-number flow, to turbulence, or to other fine structures. With limiters, you remove this laminarization and recover a second-order flow that can behave like a true high-Reynolds-number turbulent flow. This effect is something Len Margolin and I wrote about in one of our more cited papers.

“We present a rationale for the success of nonoscillatory finite volume difference schemes in modelling turbulent flows without need of subgrid scale models … certain truncation terms … have physical justification, representing the modifications to the governing equations that arise when one considers the motion of finite volumes of fluid over finite intervals of time.” – Len G. Margolin and William J. Rider,

The way I explain it is this: you need a second-order solver to get these structures, but you also need to keep that solver well behaved. Limiters are the means to do both, and that is why they produced a phase transition in the kinds of methods we use to solve hyperbolic PDEs. Now today there are a host of limiters and related methods available. For the most part, all of them are capable of doing amazing things compared to classical methods. We await the next revolution in methods. What this might be is a subject of debate. Many feel like the field is fully cooked. Nothing more is needed to improve. To me this is a patently absurd point-of-view.

“A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” – Max Planck

References

Boris, Jay P. “A fluid transport algorithm that works(Development of explicit, Eulerian finite-difference algorithm for solving continuity equation).” (1971).

Boris, Jay P. A physically motivated solution of the Alfvén problem. No. NRLMR2167. 1970.

Boris, Jay P., and David L. Book. “Flux-Corrected Transport. I. SHASTA, A Fluid Transport Algorithm That Works.” Journal of Computational Physics 11, no. 1 (1973): 38–69.

Boris, Jay P., and David L. Book. “Flux-Corrected Transport. III. Minimal-Error FCT Algorithms.” Journal of Computational Physics 20, no. 4 (1976): 397–431.

Colella, Phillip, and Paul R. Woodward. “The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations.” Journal of Computational Physics 54, no. 1 (1984): 174–201.

Godunov, Sergei K. “A Difference Scheme for Numerical Computation of Discontinuous Solutions of the Equations of Fluid Dynamics.” Matematicheskii Sbornik 47 (89), no. 3 (1959): 271–306.

Harten, Ami. “The Artificial Compression Method for Computation of Shocks and Contact Discontinuities. I. Single Conservation Laws.” Communications on Pure and Applied Mathematics 30, no. 5 (1977): 611–638.

Harten, Ami. “High Resolution Schemes for Hyperbolic Conservation Laws.” Journal of Computational Physics 49, no. 3 (1983): 357–393.

Kolgan, Vladimir P. “Application of the Principle of Minimizing the Derivative to the Construction of Finite-Difference Schemes for Computing Discontinuous Solutions of Gas Dynamics.” Uchenye Zapiski TsAGI 3, no. 6 (1972): 68–77. Translated and reprinted in Journal of Computational Physics 230, no. 7 (2011): 2384–2390.

Lax, Peter D. “Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation.” Communications on Pure and Applied Mathematics 7, no. 1 (1954): 159–193.

Lax, Peter D., and Burton Wendroff. “Systems of Conservation Laws.” Communications on Pure and Applied Mathematics 13, no. 2 (1960): 217–237.

Lax, Peter D. “Accuracy and resolution in the computation of solutions of linear and nonlinear equations.” In Recent advances in numerical analysis, pp. 107-117. Academic Press, 1978.

Margolin, Len G., and William J. Rider. “A Rationale for Implicit Turbulence Modelling.” International Journal for Numerical Methods in Fluids 39, no. 9 (2002): 821–841.

Van Leer, Bram. “Towards the Ultimate Conservative Difference Scheme. I. The Quest of Monotonicity.” In Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, edited by Henri Cabannes and Roger Temam, 163–168. Lecture Notes in Physics 18. Berlin: Springer-Verlag, 1973.

Van Leer, Bram. “Towards the Ultimate Conservative Difference Scheme. II. Monotonicity and Conservation Combined in a Second-Order Scheme.” Journal of Computational Physics 14, no. 4 (1974): 361–370.

Van Leer, Bram. “Towards the Ultimate Conservative Difference Scheme. IV. A New Approach to Numerical Convection.” Journal of Computational Physics 23, no. 3 (1977): 276–299.

Van Leer, Bram. “Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov’s Method.” Journal of Computational Physics 32, no. 1 (1979): 101–136.

Van Leer, Bram. “Upwind and High-Resolution Methods for Compressible Flow: From Donor Cell to Residual-Distribution Schemes.” Communications in Computational Physics 1, no. 2 (2006): 192–206.

Van Leer, Bram. “A Historical Oversight: Vladimir P. Kolgan and His High-Resolution Scheme.” Journal of Computational Physics 230, no. 7 (2011): 2378–2383.

Woodward, Paul, and Phillip Colella. “The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks.” Journal of Computational Physics 54, no. 1 (1984): 115–173.

Woodward, Paul R. “Trade-offs in designing explicit hydrodynamical schemes for vector computers.” In Parallel computations, pp. 153-171. Academic Press, 1982.

Woodward, Paul R. “Piecewise-parabolic methods for astrophysical fluid dynamics.” In Astrophysical Radiation Hydrodynamics, pp. 245-326. Dordrecht: Springer Netherlands, 1986.