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. Author manuscript; available in PMC: 2015 Oct 1.

Published in final edited form as: J Comput Phys. 2014 Oct 1;274:95–121. doi: 10.1016/j.jcp.2014.06.003

Plots of the density at t = 0, ρ₀, on the coarsest nonuniform grid, 25 × 25 cells (left) and p-norm of the error between the density at t = 0 and t = 1, ρ₁, as a function of the grid size, N × N (right), for the stationary isentropic vortex problem given by Eq. (37). The grid convergence of the L₁- and L_∞-norm is shown for both when the primitive variables are reconstructed at the cell boundaries from their second-order accurate cell averages (– – and – –, respectively), following Johnsen and Colonius [30], and their fourth-order accurate cell averages (– – and – –, respectively), following the present scheme. Reference slopes for the second-order (—) and fifth-order (—) convergence rates are also included.

Inline graphic — Plots of the density at t = 0, ρ₀, on the coarsest nonuniform grid, 25 × 25 cells (left) and p-norm of the error between the density at t = 0 and t = 1, ρ₁, as a function of the grid size, N × N (right), for the stationary isentropic vortex problem given by Eq. (37). The grid convergence of the L₁- and L_∞-norm is shown for both when the primitive variables are reconstructed at the cell boundaries from their second-order accurate cell averages (– – and – –, respectively), following Johnsen and Colonius [30], and their fourth-order accurate cell averages (– – and – –, respectively), following the present scheme. Reference slopes for the second-order (—) and fifth-order (—) convergence rates are also included.