Asymptotic analysis of the Riemann problem for constant coefficient hyperbolic systems with relaxation

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 84; no. 7; pp. 452 - 471
• Main Authors: Hittinger, J.A.F., Roe, P.L.
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag 01.07.2004; WILEY‐VCH Verlag; Wiley-VCH
• Subjects: asymptotic analysis; Exact sciences and technology; hyperbolic systems; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; relaxing flow; Riemann problem; Sciences and techniques of general use; Relaxation; Source terms; Hyperbolic system; Applied mechanics; Linear system; Riemann problem; Applied mathematics; Multiscale method; Initial value problem; Asymptotic behavior; Exact solution
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.200310114

The discontinuous Riemann initial value problem is considered for dissipative, constant coefficient hyperbolic systems with relaxation source terms. A short‐time asymptotic expansion, a specialization of the results of Le Floch and Raviart [11], is constructed for the general linear system. Exact rates for the decay of the initial discontinuities are found to be in agreement with recent results from other approaches. A multiple scales analysis is used to identify the long‐time asymptotic behavior. Many of the results can be exemplified in a simple model problem for which the exact solution can be found.