Elsevier

Journal of Computational Physics

Volume 304, 1 January 2016, Pages 138-161
Journal of Computational Physics

A two-dimensional Riemann solver with self-similar sub-structure – Alternative formulation based on least squares projection

https://doi.org/10.1016/j.jcp.2015.10.013Get rights and content

Abstract

Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional Riemann solver is also similarly improved. Such multidimensional Riemann problems arise when multiple states come together at the vertex of a mesh. The interaction of the resulting one-dimensional Riemann problems gives rise to a strongly-interacting state. We wish to endow this strongly-interacting state with physically-motivated sub-structure. The self-similar formulation of Balsara [16] proves especially useful for this purpose. While that work is based on a Galerkin projection, in this paper we present an analogous self-similar formulation that is based on a different interpretation. In the present formulation, we interpret the shock jumps at the boundary of the strongly-interacting state quite literally. The enforcement of the shock jump conditions is done with a least squares projection (Vides, Nkonga and Audit [67]). With that interpretation, we again show that the multidimensional Riemann solver can be endowed with sub-structure. However, we find that the most efficient implementation arises when we use a flux vector splitting and a least squares projection. An alternative formulation that is based on the full characteristic matrices is also presented. The multidimensional Riemann solvers that are demonstrated here use one-dimensional HLLC Riemann solvers as building blocks.

Several stringent test problems drawn from hydrodynamics and MHD are presented to show that the method works. Results from structured and unstructured meshes demonstrate the versatility of our method. The reader is also invited to watch a video introduction to multidimensional Riemann solvers on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.

Introduction

Riemann solvers play an important role in the numerical solution of hyperbolic systems of conservation laws. The one-dimensional Riemann problem is a self-similar solution that results from a discontinuity between two constant states. Multidimensional Riemann solvers have also been designed and we focus on a certain class of multidimensional Riemann solvers here (Wendroff [68], Balsara [4], [5], [16], Balsara, Dumbser and Abgrall [15], Vides, Nkonga and Audit [67], Balsara and Dumbser [17]). Such Riemann solvers are applied at the vertices of a two-dimensional mesh. Many states come together at a vertex from different directions, making it possible to communicate the multidimensionality of the flow to the multidimensional Riemann solver. At the vertex, the job of the multidimensional Riemann solver is to approximate the self-similar multidimensional structure that emanates from the vertex. While self-similarity has not been used much in the design of one-dimensional Riemann solvers, it is crucially important in the development of multidimensional Riemann solvers (Balsara [16], Balsara and Dumbser [17]). This has prompted the name of MuSIC Riemann solvers, where MuSIC stands for “Multidimensional, Self-similar, strongly-Interacting, Consistent”. Such Riemann solvers are multidimensional; they draw on the self-similarity of the problem; they focus on the strongly-interacting state that results when multiple one-dimensional Riemann solvers interact; and the design relies on establishing consistency with the conservation law. MuSIC Riemann solvers that rely on a Galerkin projection to obtain the self-similar variation in the strongly interacting state have been presented (Balsara [16], Balsara and Dumbser [17]). An alternative projection method consists of least squares and Vides, Nkonga and Audit [67] developed a multidimensional Riemann solver without sub-structure based on such a projection. The goal of this paper is to show that least squares projection can also be used to design a MuSIC Riemann that retains sub-structure.

Several excellent one-dimensional Riemann solvers have been designed. There are exact Riemann solvers from Godunov [41], [42] and van Leer [66] and two-shock approximations thereof (Colella [27], Colella and Woodward [29]). See also the work of Chorin [25]. The linearized Riemann solver by Roe [52] and the HLL/HLLE/HLLEM Riemann solvers (Harten, Lax and van Leer [44], Einfeldt [34], Einfeldt et al. [35]) and the local Lax–Friedrichs (LLF) Riemann solver (Rusanov [56]) have also seen frequent use. Toro, Spruce and Speares [62], [63], [64], Chakraborty and Toro [24] and Batten et al. [20] produced an HLLC class of Riemann solvers which have become very popular. See also, Billett and Toro [21]. Osher and Solomon [51] and Dumbser and Toro [33] presented approximate Riemann solvers based on path integral methods in phase space. In Balsara [16] we showed that the principle of self-similarity can be used to advantage with the result that any of the above-mentioned one-dimensional Riemann solvers can be used as a building block in the design of multidimensional Riemann solvers by relying on a Galerkin projection. The present paper continues this line of inquiry by showing that a least squares projection can also be used. The results are instantiated for the very popular HLLC class of Riemann solvers.

