Lax–Friedrichs

Lax–Friedrichs
Name	Lax–Friedrichs
Inventor	Peter Lax, Kurt Otto Friedrichs
Year	1954
Field	Numerical analysis
Application	Hyperbolic partial differential equations, conservation laws, computational fluid dynamics

Lax–Friedrichs Lax–Friedrichs is an explicit finite-difference scheme introduced by Peter Lax and Kurt Otto Friedrichs for approximating solutions of hyperbolic partial differential equations arising in fluid dynamics, aerodynamics, and wave propagation problems. Developed in the mid-20th century, the method provided an early robust tool for approximating nonlinear conservation laws and influenced later schemes such as Godunov's scheme, Rusanov scheme, and techniques used in computational fluid dynamics codes. Originating in the context of work at institutions like the Courant Institute of Mathematical Sciences and the New York University applied mathematics community, the scheme is notable for its simplicity and dissipation properties.

Introduction

The scheme was presented by Peter Lax and Kurt Otto Friedrichs in studies related to numerical solutions of hyperbolic systems and linear advection problems during the 1950s, contemporaneous with developments by John von Neumann, Richard Courant, Kurt Friedrichs's collaborators, and later researchers such as Sergei Godunov. It applies a centered spatial discretization combined with explicit temporal averaging to produce a first-order accurate, robust algorithm widely cited in texts by authors like Hermann Weyl-era analysts and modern treatises by Randall LeVeque and Chi-Wang Shu. The method became a reference point in comparisons with higher-order schemes developed at institutions such as Massachusetts Institute of Technology and Princeton University.

Formulation

The Lax–Friedrichs update for a scalar conservation law uses nodal values on a uniform grid and advances with a time step satisfying a Courant–Friedrichs–Lewy condition derived from stability analyses by Lax and colleagues. The discrete formula replaces the value at the new time level by the average of neighboring states and a centered approximation of the flux derivative, reflecting techniques seen in earlier work by Richard Courant and John von Neumann. Implementations in software packages from groups at Los Alamos National Laboratory and Sandia National Laboratories typically express the scheme in conservative form to preserve integral invariants, echoing principles used in NASA computational tools and in algorithms described by Stanford University researchers.

Numerical Properties

Lax–Friedrichs is first-order accurate in both space and time and exhibits artificial dissipation proportional to the mesh spacing and time step, a feature compared against dissipation in schemes by Rusanov and Lax–Wendroff. The scheme's numerical viscosity smooths sharp discontinuities, offering robustness similar to that seen in early Godunov-type solvers but at the cost of smeared shock profiles as observed in studies from Imperial College London and École Polytechnique. Benchmarks performed by research teams at University of Cambridge, ETH Zurich, and California Institute of Technology highlight its simplicity and predictability for linear advection and scalar Burgers-type problems.

Stability and Convergence

Stability of the Lax–Friedrichs scheme is governed by a CFL constraint analogous to conditions derived in analyses by Lax and Turing-era numerical analysts, with proofs of convergence relying on monotonicity and total variation diminishing arguments developed later by researchers such as Harten and Berman. For linear problems, spectral analyses drawing on work at Bell Labs and the Institute for Advanced Study establish conditional stability, while for nonlinear conservation laws convergence to entropy solutions connects to theoretical frameworks advanced by Oleinik and Kruzhkov. Mathematicians at Princeton University, University of Chicago, and Yale University have refined error estimates that quantify the trade-off between dissipation and accuracy.

Extensions and Variants

Variants include the multidimensional Lax–Friedrichs-flavored schemes used in meteorology models at European Centre for Medium-Range Weather Forecasts and the Rusanov flux, sometimes called local Lax–Friedrichs, developed in contexts involving researchers from Moscow State University and Lomonosov University. Hybrid approaches combine Lax–Friedrichs smoothing with higher-order reconstructions inspired by work of Godunov, ENO and WENO methods formulated by teams at Los Alamos and Princeton. Adaptive mesh refinement strategies integrating Lax–Friedrichs-inspired fluxes have been implemented in community codes from Argonne National Laboratory and Lawrence Berkeley National Laboratory.

Applications

The scheme has been applied in prototype studies of shock formation and rarefaction waves in classical problems like the Riemann problem and in practical simulations in aeronautical engineering curricula at Imperial College, MIT, and École Polytechnique Fédérale de Lausanne. It appears as a pedagogical example in courses at Columbia University, University of Oxford, and National University of Singapore and serves as a baseline in comparative studies by research groups at Duke University, Tokyo Institute of Technology, and Seoul National University. Beyond academia, Lax–Friedrichs-inspired fluxes influence production codes used by NASA, NOAA, and computational groups at Boeing and Airbus for preliminary design studies.

Category:Numerical methods