Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables
- Olindo Zanotti^{1}Email author and
- Michael Dumbser^{1}
https://doi.org/10.1186/s40668-015-0014-x
© Zanotti and Dumbser 2016
Received: 2 November 2015
Accepted: 19 December 2015
Published: 13 January 2016
Abstract
We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. To obtain a conservative method, the underlying finite volume scheme is still written in terms of the cell averages of the conserved quantities. Therefore, our new approach performs the spatial WENO reconstruction twice: the first WENO reconstruction is carried out on the known cell averages of the conservative variables. The WENO polynomials are then used at the cell centers to compute point values of the conserved variables, which are subsequently converted into point values of the primitive variables. This is the only place where the conversion from conservative to primitive variables is needed in the new scheme. Then, a second WENO reconstruction is performed on the point values of the primitive variables to obtain piecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials are subsequently evolved in time with a novel space-time finite element predictor that is directly applied to the governing PDE written in primitive form. The resulting space-time polynomials of the primitive variables can then be directly used as input for the numerical fluxes at the cell boundaries in the underlying conservative finite volume scheme. Hence, the number of necessary conversions from the conserved to the primitive variables is reduced to just one single conversion at each cell center. We have verified the validity of the new approach over a wide range of hyperbolic systems, including the classical Euler equations of gas dynamics, the special relativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD) equations, as well as the Baer-Nunziato model for compressible two-phase flows. In all cases we have noticed that the new ADER schemes provide less oscillatory solutions when compared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for the RMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the overall accuracy is improved and the CPU time is reduced by about 25 %. Because of its increased accuracy and due to the reduced computational cost, we recommend to use this version of ADER as the standard one in the relativistic framework. At the end of the paper, the new approach has also been extended to ADER-DG schemes on space-time adaptive grids (AMR).
Keywords
1 Introduction
Since their introduction by Toro and Titarev (Toro et al. 2001; Titarev and Toro 2002; Toro and Titarev 2002; Titarev and Toro 2005; Toro and Titarev 2006), ADER (arbitrary high order derivatives) schemes for hyperbolic partial differential equations (PDE) have been improved and developed along different directions. A key feature of these methods is their ability to achieve uniformly high order of accuracy in space and time in a single step, without the need of intermediate Runge-Kutta stages (Pareschi et al. 2005; Pidatella et al. 2015), by exploiting the approximate solution of a Generalized Riemann Problem (GRP) at cell boundaries. ADER schemes have been first conceived within the finite volume (FV) framework, but they were soon extended also to the discontinuous Galerkin (DG) finite element framework (Dumbser and Munz 2006; Taube et al. 2007) and to a unified formulation of FV and DG schemes, namely the so-called \(\mathbb{P}_{N}\mathbb{P}_{M}\) approach (Dumbser et al. 2008a). In the original ADER approach by Toro and Titarev, the approximate solution of the GRP is obtained through the solution of a conventional Riemann problem between the boundary-extrapolated values, and a sequence of linearized Riemann problems for the spatial derivatives. The required time derivatives in the GRP are obtained via the so-called Cauchy-Kowalevski procedure, which consists in replacing the time derivatives of the Taylor expansion at each interface with spatial derivatives of appropriate order, by resorting to the strong differential form of the PDE. Such an approach, though formally elegant, becomes prohibitive or even impossible as the complexity of the equations increases, especially for multidimensional problems and for relativistic hydrodynamics and magneto-hydrodynamics. On the contrary, in the modern reformulation of ADER (Dumbser et al. 2008b; Dumbser et al. 2008a; Balsara et al. 2013), the approximate solution of the GRP is achieved by first evolving the data locally inside each cell through a local space-time discontinuous Galerkin predictor (LSDG) step that is based on a weak form of the PDE, and, second, by solving