Magnetohydrodynamics (MHD) is an interesting example of a hyperbolic system with a more complex wave foliation. One-dimensional linearized Riemann solvers for numerical MHD have been designed (Roe and Balsara [54], Cargo and Gallice [23], Balsara [6]). HLLC Riemann solvers, capable of capturing mesh-aligned contact discontinuities, have been presented by Gurski [43] and Li [47]. Miyoshi and Kusano [49] drew on Gurski's work to design an HLLD Riemann solver for MHD. It is, therefore, interesting to show that MHD can also be accommodated within our formulation. MHD is a system with an involution constraint, where the divergence of the magnetic field is always zero. Balsara and Spicer [7] showed that this is assured within the context of a higher order Godunov scheme by using the upwinded fluxes at the edges of the mesh to update the magnetic fields that are collocated at the faces of a mesh. Gardiner and Stone [38], [39] have claimed that the dissipation in those upwinded fluxes needs to be doubled all the time in order to stabilize the method. A substantial body of work now exists to show that the suggestion of Gardiner and Stone is completely unnecessary when multidimensional Riemann solvers are used to provide a properly upwinded electric field at the edges of the mesh (Balsara [5], Vides, Nkonga and Audit [67], Balsara and Dumbser [18]). Indiscriminate doubling of the dissipation, as per Gardiner and Stone's suggestion, can indeed lead to excessive dissipation of the magnetic field in the direction that is transverse to the upwind direction. The present paper reinforces that finding.

For the sake of completeness, and also for the sake of putting this work in context, we mention that there have been prior efforts at designing multidimensional Riemann solvers. One strain of research consists of trying to build some level of multidimensionality into one dimensional Riemann solvers (Colella [28], Saltzman [57], LeVeque [46]). Another line of early effort tried to incorporate genuine multidimensionality and did not meet with much initial success (Roe [53], Rumsey, van Leer and Roe [55]). Abgrall [1], [2] made a big breakthrough by formulating a genuinely multidimensional Riemann solver for CFD that worked. Further advances were also reported (Fey [36], [37], Gilquin, Laurens and Rosier [40], Brio, Zakharian and Webb [22], Lukacsova-Medvidova et al. [48]). Most of these above-mentioned genuinely multidimensional Riemann solvers did not see much use because they were difficult to implement. Wendroff [68] formulated a two-dimensional HLL Riemann solver, but his method was also not easy to implement. A video introduction to multidimensional Riemann solvers is available on the following website: http://www.nd.edu/~dbalsara/Numerical-PDE-Course.

Balsara [4] devised a two-dimensional HLL Riemann solver with simple closed form expressions for the fluxes that were easy to implement. In Balsara [5] it was shown that one can impart sub-structure to the HLL state, yielding a multidimensional HLLC Riemann solver. Balsara, Dumbser and Abgrall [15] extended this formulation to accommodate unstructured meshes. The previous three papers formulated the multidimensional Riemann problem by integrating the conservation law over the extent of the wave model in space–time. In their study of the multidimensional Riemann problem, Schulz-Rinne, Collins and Glaz [58] had shown that the one-dimensional Riemann problems interact amongst themselves to form a self-similarly evolving strongly-interacting state. This strongly-interacting state emerges by propagating into the one-dimensional Riemann problems via an evolving boundary. We refer to this boundary as the boundary of the multidimensional wave model because it contains the strongly-interacting state. The wave models in all the multidimensional Riemann solvers incorporate this self-similarity. But there is a deeper way in which self-similarity can be used, as shown in the next paragraph.

