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The derivative Riemann problem for the Baer-Nunziato equations. (English) Zbl 1372.35184
Benzoni-Gavage, Sylvie (ed.) et al., Hyperbolic problems. Theory, numerics and applications. Proceedings of the 11th international conference on hyperbolic problems, Ecole Normale Supérieure, Lyon, France, July 17–21, 2006. Berlin: Springer (ISBN 978-3-540-75711-5/hbk). 1045-1052 (2008).
Summary: We solve the derivative Riemann problem (DRP) for the Baer-Nunziato (BN) equations for compressible two phase flows [M. R. Baer and J. W. Nunziato, Int. J. Multiphase Flow 12, 861–889 (1986; Zbl 0609.76114)]. The DRP is the Cauchy problem in which the initial condition consists of two smooth vectors, typically high-degree polynomials, with a discontinuity at the origin. In the classical Riemann problem these polynomials are two constant vectors. The technique to solve the DRP for the BN equations is an extension of that reported by the author and V. A. Titarev in [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458, No. 2018, 271–281 (2002; Zbl 1019.35061)] and [J. Comput. Phys. 212, No. 1, 150–165 (2006; Zbl 1087.65590)]. The solution \(Q_{\text{LR}}(\tau)\) is sought at the interface as a function of time. It is assumed that \(Q_{\text{LR}}(\tau)\) may be expressed as a time series expansion in which the leading term \(Q(0, 0_+)\) is the solution of the classical Riemann problem, evaluated at the interface, for \(t = 0_+\). The coefficients of the higher order terms are time derivatives of the vector of unknowns, all to be evaluated at \(x = 0\) and \(t = 0_+\). Use of the Cauchy-Kowalewski method allows us to express all time derivatives as functions of space derivatives. These spatial derivatives at \(x = 0\) and \(t = 0_+\) are found by first defining new evolution equations for spatial derivatives and then solving classical Riemann problems. The scheme reduces the solution of the derivative Riemann problem with polynomial data of two polynomials of degree at most \(K\) to the problem of solving one classical nonlinear Riemann problem for the leading term and K classical linear Riemann problems for spatial derivatives.
For the entire collection see [Zbl 1126.35003].
MSC:
35L60 First-order nonlinear hyperbolic equations
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