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

Abstract

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

A comparison of explicit and semi‐implicit finite volume schemes for viscous compressible flows in elastic pipes in fast transient regime

“…Furthermore, as shown in [13,22], the path-conservative HLLEM scheme and the path-conservative Osher Riemann solver are well balanced for certain classes of non-conservative PDE systems. In particular, these schemes exactly preserve steady state solutions with zero velocity and constant pressure in ducts with variable reference cross section (A 0 = πR 2 0 ) Fig.…”

“…The finite volume update based on a second order space-time TVD reconstruction for a non-conservative system of hyperbolic equations reads (see [13])…”

“…where the path integral has been approximated using a Gauss-Legendre integration rule with nodes s j and weights w j on the unit interval [0, 1]. For systems composed of only two PDEs, as the one considered in (1), the HLLEM Riemann solver proposed in [13] reduces to the HLL method, for which the jumps term (fluctuations) are given by the expressions…”

“…The Riemann problem for averaged 1D models has been discussed in great detail in [19,29]. In our particular case, the numerical solution of the governing PDE system is found at the aid of two approximate Riemann solvers that have been very recently extended to the path-conservative framework, namely the Osher-Solomon Riemann solver and the HLLEM Riemann solver, see [13,16,22]. A new interesting variant of Osher-type Riemann solvers that does not need the full eigenstructure of the PDE system but which relies on a polynomial viscosity matrix (PVM) has been very recently forwarded in [6].…”

Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences

“…These states are fed into a multi-state two-dimensional Riemann solver (Balsara, 2010(Balsara, , 2012(Balsara, , 2014Balsara and Dumbser, 2015b) generating the electric field at the edges (the remaining flux components are discarded). The 2D Riemann solver used here is of the HLLI type (Dumbser and Balsara, 2016) that can include every MHD wave, including the Alfvén and slow magnetosonic waves. The face-based magnetic field is updated via the same Runge-Kutta procedure.…”