A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems
Introduction
Riemann solvers are an important building block of modern numerical schemes for hyperbolic systems. For hyperbolic systems in conservation form, a large number of Riemann solvers (RS) are available. For exact Riemann solvers, see Godunov [51] and van Leer [94]. Approximate RS based on the two-shock formulation are presented in Colella [29], Colella & Woodward [30], Chorin [28]. In [76] Roe presented a very popular Riemann solver that is based on a special linearization of the nonlinear system of governing PDE. It was later reformulated by Toumi in [93] using a weak integral form, which also allows for an extension to non-conservative hyperbolic systems and which can be seen as a predecessor of the family of path-conservative schemes introduced by Parés and Castro in the seminal papers [70] and [25]. Riemann solvers whose numerical dissipation term is based on a path integral in phase space were first proposed by Osher and Solomon in [69] and have been recently generalized by Dumbser & Toro to general nonlinear hyperbolic conservation laws and to non-conservative hyperbolic systems in [45], [44]. In this context, we also would like to point out the recent reformulation of the Osher-type solver [45] in the context of polynomial viscosity methods (PVM) [34] by Castro et al., see [24]. The HLL/HLLE/HLLEM/HLLC Riemann solver (Harten, Lax & van Leer [57], Einfeldt [46], Einfeldt et al. [47], Toro, Spruce and Speares [88], Batten et al. [16], Billett & Toro [19]) have gained considerable popularity owing to their simplicity. The local Lax–Friedrichs (LLF) method (Rusanov [78]) is even simpler, but often carries the penalty of a high level of numerical dissipation, as well as centered fluxes, like the FORCE method of Toro and Billet [91].
Out of all these methods, the HLLEM Riemann solver of Einfeldt [46] and Einfeldt et al. [47] has seen the least further development and extensions to more general nonlinear systems of conservation laws. Yet, it has some very desirable features that have gone unappreciated. Like HLLC, it can be built on top of an existing HLL Riemann solver, thereby ensuring that it has a positivity preserving property and that it is entropy enforcing. Only the intermediate eigenvalues and the associated left and right eigenvectors need to be evaluated in order to implement an HLLEM RS. Since these are usually easier to compute analytically, the HLLEM RS has the additional advantage over the Riemann solvers of Roe [76] and Osher [69], [45], [44] that it is not necessary to know the entire eigenstructure of the hyperbolic system. It nevertheless produces solutions that are competitive with Roe-type, Osher-type and HLLC/HLLD-type RS. It furthermore does not require an entropy fix, unlike the original Riemann solver of Roe [76]. A first goal of this paper is therefore to bring these features to the forefront by showing that the HLLEM RS accommodates very well to general nonlinear hyperbolic conservation laws, for example the MHD system or the Euler system of compressible gas dynamics with general equation of state (EOS), or the Godunov–Romenski model for nonlinear elasticity [52], [53], [54].
Extensive recent progress has been made on path-conservative methods for treating non-conservative hyperbolic systems, see for example the following list of references, which does not pretend to be complete: Parés et al. [70], [66], Castro et al. [25], [49], [23], [27], [21], [22], Morales de Luna et al. [32], Dumbser et al. [37], [40], [44]. The progress has been made within the context of the DLM theory for non-conservative systems (Dal Maso, LeFloch & Murat [63]). Therefore, when a conservation form exists, the family of path-conservative schemes reproduces well-known Riemann solvers like the Roe RS, HLL, HLLC, FORCE, GFORCE, Osher, and so on. However, the optimal choice of the path that needs to be defined in order to connect the two states of the RP is still to a certain extent ambiguous. Usually, the path is chosen in such a way that other important properties are ensured, like the well-balancedness of the scheme, see [25], [67]. For a detailed discussion of open problems related to path-conservative methods, see [26], [2]. However, most physical systems of interest, like shallow water-type systems and multiphase flow models, tend to be in quasi-conservative form, where the non-conservative terms act only on linearly degenerate fields. This fact makes the choice of a particular path less delicate.
The study of one-dimensional Riemann solvers for non-conservative hyperbolic systems has, by now, reached a high level of sophistication. Even so, to the best of our knowledge, the HLLEM RS has never been adapted to non-conservative systems. The second goal of this paper is therefore to show that the HLLEM scheme can be easily adapted to hyperbolic systems with non-conservative products. For shallow water-type equations, the HLLEM RS is also shown to be well-balanced, owing to the fact that it resolves the intermediate waves associated with the bottom jump exactly.
The rest of this paper is organized as follows: Section 2 presents the new formulation of the HLLEM scheme in similarity variables. Since all Riemann problems have a self-similar structure, we show the utility of similarity variables in formulating the one-dimensional HLLEM RS by following Balsara [9], [10], [11], [12], Balsara, Dumbser & Abgrall [14] and Balsara & Dumbser [13]. In Section 3 we present computational results for a large set of different conservative and non-conservative hyperbolic systems, in particular for the single and two-layer shallow water equations; the multi-phase debris flow model of Pitman and Le [72], the Baer–Nunziato model of compressible multi-phase flows [6], the Euler equations of compressible gas dynamics with ideal and real equation of state, the magnetohydrodynamics system (MHD & RMHD) and finally the nonlinear elasticity equations according to Godunov and Romenski [52]. Section 4 contains conclusions and an outlook to future developments. Several FORTRAN sample codes are given in the appendix, to show the simplicity of the proposed algorithm and to facilitate the practical implementation of the HLLEM Riemann solver.
For the reader who is only interested in the purely conservative case, an explicit expression of the HLLEM flux is given in Eqn. (30) of Section 2.3, together with a detailed FORTRAN sample code in Appendix C.
Section snippets
Self-similar formulation of the HLLEM Riemann solver for conservative and non-conservative systems
We consider nonlinear hyperbolic systems of the form where is the state vector and is the state-space or phase-space. The conservative part of the system is contained in the nonlinear flux vector and the non-conservative terms are grouped together in the non-conservative product . The computational domain is denoted by . The above PDE can also be cast into the following alternative quasi-linear form, where the
Numerical results
All computational results shown in this section have been produced by using the path-conservative HLLEM scheme inside a standard second order TVD finite volume framework [90], based on the minmod slope limiter. The domain Ω is divided into equidistant control volumes with mesh spacing . The time step is computed according to the usual CFL condition as An explicit second order TVD finite volume discretization of PDE (1) based on the
Conclusions
This is the first time that the one-dimensional HLLEM Riemann solver has been extended to non-conservative hyperbolic systems and to general nonlinear hyperbolic systems of conservation laws. The proposed approach is surprisingly simple and universal. In the original papers of Einfeldt [46] and Einfeldt, Munz et al. [47] the HLLEM method has been developed for the Euler equations of compressible gas dynamics and the scaling factor δ has been derived only for this very particular system. In the
Acknowledgements
MD has been financed by 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.
DSB acknowledges support via NSF grants NSF-AST-1009091, NSF-ACI-1307369, NSF-DMS-1361197 and NSF-ACI-1533850. DSB also acknowledges support via NASA grants from the Fermi program as well as NASA-NNX 12A088G.
The authors would like to cordially thank the two anonymous referees for their helpful
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