1 Introduction
2 Numerical method
2.1 Formulation of the equations
2.2 The finite volume scheme
2.3 A novel WENO reconstruction in primitive variables
2.4 A local spacetime DG predictor in primitive variables
2.4.1 Description of the predictor
2.4.2 An efficient initial guess for the predictor
MUSCLCN

AdamsBashforth
 

\(\mathbb{P}_{0}\mathbb{P}_{2}\)
 1.0  0.64 
\(\mathbb{P}_{0}\mathbb{P}_{3}\)
 1.0  0.75 
\(\mathbb{P}_{0}\mathbb{P}_{4}\)
 1.0  0.72 
3 Numerical tests with the new ADERWENO finite volume scheme in primitive variables
3.1 Euler equations
3.1.1 2D isentropic vortex
2D isentropic vortex problem
 

\(\boldsymbol{N}_{\boldsymbol{x}}\)

ADERPrim

ADERCons

ADERChar

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  4.060E03    5.028E03    5.010E03    3 
120  2.359E03  2.98  2.974E03  2.88  2.968E03  2.87  
140  1.489E03  2.98  1.897E03  2.92  1.893E03  2.92  
160  9.985E04  2.99  1.281E03  2.94  1.279E03  2.94  
200  5.118E04  2.99  6.612E04  2.96  6.607E04  2.96  
\(\mathbb{P}_{0}\mathbb{P}_{3}\)
 50  2.173E03    4.427E03    5.217E03    4 
60  8.831E04  4.93  1.721E03  5.18  2.232E03  4.65  
70  4.177E04  4.85  8.138E04  4.85  1.082E03  4.69  
80  2.194E04  4.82  4.418E04  4.57  5.746E04  4.74  
100  7.537E05  4.79  1.605E04  4.53  1.938E04  4.87  
\(\mathbb{P}_{0}\mathbb{P}_{4}\)
 50  2.165E03    3.438E03    3.416E03    5 
60  6.944E04  6.23  1.507E03  4.52  1.559E03  4.30  
70  3.292E04  4.84  7.615E04  4.43  7.615E04  4.65  
80  1.724E04  4.84  4.149E04  4.55  4.148E04  4.55  
100  5.884E05  4.82  1.449E04  4.71  1.448E04  4.72 
3.1.2 Sod’s Riemann problem
ADERPrim

ADERCons

ADERChar
 

\(\mathbb{P}_{0}\mathbb{P}_{2}\)
 1.0  0.74  0.81 
\(\mathbb{P}_{0}\mathbb{P}_{3}\)
 1.0  0.74  0.80 
\(\mathbb{P}_{0}\mathbb{P}_{4}\)
 1.0  0.77  0.81 
3.1.3 Interacting blast waves
3.1.4 Double Mach reflection problem
3.2 Relativistic hydrodynamics and magnetohydrodynamics
3.2.1 RHD Riemann problems
Problem

γ

ρ

\(\boldsymbol{v}_{\boldsymbol{x}}\)

p

\(\boldsymbol{t}_{\boldsymbol{f}}\)
 

RHDRP1 
x>0  5/3  1  −0.6  10  0.4 
x ≤ 0  10  0.5  20  
RHDRP2 
x>0  5/3  10^{−3}
 0.0  1  0.4 
x ≤ 0  10^{−3}
 0.0  10^{−5}

ADERPrim

ADERCons

ADERChar
 

\(\mathbb{P}_{0}\mathbb{P}_{2}\)
 1.0  1.26  1.40 
\(\mathbb{P}_{0}\mathbb{P}_{3}\)
 1.0  1.13  1.24 
\(\mathbb{P}_{0}\mathbb{P}_{4}\)
 1.0  1.04  1.06 
3.2.2 RHD KelvinHelmholtz instability
3.2.3 RMHD Alfvén wave
2D circularly polarized Alfvén wave
 

\(\boldsymbol{N}_{\boldsymbol{x}}\)

\(\boldsymbol{L}_{\boldsymbol{1}}\)
error

\(\boldsymbol{L}_{\boldsymbol{1}}\)
order

\(\boldsymbol{L}_{\boldsymbol{2}}\)
error

\(\boldsymbol{L}_{\boldsymbol{2}}\)
order

Theor.
 
