Bounds for Wave Speeds in the Riemann Problem: Direct Theoretical Estimates

• Published in: Computers & fluids Vol. 209; p. 104640
• Main Authors: Toro, E.F., Müller, L.O., Siviglia, A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 15-09-2020; Elsevier BV
• Subjects: Algorithms; Arteries; Blood flow; Bounds for wave speeds; Cauchy problems; Courant condition; Estimates; Euler-Lagrange equation; Exact solutions; Flow equations; Fluxes; Gas dynamics; Hyperbolic equations; Hyperbolic systems; Iterative methods; Riemann problem; Shallow water equations; Stability; Waves; Hyperbolic equations; Stability; Courant condition; Riemann problem; Waves; Bounds for wave speeds
• ISSN: 0045-7930; 1879-0747
• DOI: 10.1016/j.compfluid.2020.104640

•Largely unexplored topic of wave speed bounds for hyperbolic equations is addressed.•Existing results in the field are compiled and reviewed.•Theoretical bounds for wave speeds arising from the Riemann problem are presented.•Approach applied to: the Euler, the shallow water and the blood flow equations.•All (old and newly proposed) approaches are assessed via selected test problems.
In this paper we provide bound estimates for the two fastest wave speeds emerging from the solution of the Riemann problem for three well-known hyperbolic systems, namely the Euler equations of gas dynamics, the shallow water equations and the blood flow equations for arteries. Several approaches are presented, all being direct, that is non-iterative. The resulting bounds range from crude but simple estimates to accurate but sophisticated estimates that make limited use of information from the solution of the Riemann problem. Through a carefully chosen suite of test problems we asses our wave speed estimates against exact solutions and against previously proposed wave speed estimates. The results confirm that the derived theoretical bounds are actually so, from below and above, for minimal and maximal wave speeds respectively. The results also show that popular previously proposed estimates do not bound the true wave speeds in general. Applications in mind, but not pursued here, include (i) reliable implementation of the Courant condition to determine a stable time step in all explicit methods for hyperbolic equations; (ii) use in local time stepping algorithms and (iii) construction of HLL-type numerical fluxes for hyperbolic equations.