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 Upstream-centered 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.
Linear reconstruction
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,
u t + F x ( u ) = 0. {\displaystyle u_{t}+F_{x}\left(u\right)=0.\,}
Where u {\displaystyle u} represents a state variable and F {\displaystyle F} represents a flux variable. The basic scheme of Godunov uses piecewise constant approximations for each cell, and results in a first-order upwind discretisation of the above problem with cell centres indexed as i {\displaystyle i} . A semi-discrete scheme can be defined as follows,
d u i d t + 1 Δ x i [ F ( u i ) − F ( u i − 1 ) ] = 0. {\displaystyle {\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}+{\frac {1}{\Delta x_{i}}}\left[F\left(u_{i}\right)-F\left(u_{i-1}\right)\right]=0.}
This basic scheme is not able to handle shocks or sharp discontinuities as they tend to become smeared. An example of this effect is shown in the diagram opposite, which illustrates a 1D advective equation with a step wave propagating to the right. The simulation was carried out with a mesh of 200 cells and used a 4th order Runge–Kutta time integrator (RK4). To provide higher resolution of discontinuities, Godunov's scheme can be extended to use piecewise linear approximations of each cell, which results in a central difference scheme that is second-order accurate in space. The piecewise linear approximations are obtained from
u ( x ) = u i + ( x − x i ) ( x i + 1 − x i ) ( u i + 1 − u i ) ∀ x ∈ ( x i , x i + 1 ] . {\displaystyle u\left(x\right)=u_{i}+{\frac {\left(x-x_{i}\right)}{\left(x_{i+1}-x_{i}\right)}}\left(u_{i+1}-u_{i}\right)\qquad \forall x\in (x_{i},x_{i+1}].}
Thus, evaluating fluxes at the cell edges we get the following semi-discrete scheme
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