Abstract The paper addresses a new method for approximately solving the Riemann problem for an arbitrary non-conservative hyperbolic system of partial differential equations. The method is based on the reconstruction of the complete wave structure in the Riemann problem with using the Borel measures interpretation (path-representation) of the non-conservative products. In contrast to alternative approximate methods of the Riemann problem solution, e.g., HLL-type methods, the method proposed takes into account the complete wave structure of the solution and requires the calculation of only eigenvalues and eigenvectors of the matrix of the system to be considered and therefore can be easy realized for any hyperbolic system. In the case of conservative systems, the method gives the exact Riemann problem solution with any required accuracy. It can be applied, for example, to compute the exact Rieman problem solution for compressible Euler equations with an arbitrary equation of state. The method is applied to solving the Rieman problem for the Euler equations, the two-phase Baer-Nunziato equations, the elasto-plastic equations. The results obtained concern the convergence and accuracy propreties of the method.

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关于求解偏微分方程非保守双曲系统的黎曼问题

摘要 本文提出了一种近似求解偏微分方程的任意非保守双曲系统的黎曼问题的新方法。该方法基于使用非保守乘积的 Borel 测度解释（路径表示）重建黎曼问题中的完整波结构。与黎曼问题解的替代近似方法（例如 HLL 型方法）相比，所提出的方法考虑了解的完整波结构，并且只需要计算要考虑的系统矩阵的特征值和特征向量因此对于任何双曲线系统都可以轻松实现。在保守系统的情况下，该方法以任何所需的精度给出精确的黎曼问题解。例如，它可以用于计算具有任意状态方程的可压缩欧拉方程的精确黎曼问题解。该方法应用于求解欧拉方程、两相Baer-Nunziato方程、弹塑性方程的黎曼问题。获得的结果与该方法的收敛性和准确性有关。