Research Work
Professor van Leer developed MUSCL scheme, which stands for Monotone Upstream-centered Schemes for Conservation Laws, and the term was introduced by him in a seminal paper (van Leer, 1979). In this paper he constructed the first high-order, total variation diminishing (TVD) scheme where he obtained second order spatial and temporal accuracy. It is a finite volume method that provides high accuracy numerical solutions to partial differential equations which can involve solutions that exhibit shocks, discontinuities, or steep gradients. In 1982 he published another important paper that introduced a computationally efficient alternative, based on flux-vector splitting, to the exact Riemann solver in Godunov algorithm (van Leer, 1982).
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 are, in turn, used as input to the (approximate) Riemann solver. The Riemann solver solutions are averaged and used to advance the solution in time.
Read more about this topic: Bram Van Leer
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