An accurate and efﬁcient Riemann solver with tangent velocities for Godunov schemes in special relativistic hydrodynamics

Sep 21, 2026

Uploaded by Quinn

A low-computational-cost exact Riemann solver with tangent velocities is proposed for numerically integrating special relativistic hydrodynamics (SRHD) within a Godunov scheme. The method uses a Newton iterative procedure in pressure, coupled with preliminary wave-pattern detection and an ODE integration for the normal velocity across a rarefaction. Compared with the Martì and M¨uller solver, it reduces computational cost by about 90%. The solver is tested in a first-order Godunov scheme for high jumps of primitive variables and applied to a two-dimensional mixing layer, with results compared to an HLL method.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages22

Reading Time55 min

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Summary

Introduction

Riemann problem in SRHD and wave structure

Previous numerical approaches and motivation

Paper structure

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2010; 64:1–22 Published online 17 August 2009 in Wiley InterScience ([www.interscience.wiley.com](www.interscience.wiley.com)). DOI: 10.1002/ﬂd.2133 An accurate and efﬁcient Riemann solver with tangent velocities for Godunov schemes in special relativistic hydrodynamics Danilo Durante 1 , ∗ ,† and Giorgio Riccardi2 1I.N.S.E.A.N. , The Italian Ship Model Basin, via di Vallerano, 139-00128 Roma, Italy 2 Department of Aerospace and Mechanical Engineering, Second University of Naples, via Roma, 29-81031 Aversa (Ce), Italy SUMMARY In order to numerically integrate the special relativistic hydrodynamics (SRHD) equations through a Godunov scheme, the use of an exact Riemann solver having a low computational cost is proposed. The solver works through a Newton iterative method in pressure, by using a preliminary detection of the wave pattern and an integration by series of the ODE for the normal velocity across a rarefaction wave. A comparison with the solver of Mart`ı and M¨uller shows about a 90% reduction of the computational cost. The present Riemann solver has been used inside a ﬁrst-order Godunov scheme, in order to test the solver for high jumps of the primitive variables, perhaps avoiding certain numerical manipulations (e.g. limiting ﬁlters and multi-step integration schemes) . The time evolution of a two-dimensional mixing layer is ﬁnally considered and a comparison with the solution obtained from a HLL method is carried out. The main result of this analysis lies in showing how the present exact Riemann solver can be successfully employed in the effective numerical integration of the SRHD equations. Copyright q 2009 John Wiley & Sons, Ltd. Received 9 September 2008; Revised 14 May 2009; Accepted 20 June 2009 KEY WORDS: SRHD Riemann problem; Godunov integration scheme; rarefaction waves with tangent velocities 1. INTRODUCTION The present paper deals with the numerical integration of the equations of motion of an inviscid ﬂuid in the special relativistic hydrodynamics (SRHD) framework. Encouraged by the important paper of Pons et al. [1], where the exact solution of the Riemann problem in the presence of tangent velocities is deduced, an exact Riemann solver has been built in order to implement it ina multidimensional Godunov integration scheme. ∗Correspondence to: Danilo Durante, I.N.S.E.A.N., The Italian Ship Model Basin, via di Vallerano, 139-00128 Roma, Italy. †E-mail: [durante79@gmail.com](durante79@gmail.com), [d.durante@insean.it](d.durante@insean.it)[ ](d.durante@insean.it)Contract/grant sponsor: European Defence Agency Copyright q 2009 John Wiley & Sons, Ltd. 2 D. DURANTE AND G. RICCARDI As is well known, the cartesian Riemann problem investigates the waves produced by the relaxation of a ﬁeld which is initially discontinuous across a plane (the plane (x 2 , x 3), for example): the ﬂow develops in a self-similar way along the third coordinate direction (x 1), by retaining its planar symmetry. At the initial time, the ﬂuid in the left half-space (x 1<0, state 1) has constant properties, as well as the ﬂuid in the right half-space (x 1>0, state 4) . Once the ﬂow is allowed to relax, two waves (rarefactions, indicated by ‘R’, and/or shocks, ‘S’) of different families are produced. They are separated by a contact discontinuity (‘C’), across which only the rest mass density (internal energy and enthalpy, as well) and the tangent components of the velocity can have ﬁnite jumps. Two uniform states lie also in the regions between C and the back fronts of these waves, respectively on the left (state 2) and on the right (state 3) of C. The SRHD Riemann problem has been set up by Taub [2], who deduced the Rankine–Hugoniot relations at the end of the forties of the past century, which were solved by Mart´ı and M¨uller [3] without considering tangent velocities. As for the classical hydrodynamics, the pressure in the

FAQs

What problem does the proposed solver address in SRHD?

It addresses numerical integration of the SRHD equations using a Godunov scheme by building an exact Riemann solver that includes tangent velocities.

How is the Riemann solver computed?

It employs a Newton iterative method in pressure, with wave-pattern detection and ODE integration of the normal velocity across a rarefaction wave.

What performance and validation results are reported?

The method shows about a 90% reduction in computational cost versus the Martì and M¨uller solver, is tested for high primitive-variable jumps in a first-order Godunov scheme, and is applied to a two-dimensional mixing layer with comparison to an HLL method.

Sep 21, 2026

Uploaded by Quinn