Combination of Godunov's concept
(local solution of
fully non-linear Riemann solvers)
high-order reconstruction (solution averaging):
- Godunov (1959):
Exact Riemann solver (with
- Van Leer (1979):
(Monotone Upwind Schemes for Scalar Conservation Laws):
approximation of piecewise-linear Riemann problems by piecewise-constant Riemann problems
solution of the Lagrange equations,
- Colella & Woodward (1984):
(Piecewise Parabolic Method):
piecewise parabolic reconstruction
via primitive functions,
Godunov-type schemes are conceptionally appealing.
The improved high-resolution methods give excellent results.
- However, they are relatively complex and
require a lot of operations per grid cell.
- Approximate (linearized) Riemann solvers may serve as well in
splitting the flow into waves with different characteristic velocities
and upwind directions.