A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non Stiff Source Terms
A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non Stiff Source Terms
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发表时间：
2006
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通讯作者：
Shamsul Qamar;G. Warnecke
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作者：
Shamsul Qamar;G. Warnecke
本文提出了一种高阶时空守恒方法，用于求解具有刚性源项和非刚性源项的双曲型方程组以及松弛方程组。我们称该方案为斜率传播(SP)方法。它是我们的齐次双曲组格式的推广[1]。在目前的非均匀体系中，弛豫时间可以从一个数量级变化到一个很小的值。这些较小的值使松弛项变得更强、更僵硬。在这种情况下，欠分辨的数值格式可能会产生虚假的数值结果。然而，即使初始层和小的松弛时间没有被数值求解，我们的方案仍然能够以高精度正确地捕捉到物理现象的行为。该方案以统一的方式对待空间和时间。流动变量及其坡度是该格式中的基本未知数。源项通过其在时空控制体上的体积积分来处理，并且是整个时空通量平衡的直接部分。我们使用两种方法来计算流量变量的斜率，第一种方法直接由控制体积上的通量平衡得到，而第二种方法则使用有限差分方法。该格式的主要特点是它的简单，它的雅可比和黎曼方程，以及它的效率和高的顺序精度。特别地，我们证明了该格式具有连续渐近极限的离散模拟。我们已经对文献中的各种测试模型实施了我们的方案，例如Broadwell模型、扩展的热力学方程、浅水方程、交通流和具有热传递的欧拉方程。数值结果验证了该方法的准确性、通用性和稳健性。
In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems. We call the scheme a slope propagation (SP) method. It is an extension of our scheme derived for homogeneous hyperbolic systems [1]. In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value. These small values make the relaxation term stronger and highly stiff. In such situations underresolved numerical schemes may produce spurious numerical results. However, our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved. The scheme treats the space and time in a unified manner. The flow variables and their slopes are the basic unknowns in the scheme. The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance. We use two approaches for the slope calculations of the flow variables, the first one results directly from the flux balance over the control volumes, while in the second one we use a finite difference approach. The main features of the scheme are its simplicity, its Jacobian-free and Riemann solver-free recipe, as well as its efficiency and high of order accuracy. In particular we show that the scheme has a discrete analog of the continuous asymptotic limit. We have implemented our scheme for various test models available in the literature such as the Broadwell model, the extended thermodynamics equations, the shallow water equations, traffic flow and the Euler equations with heat transfer. The numerical results validate the accuracy, versatility and robustness of the present scheme.