\(\mathbb{P}_{0}\mathbb{P}_{2}\)
 50  5.387E02    9.527E03    3 
60  3.123E02  2.99  5.523E03  2.99  
70  1.969E02  2.99  3.481E03  2.99  
80  1.320E02  2.99  2.334E03  2.99  
100  6.764E03  3.00  1.196E03  3.00  
\(\mathbb{P}_{0}\mathbb{P}_{3}\)
 50  2.734E04    4.888E05    4 
60  1.153E04  4.73  2.061E05  4.74  
70  5.622E05  4.66  1.004E05  4.66  
80  3.043E05  4.60  5.422E06  4.61  
100  1.108E05  4.53  1.968E06  4.54  
\(\mathbb{P}_{0}\mathbb{P}_{4}\)
 30  2.043E03    3.611E04    5 
40  4.873E04  4.98  8.615E05  4.98  
50  1.603E04  4.98  2.846E05  4.96  
60  6.491E05  4.96  1.168E05  4.88  
70  3.173E05  4.64  6.147E06  4.16 
3.2.4 RMHD Riemann problems
Problem

γ

ρ

\(\boldsymbol{(v}_{\boldsymbol{x}}\)

\(\boldsymbol{v}_{\boldsymbol{y}} \)

\(\boldsymbol{v}_{\boldsymbol{z}}\boldsymbol{)}\)

p

\(\boldsymbol{(B}_{\boldsymbol{x}} \)

\(\boldsymbol{B}_{\boldsymbol{y}} \)

\(\boldsymbol{B}_{\boldsymbol{z}}\boldsymbol{)}\)

\(\boldsymbol{t}_{\boldsymbol{f}}\)
 

RMHDRP1 
x>0  2.0  0.125  0.0  0.0  0.0  0.1  0.5  −1.0  0.0  0.4 
x ≤ 0  1.0  0.0  0.0  0.0  1.0  0.5  1.0  0.0  
RMHDRP2 
x>0  5/3  1.0  −0.45  −0.2  0.2  1.0  2.0  −0.7  0.5  0.55 
x ≤ 0  1.08  0.4  0.3  0.2  0.95  2.0  0.3  0.3 
3.2.5 RMHD rotor problem
3.3 The BaerNunziato equations
\(\boldsymbol{\rho}_{\boldsymbol{s}}\)

\(\boldsymbol{u}_{\boldsymbol{s}}\)

\(\boldsymbol{p}_{\boldsymbol{s}}\)

\(\boldsymbol{\rho}_{\boldsymbol{g}}\)

\(\boldsymbol{u}_{\boldsymbol{g}}\)

\(\boldsymbol{p}_{\boldsymbol{g}}\)

\(\boldsymbol{\phi}_{\boldsymbol{s}}\)

\(\boldsymbol{t}_{\boldsymbol{e}}\)
 

BNRP1 (Deledicque and Papalexandris 2007): \(\gamma_{s} = 1.4\), \(\pi_{s} = 0\), \(\gamma _{g} = 1.4\), \(\pi_{g} = 0\)
 
L  1.0  0.0  1.0  0.5  0.0  1.0  0.4  0.10 
R  2.0  0.0  2.0  1.5  0.0  2.0  0.8  
BNRP2 (Deledicque and Papalexandris 2007): \(\gamma_{s} = 3.0\), \(\pi_{s} = 100\), \(\gamma_{g} = 1.4\), \(\pi_{g} = 0\)
 
L  800.0  0.0  500.0  1.5  0.0  2.0  0.4  0.10 
R  1,000.0  0.0  600.0  1.0  0.0  1.0  0.3  
BNRP3 (Deledicque and Papalexandris 2007): \(\gamma_{s} = 1.4\), \(\pi_{s} = 0\), \(\gamma _{g} = 1.4\), \(\pi_{g} = 0\)
 
L  1.0  0.9  2.5  1.0  0.0  1.0  0.9  0.10 
R  1.0  0.0  1.0  1.2  1.0  2.0  0.2  
BNRP5 (Schwendeman et al. 2006): \(\gamma_{s} = 1.4\), \(\pi_{s} = 0\), \(\gamma _{g} = 1.4\), \(\pi_{g} = 0\)
 
L  1.0  0.0  1.0  0.2  0.0  0.3  0.8  0.20 
R  1.0  0.0  1.0  1.0  0.0  1.0  0.3  
BNRP6 (Andrianov and Warnecke 2004): \(\gamma_{s} = 1.4\), \(\pi_{s} = 0\), \(\gamma _{g} = 1.4\), \(\pi_{g} = 0\)
 
L  0.2068  1.4166  0.0416  0.5806  1.5833  1.375  0.1  0.10 
R  2.2263  0.9366  6.0  0.4890  −0.70138  0.986  0.2 