a sequence of classical Riemann problems along the time axis at each element interface. This approach has the additional benefit that it can successfully cope with stiff source terms in the equations, a fact which is often encountered in physical applications. For these reasons, ADER schemes have been applied to real physical problems mostly in their modern version. Notable examples of applications include the study of Navier-Stokes equations, with or without chemical reactions (Hidalgo and Dumbser 2011; Dumbser 2010), geophysical flows (Dumbser et al. 2009), complex three-dimensional free surface flows (Dumbser 2013), relativistic magnetic reconnection (Dumbser and Zanotti 2009; Zanotti and Dumbser 2011), and the study of the Richtmyer-Meshkov instability in the relativistic regime (Zanotti and Dumbser 2015). In the last few years, ADER schemes have been enriched with several additional properties, reaching a high level of flexibility. First of all, ADER schemes have been soon extended to deal with non-conservative systems of hyperbolic PDE (Toro and Hidalgo 2009; Dumbser et al. 2009; Dumbser et al. 2014), by resorting to path-conservative methods (Parés and Castro 2004; Pares 2006). ADER schemes have also been extended to the Lagrangian framework, in which they are currently applied to the solution of multidimensional problems on unstructured meshes for various systems of equations, (Boscheri and Dumbser 2013; Dumbser and Boscheri 2013; Boscheri et al. 2014a; Boscheri et al. 2014b; Boscheri and Dumbser 2014). On another side, ADER schemes have been combined with Adaptive Mesh Refinement (AMR) techniques (Dumbser et al. 2013; Zanotti and Dumbser 2015), exploiting the local properties of the discontinuous Galerkin predictor step, which is applied cell-by-cell irrespective of the level of refinement of the neighbour cells. Moreover, ADER schemes have also been used in combination with discontinuous Galerkin methods, even in the presence of shock waves and other discontinuities within the flow, thanks to a novel a posteriori sub-cell finite volume limiter technique based on the MOOD approach (Clain et al. 2011; Diot et al. 2012), that is designed to stabilize the discrete solution wherever the DG approach fails and produces spurious oscillations or negative densities and pressures (Dumbser et al. 2014; Zanotti et al. 2015a; Zanotti et al. 2015b).
The various implementations of ADER schemes mentioned so far differ under several aspects, but they all share the following common features: they apply the local space-time discontinuous Galerkin predictor to the conserved variables, which in turn implies that, if a WENO finite volume scheme is used, the spatial WENO reconstruction is also performed in terms of the conserved variables. Although this may be regarded as a reasonable choice, it has two fundamental drawbacks. The first one has to do with the fact that, as shown by Munz (1986), the reconstruction in conserved variables provides the worst shock capturing fidelity when compared to the reconstruction performed either in primitive or in characteristic variables. The second drawback is instead related to computational performance. Since the computation of the numerical fluxes requires the calculation of integrals via Gaussian quadrature, the physical fluxes must necessarily be computed at each space-time Gauss-Legendre quadrature point. However, there are systems of equations (e.g. the relativistic hydrodynamics or magnetohydrodynamics equations) for which the physical fluxes can only be written in terms of the primitive variables. As a result, a conversion from the conserved to the primitive variables is necessary for the calculations of the fluxes, and this operation, which is never analytic for such systems of equations, is rather expensive. For these reasons it would be very desirable to have an ADER scheme in which both the reconstruction and the subsequent local space-time discontinuous Galerkin predictor are performed in primitive variables. It is the aim of the present paper to explore this possibility. It is also worth stressing that in the context of high order finite difference Godunov methods, based on traditional Runge-Kutta discretization in time, the reconstruction in primitive variables has been proved to be very successful by Del Zanna et al. (2007) in their ECHO general relativistic code (see also Bucciantini and Del Zanna 2011; Zanotti et al. 2011). In spite of the obvious differences among the numerical schemes adopted, the approach that we propose here and the ECHO-approach share the common feature of requiring a single (per cell) conversion from the conserved to the primitive variables.
