# 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 Monotone Upstream-centered Schemes for Conservation Laws, 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, such as the Kurganov and Tadmor scheme outlined below.

Read more about MUSCL Scheme:  Linear Reconstruction, Kurganov and Tadmor Central Scheme, Piecewise Parabolic Reconstruction, Example: 1D Euler Equations

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MUSCL Scheme - Example: 1D Euler Equations
... With the edge fluxes known, we can now construct the semi-discrete scheme, i.e The solution can now proceed by integration using standard numerical techniques ... The above illustrates the basic idea of the MUSCL scheme ... for a practical solution to the Euler equations, a suitable scheme (such as the above KT scheme), also has to be chosen in order to define the ...

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Your scheme must be the framework of the universe; all other schemes will soon be ruins.
Henry David Thoreau (1817–1862)