When solving compressible multi-material flow problems, an unresolved challenge is the computation of advective fluxes across material interfaces that separate drastically different thermodynamic states and relations. A popular idea in this regard is to locally construct bimaterial Riemann problems, and to apply their exact solutions in flux computation. For general equations of state, however, finding the exact solution of a Riemann problem is expensive as it requires nested loops. Multiplied by the large number of Riemann problems constructed during a simulation, the computational cost often becomes prohibitive. The work presented in this paper aims to accelerate the solution of bimaterial Riemann problems without introducing approximations or offline precomputation tasks. The basic idea is to exploit some special properties of the Riemann problem equations, and to recycle previous solutions as much as possible. Following this idea, four acceleration methods are developed, including (1) a change of integration variable through rarefaction fans, (2) storing and reusing integration trajectory data, (3) step size adaptation, and (4) constructing an R-tree on the fly to generate initial guesses. The performance of these acceleration methods are assessed using four example problems in underwater explosion, laser-induced cavitation, and hypervelocity impact. These problems exhibit strong shock waves, large interface deformation, contact of multiple (>2) interfaces, and interaction between gases and condensed matters. In these challenging cases, the solution of bimaterial Riemann problems is accelerated by 37 to 83 times. As a result, the total cost of advective flux computation, which includes the exact Riemann problem solution at material interfaces and the numerical flux calculation over the entire computational domain, is accelerated by 18 to 79 times.

中文翻译：

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可压缩多材料流模拟中双材料黎曼问题的有效求解

在解决可压缩的多材料流动问题时，一个未解决的挑战是计算跨材料界面的平流通量，这些界面将截然不同的热力学状态和关系分开。在这方面一个流行的想法是局部构造双材料黎曼问题，并将它们的精确解应用于通量计算。然而，对于一般状态方程，找到黎曼问题的精确解是昂贵的，因为它需要嵌套循环。乘以在模拟过程中构建的大量黎曼问题，计算成本往往变得令人望而却步。本文介绍的工作旨在加速解决双材料黎曼问题，而无需引入近似值或离线预计算任务。基本思想是利用黎曼问题方程的一些特殊性质，并尽可能地回收以前的解。按照这个想法，开发了四种加速方法，包括（1）通过稀疏风扇改变积分变量，（2）存储和重用积分轨迹数据，（3）步长自适应，以及（4）在苍蝇产生初步猜测。这些加速方法的性能使用水下爆炸、激光诱导空化和超高速冲击中的四个示例问题进行评估。这些问题表现出强烈的冲击波、大的界面变形、多个 (>2) 界面的接触以及气体与凝聚态物质之间的相互作用。在这些具有挑战性的案例中，双材料黎曼问题的求解速度提高了 37 到 83 倍。结果，平流通量计算的总成本（包括材料界面处的精确黎曼问题求解和整个计算域的数值通量计算）加速了 18 到 79 倍。