The plan of the paper is the following: in Section 2 we describe the numerical method, with particular emphasis on Section 2.3 and on Section 2.4, where the spatial reconstruction strategy and the local space-time discontinuous Galerkin predictor in primitive variable are described. The results of our new approach are presented in Section 3 for a set of four different systems of equations. In Section 4 we show that the new strategy can also be extended to pure discontinuous Galerkin schemes, even in the presence of space-time adaptive meshes (AMR). Finally, Section 5 is devoted to the conclusions of the work.
2 Numerical method
We present our new approach for purely regular Cartesian meshes, although there is no conceptual reason preventing the extension to general curvilinear or unstructured meshes, which may be considered in future studies.
2.1 Formulation of the equations
2.2 The finite volume scheme
2.3 A novel WENO reconstruction in primitive variables
- 1.
We perform a first standard spatial WENO reconstruction of the conserved variables starting from the cell averages \(\bar{\mathbf{Q} }_{ijk}^{n}\). This allows to obtain a reconstructed polynomial \(\mathbf{w} _{h}(x,y,z,t^{n})\) in conserved variables valid within each cell.
- 2.
Since \(\mathbf{w}_{h}(x,y,z,t^{n})\) is defined at any point inside the cell, we simply evaluate it at the cell center in order to obtain the point value \(\mathbf{Q}_{ijk}^{n}= \mathbf{w}_{h}(x_{i},y_{j},z_{k},t^{n})\). This conversion from cell averages \(\bar{\mathbf{Q}}_{ijk}^{n}\) to point values \(\mathbf{Q}_{ijk}^{n}\) is the main key idea of our new method, since the simple identity \(\mathbf{Q}_{ijk}^{n} = \bar{\mathbf{Q}}_{ijk}^{n}\) is valid only up to second order of accuracy! After that, we perform a conversion from the point-values of the conserved variables to the point-values in primitive variables, i.e. we apply Eq. (5), thus obtaining the corresponding primitive variables \(\mathbf{V}_{ijk}^{n} = \mathbf{V} (\mathbf{Q}_{ijk}^{n})\) at each cell center. This is the only step in the entire algorithm that needs a conversion from the conservative to the primitive variables.
- 3.
Finally, from the point-values of the primitive variables at the cell centers, we perform a second WENO reconstruction to obtain a reconstruction polynomial in primitive variables, denoted as \(\mathbf{p}_{h}(x,y,z,t^{n})\). This polynomial is then used as the initial condition for the new local space-time DG predictor in primitive variables described in Section 2.4.
2.4 A local space-time DG predictor in primitive variables
2.4.1 Description of the predictor
As already remarked, the computation of the fluxes through the integrals (11)-(13) is more conveniently performed if the primitive variables are available at each space-time quadrature point. In such a case, in fact, no conversion from the conserved to the primitive variables is required. According to the discussion of the previous Section, it is possible to obtain a polynomial \(\mathbf{p}_{h}(x,y,z,t^{n})\) in primitive variables at the reference time \(t^{n}\). This is however not enough for a high accurate computation of the numerical fluxes, and \(\mathbf{p}_{h}(x,y,z,t^{n})\) must be evolved in time, locally for each cell, in order to obtain a polynomial \(\mathbf{v}_{h}(x,y,z,t)\) approximating the solution at any time in the range \([t^{n};t^{n+1}]\).
