Convergence of the multigrid scheme for the Sod shock tube problem as computed with the ESDIRK23 discretization. The logarithm of the L₂-norm of the defect for the density field (normalized with respect to its initial value at the beginning of a multigrid iteration) is depicted as a function of the total number of F(1, 0) multigrid cycles performed. The first five (physical) time steps are shown, separated by dotted lines. Each ESDIRK23 time step contains two implicit stages. For each of these stages, an implicit system given by Eq. (47) has to be solved by the multigrid solver. These stages can be discerned by vertical jumps in the defect, which signify the start of the multigrid iteration for the next implicit stage.