EngquistOsherscheman

Engquist–Osher scheme, written here as EngquistOsherscheman, refers to a monotone numerical flux used in finite-volume methods for solving hyperbolic conservation laws. It was developed by Björn Engquist and Stanley Osher in the early 1980s as a robust approach to approximate solutions of nonlinear hyperbolic equations, particularly in the presence of shocks and discontinuities.

Conceptually, the Engquist–Osher flux splits the flux function into positive and negative contributions based on the sign of characteristic speeds. For a scalar conservation law ∂t u + ∂x f(u) = 0, the numerical flux F between left and right states uL and uR is given by a decomposition that integrates the positive part of the derivative f′ and the negative part separately. A common mathematical form is F(uL, uR) = ∫0^{uL} max(f′(s), 0) ds + ∫0^{uR} min(f′(s), 0) ds. This construction yields a conservative, monotone, and entropy-satisfying flux, which helps prevent non-physical oscillations near discontinuities.

Applications and extensions: The Engquist–Osher scheme is primarily applied to one-dimensional scalar conservation laws and serves as a foundational flux in more complex finite-volume frameworks. It can be extended to systems of equations and multidimensional problems through dimensional splitting or unsplit multidimensional formulations. In practice, it is valued for its robustness and simplicity, though it can be more diffusive than some modern Riemann-solvers for certain problems.

Relationship to other methods: It is one of several monotone, upwind-type fluxes used in computational fluid dynamics, often contrasted with Roe, HLL, and Lax–Friedrichs schemes. It provides a reliable baseline method especially when exact Riemann solvers are impractical.

See also: hyperbolic conservation laws; numerical flux; finite-volume method; Engquist–Osher flux.