2.4.2 An efficient initial guess for the predictor
CPU time comparison among different versions of the initial guesses for the LSDG predictor
MUSCL-CN | Adams-Bashforth | |
---|---|---|
\(\mathbb{P}_{0}\mathbb{P}_{2}\) | 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 ADER-WENO finite volume scheme in primitive variables
In the following we explore the properties of the new ADER-WENO finite volume scheme by solving a wide set of test problems belonging to four different systems of equations: the classical Euler equations, the relativistic hydrodynamics (RHD) and magnetohydrodynamics (RMHD) equations and the Baer-Nunziato equations for compressible two-phase flows. For the sake of clarity, we introduce the notation ‘ADER-Prim’ to refer to the novel approach of this work for which both the spatial WENO reconstruction and the subsequent LSDG predictor are performed on the primitive variables. On the contrary, we denote the traditional ADER implementation, for which both the spatial WENO reconstruction and the LSDG predictor are performed on the conserved variables, as ‘ADER-Cons’. In a few circumstances, we have also compared with the ‘ADER-Char’ scheme, namely a traditional ADER scheme in which, however, the spatial reconstruction is performed on the characteristic variables. In this Section we focus our attention on finite volume schemes, which, according to the notation introduced in Dumbser et al. (2008a), are denoted as \(\mathbb{P}_{0}\mathbb{P}_{M}\) methods, where M is the degree of the approximating polynomial. In Section 4 a brief account is given to discontinuous Galerkin methods, referred to as \(\mathbb{P}_{N}\mathbb{P}_{N}\) methods, for which an ADER-Prim version is also possible.
3.1 Euler equations
3.1.1 2D isentropic vortex
\(\pmb{L_{2}}\) 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. | ||||
\(\boldsymbol{L}_{\boldsymbol{2}}\) 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 |
In addition to the convergence properties, we have compared the performances of the Adams-Bashforth version of the initial guess for the LSDG predictor with the traditional version based on the MUSCL-CN algorithm. The comparison has been performed over a \(100\times100\) uniform grid. The results are shown in Table 1, from which we conclude that the Adams-Bashforth initial guess is indeed computationally more efficient in terms of CPU time. However, we have also experienced that it is typically less robust, and in some of the most challenging numerical tests discussed in the rest of the paper we had to use the more traditional MUSCL-CN initial guess.
3.1.2 Sod’s Riemann problem
CPU time comparison among different ADER implementations for the Sod Riemann problem
ADER-Prim | ADER-Cons | ADER-Char | |
---|---|---|---|
\(\mathbb{P}_{0}\mathbb{P}_{2}\) | 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
As a tentative conclusion about the performances of ADER-Prim for the Euler equations, we may say that, although it is the most accurate on smooth solutions (see Table 2), and comparable to a traditional ADER with reconstruction in characteristic variables, it is computationally more expensive than ADER-Cons and ADER-Char. Hence, ADER-Prim will rarely become the preferred choice in standard applications for the Euler equations.
3.2 Relativistic hydrodynamics and magnetohydrodynamics
The components of the electric and of the magnetic field in the laboratory frame are denoted by \(E_{i}\) and \(B_{i}\), while the Lorentz factor of the fluid with respect to this reference frame is \(W=(1-v^{2})^{-1/2}\). We emphasize that the electric field does not need to be evolved in time under the assumption of infinite electrical conductivity, since it can always be computed in terms of the velocity and of the magnetic field as \(\vec{E} = - \vec{v} \times\vec{B}\).
