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Table 2 \(\pmb{L_{2}}\) errors of the mass density and corresponding convergence rates for the 2D isentropic vortex problem

From: Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables

2D isentropic vortex problem
  \(\boldsymbol{N}_{\boldsymbol{x}}\) ADER-Prim ADER-Cons ADER-Char Theor.
\(\boldsymbol{L}_{\boldsymbol{2}}\) error \(\boldsymbol{L}_{\boldsymbol{2}}\) order \(\boldsymbol{L}_{\boldsymbol{2}}\) error \(\boldsymbol{L}_{\boldsymbol{2}}\) order \(\boldsymbol{L}_{\boldsymbol{2}}\) error \(\boldsymbol{L}_{\boldsymbol{2}}\) order
\(\mathbb{P}_{0}\mathbb{P}_{2}\) 100 4.060E-03 - 5.028E-03 - 5.010E-03 - 3
120 2.359E-03 2.98 2.974E-03 2.88 2.968E-03 2.87
140 1.489E-03 2.98 1.897E-03 2.92 1.893E-03 2.92
160 9.985E-04 2.99 1.281E-03 2.94 1.279E-03 2.94
200 5.118E-04 2.99 6.612E-04 2.96 6.607E-04 2.96
\(\mathbb{P}_{0}\mathbb{P}_{3}\) 50 2.173E-03 - 4.427E-03 - 5.217E-03 - 4
60 8.831E-04 4.93 1.721E-03 5.18 2.232E-03 4.65
70 4.177E-04 4.85 8.138E-04 4.85 1.082E-03 4.69
80 2.194E-04 4.82 4.418E-04 4.57 5.746E-04 4.74
100 7.537E-05 4.79 1.605E-04 4.53 1.938E-04 4.87
\(\mathbb{P}_{0}\mathbb{P}_{4}\) 50 2.165E-03 - 3.438E-03 - 3.416E-03 - 5
60 6.944E-04 6.23 1.507E-03 4.52 1.559E-03 4.30
70 3.292E-04 4.84 7.615E-04 4.43 7.615E-04 4.65
80 1.724E-04 4.84 4.149E-04 4.55 4.148E-04 4.55
100 5.884E-05 4.82 1.449E-04 4.71 1.448E-04 4.72
  1. A comparison is shown among the reconstruction in primitive variables (ADER-Prim), in conserved variables (ADER-Cons) and in characteristic variables (ADER-Char). The Osher-type numerical flux has been used.