This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

中文翻译：

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多孔介质三相流混合双曲椭圆柯西问题的半解析处理

这项工作提供了剩余幂级数方法 (RPSM) 的技术应用描述，以开发一种快速准确的算法，用于具有黎曼初始数据的守恒律混合双曲椭圆系统。RPSM 不需要离散化，将系统简化为代数方程的显式系统，从而进行大量复杂的计算，并以封闭形式精确解（如果存在）的泰勒幂级数展开形式提供解。从理论上讲，讨论了收敛假设，并推导出了指数率的误差界限。在数值上，混合型系统实现了近似解的收敛性和稳定性。报告的结果适用于在物理学和工程学的不同分支中出现的一般柯西问题，揭示了可靠性，