Although formally very similar to the classical gas dynamics equations, their relativistic counterpart present two fundamental differences. The first one is that, while the physical fluxes \(\mathbf{f}^{i}\) of the classical gas dynamics equations can be written analytically in terms of the conserved variables, i.e. \(\mathbf{f}^{i}=\mathbf{f}^{i}(\mathbf{Q})\), those of the relativistic hydrodynamics (or magnetohydrodynamics) equations need the knowledge of the primitive variables, i.e. \(\mathbf{f}^{i}=\mathbf{f}^{i}(\mathbf{V})\) for RMHD. The second difference is that, in the relativistic case, the conversion from the conserved to the primitive variables, i.e. the operation \((D,S_{j},U,B_{j})\rightarrow(\rho,v_{i},p,B_{i})\), is not analytic, and it must be performed numerically through some appropriate iterative procedure. Since in an ADER scheme such a conversion must be performed in each space-time degree of freedom of the space-time DG predictor and at each Gaussian quadrature point for the computation of the fluxes in the finite volume scheme, we may expect a significant computational advantage by performing the WENO reconstruction and the LSDG predictor directly on the primitive variables. In this way, in fact, the conversion \((D,S_{j},U,B_{j})\rightarrow(\rho,v_{i},p,B_{i})\) is required only once at the cell center (see Section 2.3), and not in each space-time degree of freedom of the predictor and at each Gaussian point for the quadrature of the numerical fluxes. We emphasize that the choice of the variables to reconstruct for the relativistic velocity is still a matter of debate. The velocity \(v_{i}\) may seem the most natural one, but, as first noticed by Komissarov (1999), reconstructing \(W v_{i}\) can increase the robustness of the scheme. However, this is not always the case (see Section 3.2.5 below) and in our tests we have favored either the first or the second choice according to convenience. Concerning the specific strategy adopted to recover the primitive variables, in our numerical code we have used the third method reported in Section 3.2 of Del Zanna et al. (2007). Alternative methods can be found in Noble et al. (2006), Rezzolla and Zanotti (2013).
In the following, we first limit our attention to a few physical systems for which \(B_{i}=E_{i}=0\), hence to relativistic hydrodynamics, and then we consider truly magnetohydrodynamics tests with \(B_{i}\neq0\).
3.2.1 RHD Riemann problems
Left and right states of the one-dimensional RHD Riemann problems
Problem | γ | ρ | \(\boldsymbol{v}_{\boldsymbol{x}}\) | p | \(\boldsymbol{t}_{\boldsymbol{f}}\) | |
---|---|---|---|---|---|---|
RHD-RP1 | x>0 | 5/3 | 1 | −0.6 | 10 | 0.4 |
x ≤ 0 | 10 | 0.5 | 20 | |||
RHD-RP2 | x>0 | 5/3 | 10^{−3} | 0.0 | 1 | 0.4 |
x ≤ 0 | 10^{−3} | 0.0 | 10^{−5} |
In the first Riemann problem, which was also analyzed by Mignone and Bodo (2005), two rarefaction waves are produced, separated by a contact discontinuity. It has been solved through a fourth order \(\mathbb{P}_{0}\mathbb{P}_{3}\) scheme, using the Rusanov Riemann solver over a uniform grid with 300 cells. As it is clear from Figure 4, the ADER-Prim scheme performs significantly better than the ADER-Cons. In particular, the overshoot and undershoot at the tail of the right rarefaction is absent. In general, the results obtained with ADER-Prim are essentially equivalent to those of ADER-Char, namely when the reconstruction in characteristic variables is adopted. This is manifest after looking at the bottom right panel of Figure 4, where a magnification of the rest mass density at the contact discontinuity is shown. Additional interesting comparisons can be made about the second Riemann problem, which can be found in Radice and Rezzolla (2012), and which is displayed in Figure 5. In this case a third order \(\mathbb {P}_{0}\mathbb{P}_{2}\) scheme has been used, again with the Rusanov Riemann solver over a uniform grid with 500 cells. The right propagating shock has a strong jump in the rest mass density, as it is visible from the bottom right panel of the figure, and the position of the shock front is better captured by the two schemes ADER-Prim and ADER-Char.
