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

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

2D isentropic vortex problem
	\(\boldsymbol{N}_{\boldsymbol{x}}\)	ADER-Prim		ADER-Cons		ADER-Char		Theor.
	\(\boldsymbol{N}_{\boldsymbol{x}}\)	\(\boldsymbol{L}_{\boldsymbol{2}}\) 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	Theor.
\(\mathbb{P}_{0}\mathbb{P}_{2}\)	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

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.