Relation between PVM schemes and simple Riemann solvers

Luna, Tomás Morales de; Díáz, Manuel; Parés, Carlos

doi:10.1002/num.21871

Cited by 10 publications

(9 citation statements)

References 16 publications

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“…The evolution to the half time level needed in (32) and (33) is directly obtained from a discrete form of the PDE (1) in the small, without coupling to the neighbor elements:…”

Section: Numerical Resultsmentioning

confidence: 99%

“…First, notice that the HLLEM RS is also amenable to a high order quadrature-free formulation in ADER-WENO finite volume schemes (Dumbser et al [43]) if the signal speeds and the viscosity matrix are frozen for each element interface. Second, Morales de Luna et al [32] have recently presented a formulation for analyzing the viscosity of any Riemann solver in terms of a polynomial expansion. Given the practical thrust of this paper, we do not carry out a PVM analysis of the HLLEM RS, but we do point out that such a study is worthwhile.…”

Section: Ideal Relativistic Mhd Equations (Rmhd)mentioning

confidence: 99%

“…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]).…”

Section: Introductionmentioning

confidence: 99%

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A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems

Dumbser

Balsara

2016

Journal of Computational Physics

161

125

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In this paper a new, simple and universal formulation of the HLLEM Riemann solver (RS) is proposed that works for general conservative and non-conservative systems of hyperbolic equations. For non-conservative PDE, a pathconservative formulation of the HLLEM RS is presented for the first time in this paper. The HLLEM Riemann solver is built on top of a novel and very robust path-conservative HLL method. It thus naturally inherits the positivity properties and the entropy enforcement of the underlying HLL scheme. However, with just the slight additional cost of evaluating eigenvectors and eigenvalues of intermediate characteristic fields, we can represent linearly degenerate intermediate waves with a minimum of smearing. For conservative systems, our paper provides the easiest and most seamless path for taking a pre-existing HLL RS and quickly and effortlessly converting it to a RS that provides improved results, comparable with those of an HLLC, HLLD, Osher or Roe-type RS. This is done with minimal additional computational complexity, making our variant of the HLLEM RS also a very fast RS that can accurately represent linearly degenerate discontinuities. Our present HLLEM RS also transparently extends these advantages to non-conservative systems. For shallow water-type systems, the resulting method is proven to be well-balanced. Several test problems are presented for shallow water-type equations and two-phase flow models, as well as for gas dynamics with real equation of state, magnetohydrodynamics (MHD & RMHD), and nonlinear elasticity. Since our new formulation accommodates multiple intermediate waves and has a broader applicability than the original HLLEM method, it could alternatively be called the HLLI Riemann solver, where the "I" stands for the intermediate characteristic fields that can be accounted for. Keywords: path-conservative HLLEM Riemann solver, resolution of linearly degenerate intermediate waves, conservation laws and general hyperbolic PDE with non-conservative terms, well-balanced scheme for single and two-layer shallow water equations, Euler equations with real equation of state and multiphase flows, RMHD/MHD equations and nonlinear elasticity

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“…The evolution to the half time level needed in (32) and (33) is directly obtained from a discrete form of the PDE (1) in the small, without coupling to the neighbor elements:…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Ideal Relativistic Mhd Equations (Rmhd)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems

Dumbser

Balsara

2016

Journal of Computational Physics

161

125

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“…One could emphasize that this family of numerical methods in which the PVM method is based on interpolation polynomials of degree 2 is equivalent to Roe and 3-wave ARS methods. 53 In the present case, the PVM method used for the comparison defines the wave velocities such that…”

Section: About the Pvm Methodsmentioning

confidence: 99%

A simple three‐wave approximate Riemann solver for the Saint‐Venant‐Exner equations

Audusse

Chalons

Ung

2018

Numerical Methods in Fluids

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Erosion and sediments transport processes have a great impact on industrial structures and on water quality. Despite its limitations, the Saint-Venant-Exner system is still (and for sure for some years) widely used in industrial codes to model the bedload sediment transport. In practice, its numerical resolution is mostly handled by a splitting technic that allows a weak coupling between hydraulic and morphodynamic distinct softwares but may suffer from important stability issues. In recent works, many authors proposed alternative methods based on a strong coupling that cure this problem but are not so trivial to implement in an industrial context. In this work, we then pursue two objectives. First we propose a very simple scheme based on an approximate Riemann solver, respecting the strong coupling framework, and we demonstrate its stability and accuracy through a number of numerical test cases. But, second, we reinterpret our scheme as a splitting technic and we extend the purpose to propose what should be the minimal coupling that ensures the stability of the global numerical process in industrial codes, at least when dealing with collocated finite volume method. The resulting splitting method is, up to our knowledge, the only one for which stability properties are fully demonstrated.

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“…As it has been pointed out in [18], there is a close connection between PVM and SRS methods. In particular, under certain assumptions, a SRS method can be interpreted as an interpolatory PVM.…”

Section: Introductionmentioning

confidence: 87%

On the efficient implementation of PVM methods and simple Riemann solvers. Application to the Roe method for large hyperbolic systems

Pimentel-García

Parés

Castro

et al. 2021

Applied Mathematics and Computation

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Relation between PVM schemes and simple Riemann solvers

Cited by 10 publications

References 16 publications

A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems

A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems

A simple three‐wave approximate Riemann solver for the Saint‐Venant‐Exner equations

On the efficient implementation of PVM methods and simple Riemann solvers. Application to the Roe method for large hyperbolic systems

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