CPU time comparison among different ADER implementations for the RHD-RP1 problem
ADER-Prim | ADER-Cons | ADER-Char | |
---|---|---|---|
\(\mathbb{P}_{0}\mathbb{P}_{2}\) | 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 Kelvin-Helmholtz instability
3.2.3 RMHD Alfvén wave
\(\pmb{L_{1}}\) and \(\pmb{L_{2}}\) errors analysis for the 2D Alfvén wave problem
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 | 5.387E-02 | - | 9.527E-03 | - | 3 |
60 | 3.123E-02 | 2.99 | 5.523E-03 | 2.99 | ||
70 | 1.969E-02 | 2.99 | 3.481E-03 | 2.99 | ||
80 | 1.320E-02 | 2.99 | 2.334E-03 | 2.99 | ||
100 | 6.764E-03 | 3.00 | 1.196E-03 | 3.00 | ||
\(\mathbb{P}_{0}\mathbb{P}_{3}\) | 50 | 2.734E-04 | - | 4.888E-05 | - | 4 |
60 | 1.153E-04 | 4.73 | 2.061E-05 | 4.74 | ||
70 | 5.622E-05 | 4.66 | 1.004E-05 | 4.66 | ||
80 | 3.043E-05 | 4.60 | 5.422E-06 | 4.61 | ||
100 | 1.108E-05 | 4.53 | 1.968E-06 | 4.54 | ||
\(\mathbb{P}_{0}\mathbb{P}_{4}\) | 30 | 2.043E-03 | - | 3.611E-04 | - | 5 |
40 | 4.873E-04 | 4.98 | 8.615E-05 | 4.98 | ||
50 | 1.603E-04 | 4.98 | 2.846E-05 | 4.96 | ||
60 | 6.491E-05 | 4.96 | 1.168E-05 | 4.88 | ||
70 | 3.173E-05 | 4.64 | 6.147E-06 | 4.16 |
3.2.4 RMHD Riemann problems
Left and right states of the one-dimensional RMHD Riemann problems
Problem | γ | ρ | \(\boldsymbol{(v}_{\boldsymbol{x}}\) | \(\boldsymbol{v}_{\boldsymbol{y}} \) | \(\boldsymbol{v}_{\boldsymbol{z}}\boldsymbol{)}\) | p | \(\boldsymbol{(B}_{\boldsymbol{x}} \) | \(\boldsymbol{B}_{\boldsymbol{y}} \) | \(\boldsymbol{B}_{\boldsymbol{z}}\boldsymbol{)}\) | \(\boldsymbol{t}_{\boldsymbol{f}}\) | |
---|---|---|---|---|---|---|---|---|---|---|---|
RMHD-RP1 | 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 | |||
RMHD-RP2 | 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 Baer-Nunziato equations
Initial states left (L) and right (R) for the Riemann problems for the Baer-Nunziato 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 |
4 Extension to discontinuous Galerkin and adaptive mesh refinement
The resulting ADER-DG scheme in primitive variables can be combined with spacetime adaptive mesh refinement (AMR), in such a way to resolve the smallest details of the solution in highly complex flows. We refer to Zanotti et al. (2015b), Zanotti et al. (2015a) for a full account of our AMR solver in the context of ADER-DG schemes. Here we want to show three representative test cases of the ability of the new ADER-Prim-DG scheme with adaptive mesh refinement, by considering the cylindrical expansion of a blast wave in a plasma with an initially uniform magnetic field (see also Komissarov 1999; Leismann et al. 2005; Del Zanna et al. 2007; Dumbser and Zanotti 2009), as well as the shock problems of Leblanc, Sedov (1959) and Noh (1987).
4.1 RMHD blast wave problem
At time \(t=0\), the rest-mass density and the pressure are \(\rho=0.01\) and \(p=1\), respectively, within a cylinder of radius \(R=1.0\), while outside the cylinder \(\rho=10^{-4}\) and \(p=5\times10^{-4}\). Moreover, there is a constant magnetic field \(B_{0}\) along the x-direction and the plasma is at rest, while a smooth ramp function between \(r=0.8\) and \(r=1\) modulates the initial jump between inner and outer values, similarly to Komissarov (1999) and Del Zanna et al. (2007).
