MUSCL

MUSCL stands for Monotone Upstream-Centered Schemes for Conservation Laws. It is a family of high-resolution finite-volume methods for solving hyperbolic conservation laws. The approach extends first-order Godunov-type schemes by reconstructing a linear profile within each computational cell and using this reconstruction to compute fluxes at cell interfaces with left and right states that are constrained to be monotone.

A key feature of MUSCL is the use of slope limiters to enforce the TVD (total variation diminishing) property, which helps prevent non-physical oscillations near shocks or discontinuities. The reconstruction yields second-order accuracy in space away from discontinuities, while the flux at each interface is obtained from a Riemann solver applied to the reconstructed left and right states.

For time accuracy, MUSCL is typically paired with predictor-corrector approaches or with the MUSCL-Hancock variant, which advances the solution in two half-time steps to achieve second-order temporal accuracy with a relatively simple implementation. The method can employ various approximate or exact Riemann solvers (such as Roe, HLL, HLLC, or Rusanov) to compute the interface fluxes.

History and usage: the scheme was introduced by B. van Leer in 1979 as a framework for high-resolution schemes in conservation laws. It has since become a standard tool in computational fluid dynamics for problems including compressible gas dynamics, shallow water flows, and traffic-flow models.

See also: Godunov method, finite-volume method, Riemann solver, slope limiter, TVD.