MUSCL scheme
In the study of partial differential equations, the MUSCL scheme is a finite volume method that can provide highly accurate numerical solutions for a given system, even in cases where the solutions exhibit shocks, discontinuities, or large gradients. MUSCL stands for Monotonic Upwind Scheme for Conservation Laws (van Leer, 1979), and the term was introduced in a seminal paper by Bram van Leer (van Leer, 1979). In this paper he constructed the first high-order, total variation diminishing (TVD) scheme where he obtained second order spatial accuracy.
The idea is to replace the piecewise constant approximation of Godunov's scheme by reconstructed states, derived from cell-averaged states obtained from the previous time-step. For each cell, slope limited, reconstructed left and right states are obtained and used to calculate fluxes at the cell boundaries (edges). These fluxes can, in turn, be used as input to a Riemann solver, following which the solutions are averaged and used to advance the solution in time. Alternatively, the fluxes can be used in Riemann-solver-free schemes, which are basically Rusanov-like schemes.
We will consider the fundamentals of the MUSCL scheme by considering the following simple first-order, scalar, 1D system, which is assumed to have a wave propagating in the positive direction,
Where represents a state variable and represents a flux variable.
...
Wikipedia