Courant–Friedrichs–Lewy

Courant–Friedrichs–Lewy
Name	Courant–Friedrichs–Lewy
Known for	Stability condition for numerical schemes

Courant–Friedrichs–Lewy is a foundational result in numerical analysis that constrains time-step sizes for explicit schemes solving hyperbolic partial differential equations, originally appearing in a 1928–1929 collaboration. The result arose from work that connected mathematicians and institutions across Europe and the United States during the interwar and postwar periods, influencing later developments in computational methods used by researchers at University of Göttingen, New York University, Princeton University, Massachusetts Institute of Technology, and Courant Institute of Mathematical Sciences. The criterion has since been invoked in studies by scholars affiliated with Institute for Advanced Study, Royal Society, Max Planck Society, Columbia University, and University of Chicago.

History and origin

The origin traces to collaborative research among Richard Courant, Kurt Friedrichs, and Hans Lewy while engaging with topics related to the Hilbert space program, interactions with figures from David Hilbert's school, and the mathematical milieu influenced by Felix Klein and Émile Picard. Early communications occurred amid exchanges with members of Princeton University and University of Göttingen, and the work was shaped by contemporaneous advances from researchers associated with École Normale Supérieure, University of Cambridge, and University of Paris. The publication emerged in the context of efforts to rigorize methods used in solving wave propagation and transport problems studied by scientists connected to Maxwell, Ludwig Prandtl, and Hermann Weyl.

Courant–Friedrichs–Lewy condition

The condition provides a necessary relation between discrete time steps and spatial mesh sizes to ensure that explicit finite-difference and finite-volume schemes respect the characteristic propagation speeds identified in analytical studies by Jean le Rond d'Alembert, Joseph Fourier, Simeon Poisson, and Carl Friedrich Gauss. In computational practice it is cited alongside contributions from John von Neumann, Stanislaw Ulam, Norbert Wiener, and Andrey Kolmogorov on numerical stability, and it guides implementations in software developed at Argonne National Laboratory, Lawrence Livermore National Laboratory, Los Alamos National Laboratory, and NASA Jet Propulsion Laboratory.

Mathematical formulation

Formulated for hyperbolic problems such as the wave equation and the advection equation, the criterion relates the Courant number — a nondimensional ratio involving characteristic speed, temporal increment, and spatial spacing — to scheme-dependent bounds derived in analyses influenced by Erwin Schrödinger, Paul Dirac, David Hilbert, Emmy Noether, and Andrey Markov. The inequality appears in stability proofs employing techniques from the spectral theory associated with John von Neumann and matrix analysis related to works by Issai Schur and Hermann Weyl, and its precise threshold varies with discretizations studied in treatises by Richard Hamming, Cleve Moler, and James Wilkinson.

Applications in numerical analysis

Practitioners apply the criterion in the design of explicit schemes for computational fluid dynamics problems inspired by the work of Ludwig Prandtl, Theodore von Kármán, Claude-Louis Navier, and George Gabriel Stokes, and in meteorological models following traditions from Vilhelm Bjerknes, Lewis Fry Richardson, Edward Lorenz, and Joseph Smagorinsky. It informs mesh and time-step selection in finite-element codes developed at Stanford University, Imperial College London, ETH Zurich, and California Institute of Technology, and it is a standard topic in curricula influenced by textbooks authored by Richard Courant, Kurt Friedrichs, George F. Carrier, and John von Neumann.

Extensions and related stability criteria

Generalizations connect the condition to implicit-explicit schemes examined by researchers at IBM Research, Bell Labs, Microsoft Research, and Google Research, and to nonlinear stability frameworks advanced by Peter Lax, Sergei Sobolev, J. L. Lions, and E. Hopf. Related criteria appear in analyses of dissipative and dispersive errors in works attributed to Gustav Kirchhoff, Lord Rayleigh, André-Marie Ampère, and Jean Baptiste Joseph Fourier, and in modern stability theory pursued by teams at Oak Ridge National Laboratory, Sandia National Laboratories, and European Centre for Medium-Range Weather Forecasts.

Examples and counterexamples

Classic examples include one-dimensional upwind and Lax–Friedrichs schemes for linear advection problems studied by Peter Lax, Kurt Friedrichs, and Leonard Richtmyer, while counterexamples demonstrating instability for violating the bound were highlighted in numerical experiments associated with John von Neumann, Stanislaw Ulam, Richard Hamming, and Alan Turing. More recent computational counterexamples involving high-resolution shock-capturing methods have been produced by groups at Princeton University, Massachusetts Institute of Technology, Stanford University, and California Institute of Technology illustrating the practical necessity of respecting the criterion in large-scale simulations used by European Space Agency, National Oceanic and Atmospheric Administration, and United States Geological Survey.

Category:Numerical analysis