The Riemann problem for first-order hyperbolic systems of partial differential equations is of fundamental importance for both theoretical and numerical purposes. Many approximate solvers have been developed for such systems; exact solution algorithms have received less attention because computation of the exact solution typically requires iterative solution of algebraic equations. Iterative algorithms may be less computationally efficient or might fail to converge in some cases. We investigate the achievable efficiency of robust iterative Riemann solvers for relatively simple systems, focusing on the shallow water and Euler equations. We consider a range of initial guesses and iterative schemes applied to an ensemble of test Riemann problems. For the shallow water equations, we find that Newton's method with a simple modification converges quickly and reliably. For the Euler equations we obtain similar results; however, when the required precision is high, a combination of Ostrowski and Newton iterations converges faster. These solvers are slower than standard approximate solvers like Roe and HLLE, but come within a factor of two in speed. We also provide a preliminary comparison of the accuracy of a finite volume discretization using an exact solver versus standard approximate solvers.

翻译：一阶双曲型偏微分方程组的黎曼问题在理论和数值计算中均具有基础重要性。针对此类方程组，已发展出多种近似求解器；而精确求解算法受到的关注较少，因为计算精确解通常需要迭代求解代数方程。迭代算法可能计算效率较低，或在某些情况下无法收敛。我们研究了针对相对简单系统（重点关注浅水方程和欧拉方程）的鲁棒迭代黎曼求解器可实现的计算效率。我们考虑了一系列初始猜测值和迭代方案，并将其应用于一组测试黎曼问题。对于浅水方程，我们发现经过简单改进的牛顿法能够快速可靠地收敛。对于欧拉方程，我们获得了类似结果；然而，当所需精度较高时，奥斯特罗夫斯基-牛顿迭代组合的收敛速度更快。这些求解器虽然比标准的近似求解器（如Roe和HLLE）慢，但速度差距可控制在两倍以内。我们还初步比较了使用精确求解器与标准近似求解器的有限体积离散格式的精度。