The Derivative Riemann Problem for the Baer–Nunziato Equations

We solve the derivative Riemann problem (DRP) for the Baer–Nunziato (BN) equations for compressible two phase flows [1]. 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 in [4] and [3]. The solution Q LR (τ) is sought at the interface as a function of time. It is assumed that Q LR (τ) 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.