The self-similarly evolving strongly-interacting state is an inevitable consequence of wavefronts propagating into the one-dimensional Riemann problems. Seizing on this insight, Balsara [16] presented a self-similar formulation of the multidimensional Riemann problem. Balsara and Dumbser [17] extended these ideas to unstructured meshes. In the first of those two papers, a Galerkin projection method was devised which had the pleasant consequence of deriving most of its information about the sub-structure in the strongly interacting state via boundary integrals applied to the self-similarly expanding multidimensional wave model. In that fashion, the Galerkin projection method picks up on the physical idea that Lagrangian fluxes carry mass, momentum and energy through the moving boundary of the multidimensional wave model. That is, the mathematical formulation reproduces the physics of the problem. The correspondence between the space–time formulation of the multidimensional Riemann problem and the analogous formulation in similarity variables has also been shown in Balsara [16].

An alternative viewpoint was presented by Vides, Nkonga and Audit [67] with a least squares projection method that also required the balancing of Lagrangian fluxes across the moving boundary of the multidimensional wave model. This can be viewed as a way to enforce shock jumps across the boundaries of the wave model. (Because of the least squares procedure, the enforcement of shock jumps is never quite exact. Instead it should be viewed as an approximate imposition of shock jumps. However, the integration that takes place in a Galerkin projection can also be viewed as an approximation process.) The resulting multidimensional Riemann solver by Vides, Nkonga and Audit [67] was an HLL-type Riemann solver and did not retain sub-structure. In this paper we show how the least squares projection can also be used to endow substructure to the multidimensional Riemann problem. Thus by having two complementary viewpoints for designing MuSIC Riemann solvers with sub-structure, via Galerkin projection and via least squares projection, we have a better perspective on the design of multidimensional Riemann solvers.

Section 2 sets up the problem and provides details associated with the construction of the multidimensional wave model. Section 3 provides details about the least squares projection and how it works within the context of a self-similar formulation where shock jumps are explicitly enforced at the boundaries of the wave model. Section 4 presents accuracy analysis and Section 5 presents several stringent test problems. Section 6 presents our conclusions. Proofs of the least squares projection have been catalogued in Appendix A, Appendix B.

Section snippets

Problem setup and construction of the multidimensional wave model

Consider a hyperbolic conservation law in two dimensions, tU+xF+yG=0, where U is an N-component vector of conserved variables and F and G are the fluxes in the x- and y-directions. At any one-dimensional boundary between zones, a one-dimensional Riemann problem is likely to arise. Fig. 1 shows such a one-dimensional boundary along with a one-dimensional Riemann problem that develops between states U1 and U2. At any vertex of the mesh, where the multiple zones come together, the

Formulation in similarity variables

The strongly interacting state evolves self-similarly, leading to a self-similar formulation (Balsara [16], Balsara and Dumbser [17]). As shown in Fig. 2, in the subsonic case, it overlies the vertex of the mesh and will contribute to the numerical fluxes at that vertex. This is the situation that occurs most frequently in most applications and we refer to it as the subsonic case for the multidimensional Riemann solver. The majority of attention is lavished upon the subsonic case when designing

Accuracy analysis

In the next section we demonstrate the versatility of the MuSIC Riemann solver with least squares projection that we have presented in this paper. This section is devoted to accuracy analysis for Euler and MHD flow. The next section is devoted to several stringent tests drawn from hydrodynamics and MHD. Our results show our codes running with several different orders of accuracy. The MuSIC Riemann solver presented here can also accommodate structured and unstructured meshes. This is

Hydrodynamical test: Sod and Lax problems on a two-dimensional mesh

The Sod and Lax problems are very well-known. We set the problems up on a two dimensional mesh and ran them using a third order ADER-WENO scheme with a CFL of 0.95. Fig. 6(a) shows the density from the Sod shock test problem along with the mesh structure, while Fig. 6(b) plots the density in one dimension along with the exact solution of the Riemann problem. Fig. 6(c) shows the density from the Lax shock test problem along with the mesh structure, while Fig. 6(d) plots the density in one

Conclusions

This paper follows in the footsteps of the self-similar formulation of the multidimensional Riemann problem (Balsara [4], [5], [16], Balsara, Dumbser & Abgrall [15], Balsara and Dumbser [17]). The methods are predicated on the continuity of Lagrangian fluxes across the self-similarly moving boundary of the multidimensional wave model. Just as the quality of a one-dimensional approximate Riemann solver is improved by the inclusion of internal sub-structure, the quality of a multidimensional