4.2 Leblanc, Sedov and Noh problem
5 Conclusions
The new version of ADER schemes introduced in Dumbser et al. (2008b) relies on a local space-time discontinuous Galerkin predictor, which is then used for the computation of high order accurate fluxes and sources. This approach has the advantage over classical Cauchy-Kovalewski based ADER schemes (Toro et al. 2001; Titarev and Toro 2002; Toro and Titarev 2002; Titarev and Toro 2005; Toro and Titarev 2006; Dumbser and Munz 2006; Taube et al. 2007) that it is in principle applicable to general nonlinear systems of conservation laws. However, for hyperbolic systems in which the conversion from conservative to primitive variables is not analytic but only available numerically, a large number of such expensive conversions must be performed, namely one for each space-time quadrature point for the integration of the numerical fluxes over the element interfaces and one for each space-time degree of freedom in the local space-time DG predictor.
Motivated by this limitation, we have designed a new version of ADER schemes, valid primarily for finite volume schemes but extendible also to the discontinuous Galerkin finite element framework, in which both the spatial WENO reconstruction and the subsequent local space-time DG predictor act on the primitive variables. In the finite volume context this can be done by performing a double WENO reconstruction for each cell. In the first WENO step, piece-wise polynomials of the conserved variables are computed from the cell averages in the usual way. Then, these reconstruction polynomials are simply evaluated in the cell centers, in order to obtain point values of the conserved variables. After that, a single conversion from the conserved to the primitive variable is needed in each cell. Finally, a second WENO reconstruction acts on these point values and provides piece-wise polynomials of the primitive variables. The local space-time discontinuous Galerkin predictor must then be reformulated in a non-conservative fashion, supplying the time evolution of the reconstructed polynomials for the primitive variables.
For all systems of equations that we have explored, classical Euler, relativistic hydrodynamics (RHD) and magnetohydrodynamics (RMHD) and the Baer-Nunziato equations, we have noticed a significant reduction of spurious oscillations provided by the new reconstruction in primitive variables with respect to traditional reconstruction in conserved variables. This effect is particularly evident for the Baer-Nunziato equations. In the relativistic regime, there is also an improvement in the ability of capturing the position of shock waves (see Figure 5). To a large extent, the new primitive formulation provides results that are comparable to reconstruction in characteristic variables.
Moreover, for systems of equations in which the conversion from the conserved to the primitive variables cannot be obtained in closed form, such as for the RHD and RMHD equations, there is an advantage in terms of computational efficiency, with reductions of the CPU time around ∼20 %, or more. We have also introduced an additional improvement, namely the implementation of a new initial guess for the LSDG predictor, which is based on an extrapolation in time, similar to Adams-Bashforth-type ODE integrators. This new initial guess is typically faster than those traditionally available, but it is also less robust in the presence of strong shocks.
We predict that the new version of ADER based on primitive variables will become the standard ADER scheme in the relativistic framework. This may become particularly advantageous for high energy astrophysics, in which both high accuracy and high computational efficiency are required.
Since we adopt Cartesian coordinates, \(\mathbf{ f}^{x}(\mathbf{Q})\), \(\mathbf{f}^{y}(\mathbf{Q})\), \(\mathbf{f}^{z}(\mathbf{Q})\) express the fluxes along the x, y and z directions, respectively.
We note that, since the spacetime is flat and we are using Cartesian coordinates, the covariant and the contravariant components of spatial vectors can be used interchangeably, namely \(A_{i}=A^{i}\), for the generic vector A⃗.
Declarations
Acknowledgements
The research presented in this paper was financed by (i) the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013) with the research project STiMulUs, ERC Grant agreement no. 278267 and (ii) it has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement no. 671698 (call FETHPC-1-2014, project ExaHyPE). We are grateful to Bruno Giacomazzo and Luciano Rezzolla for providing the numerical code for the exact solution of the Riemann problem in RMHD. We would also like to acknowledge PRACE for awarding access to the SuperMUC supercomputer based in Munich (Germany) at the Leibniz Rechenzentrum (LRZ), and ISCRA, for awarding access to the FERMI supercomputer based in Casalecchio (Italy).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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