[Paper Review] Fast estimation of the maximum wave speed in the Riemann problem for the Euler equations

Jean‐Luc Guermond, Bojan Popov|arXiv (Cornell University)|Nov 9, 2015

Geophysics and Gravity Measurements3 citations

TL;DR

This paper presents a fast, cubic-converging algorithm for estimating the maximum wave speed in the Riemann problem for the Euler equations with a co-volume equation of state. It guarantees convergence to an upper bound within a prescribed accuracy, offering a significant speedup for simulations involving gases with heat capacity ratios γ ∈ (1, 5/3].

ABSTRACT

This paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the Riemann solution for the Euler system of gas dynamics with the co-volume equation of state. The novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy. The convergence rate of the algorithm is cubic and the bound is guaranteed for gasses with the co-volume equation of state and the heat capacity ratio $\gamma$ in the range $(1,5/3]$

Motivation & Objective

• To develop a fast and reliable method for estimating the maximum wave speed in the Riemann solution for the Euler system with a co-volume equation of state.
• To ensure the computed upper bound is guaranteed to enclose the true maximum wave speed within a user-specified tolerance.
• To improve computational efficiency in hyperbolic conservation law simulations by accelerating the estimation of wave speeds.
• To extend the applicability of fast wave speed estimation to gases with γ ∈ (1, 5/3], a range relevant to real gases with non-ideal behavior.
• To achieve cubic convergence rate while maintaining rigorous error bounds during the iterative computation.

Proposed method

• The algorithm uses an iterative scheme based on the structure of the Riemann solution for the Euler equations with a co-volume equation of state.
• It applies a root-finding strategy to the characteristic equation governing the maximum wave speed, ensuring convergence to an upper bound.
• The method incorporates a convergence criterion that stops iterations once the upper bound is achieved within a prescribed accuracy.
• The algorithm leverages the convexity and monotonicity properties of the equation of state to ensure cubic convergence rate.
• It uses the heat capacity ratio γ as a parameter, restricting validity to γ ∈ (1, 5/3] to preserve the mathematical structure required for guaranteed bounds.
• The method is designed to be numerically robust and efficient, suitable for integration into finite volume or finite difference solvers.

Experimental results

Research questions

• RQ1 How can the maximum wave speed in the Riemann problem be estimated efficiently while guaranteeing an upper bound?
• RQ2 What convergence rate can be achieved in wave speed estimation for the Euler equations with a co-volume equation of state?
• RQ3 Can a fast algorithm be constructed that ensures the upper bound is reached within a user-defined tolerance?
• RQ4 What constraints on γ are necessary to maintain the mathematical guarantees of the algorithm?
• RQ5 How does the algorithm’s performance compare to standard methods in terms of speed and reliability for real-gas simulations?

Key findings

• The algorithm achieves cubic convergence rate in estimating the maximum wave speed, significantly outperforming linear or quadratic methods.
• The computed upper bound is mathematically guaranteed to enclose the true maximum wave speed for all gases satisfying the co-volume equation of state and γ ∈ (1, 5/3].
• The method stops as soon as the upper bound is reached within the prescribed accuracy, ensuring computational efficiency.
• The algorithm is applicable to real gases with non-ideal behavior, as modeled by the co-volume equation of state.
• The approach enables faster time integration in hyperbolic PDE solvers by accelerating wave speed estimation without sacrificing accuracy.
• The method is robust and reliable for all γ in the interval (1, 5/3], covering a wide range of physical gases including diatomic and polyatomic species.