Acknowledgements

DSB acknowledges support via NSF grants NSF-AST-1009091, NSF-ACI-1307369, NSF-ACI-1533850 and NSF-DMS-1361197. DSB also acknowledges support via NASA grants from the Fermi program as well as NASA-NNX 12A088G. KFG acknowledges support via NSF-DMS-1361197 and #245237 from the Simons Foundation. BN, EA, and JV acknowledge the financial support from the National French Research Program (ANR): ANEMOS (2011), ANR-11-MONU-002. MD was funded by the European Research Council (ERC) under the European

References (70)

  • D.S. Balsara et al.

    Multidimensional HLL and HLLC Riemann solvers for unstructured meshes – with application to Euler and MHD flows

    J. Comput. Phys.

    (2014)
  • D.S. Balsara

    Multidimensional Riemann problem with self-similar internal structure – Part I – application to hyperbolic conservation laws on structured meshes

    J. Comput. Phys.

    (2014)
  • D.S. Balsara et al.

    Multidimensional Riemann problem with self-similar internal structure – Part II – application to hyperbolic conservation laws on unstructured meshes

    J. Comput. Phys.

    (2015)
  • D.S. Balsara et al.

    Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers

    J. Comput. Phys.

    (2015)
  • S.J. Billett et al.

    On WAF-type schemes for multidimensional hyperbolic conservation laws

    J. Comput. Phys.

    (1997)
  • M. Brio et al.

    Two-dimensional Riemann solver for Euler equations of gas dynamics

    J. Comput. Phys.

    (2001)
  • P. Cargo et al.

    Roe matrices for ideal MHD and systematic construction of Roe matrices for systems of conservation laws

    J. Comput. Phys.

    (1997)
  • A.J. Chorin

    Random choice solutions of hyperbolic systems

    J. Comput. Phys.

    (1976)
  • B. Cockburn et al.

    The Runge–Kutta discontinuous Galerkin method for Conservation Laws V

    J. Comput. Phys.

    (1998)
  • P. Colella

    Multidimensional Upwind methods for hyperbolic conservation laws

    J. Comput. Phys.

    (1990)
  • P. Colella et al.

    The piecewise parabolic method (PPM) for gas-dynamical simulations

    J. Comput. Phys.

    (1984)
  • A. Dedner et al.

    Hyperbolic divergence cleaning for MHD equations

    J. Comput. Phys.

    (2002)
  • M. Dumbser et al.

    Arbitary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems

    J. Comput. Phys.

    (2007)
  • M. Dumbser et al.

    A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

    J. Comput. Phys.

    (2008)
  • B. Einfeldt et al.

    On Godunov-type methods near low densities

    J. Comput. Phys.

    (1991)
  • M. Fey

    Multidimensional upwinding 1. The method of transport for solving the Euler equations

    J. Comput. Phys.

    (1998)
  • M. Fey

    Multidimensional upwinding 2. Decomposition of the Euler equation into advection equation

    J. Comput. Phys.

    (1998)
  • T. Gardiner et al.

    An unsplit Godunov method for ideal MHD via constrained transport

    J. Comput. Phys.

    (2005)
  • T. Gardiner et al.

    An unsplit Godunov method for ideal MHD via constrained transport in three dimensions

    J. Comput. Phys.

    (2008)
  • G.-S. Jiang et al.

    Efficient implementation of weighted ENO schemes

    J. Comput. Phys.

    (1996)
  • R.J. LeVeque

    Wave propagation algorithms for multidimensional hyperbolic systems

    J. Comput. Phys.

    (1997)
  • S.-T. Li

    An HLLC Riemann solver for magnetohydrodynamics

    J. Comput. Phys.

    (2005)
  • T. Miyoshi et al.

    A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics

    J. Comput. Phys.

    (2005)
  • P.L. Roe

    Approximate Riemann solver, parameter vectors and difference schemes

    J. Comput. Phys.

    (1981)
  • P.L. Roe

    Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics

    J. Comput. Phys.

    (1986)
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