SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B816-B845, January 2020.
This work introduces an extension of the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve a hyperbolic system of partial differential equations. To our knowledge, it has been used only for systems with mild source terms, such as gravitation problems or shallow water equations. What we propose is an implicit-explicit (IMEX) version of the RD schemes that can resolve stiff source terms, without refining the discretization up to the stiffness scale. This can be particularly useful in various models, where the stiffness is given by topological or physical quantities, e.g., multiphase flows, kinetic models, or viscoelasticity problems. We will focus on kinetic models that are BGK approximation of hyperbolic conservation laws. The extension to more complicated problems will be carried out in future works. The provided scheme is able to catch different relaxation scales automatically, without losing accuracy; we prove that the scheme is asymptotic preserving and this guarantees that, in the relaxation limit, we recast the expected macroscopic behavior. To get a high order accuracy, we use an IMEX time discretization combined with a deferred correction procedure, while naturally RD provides high order space discretization. Finally, we show some numerical tests in one and two dimensions for stiff systems of equations.

中文翻译：

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动力学模型的高阶渐近保存递推校正隐式-显式格式

SIAM科学计算杂志，第42卷，第3期，第B816-B845页，2020年1月。
这项工作将残余分布（RD）框架扩展到刚性松弛问题。RD是一类方案，用于求解偏微分方程的双曲系统。据我们所知，它仅用于具有温和源项的系统，例如引力问题或浅水方程。我们建议的是RD方案的隐式-显式（IMEX）版本，它可以解析刚性源项，而无需将离散化精细化到刚性尺度。这在各种模型中特别有用，在这些模型中，刚度由拓扑或物理量给出，例如多相流，动力学模型或粘弹性问题。我们将专注于动力学模型，它是双曲线守恒定律的BGK近似。对更复杂问题的扩展将在以后的工作中进行。所提供的方案能够自动捕获不同的松弛标度，而不会失去准确性；我们证明了该方案是渐近保持的，并保证了在松弛极限内，我们重铸了预期的宏观行为。为了获得高阶精度，我们将IMEX时间离散与延迟校正程序结合使用，而自然RD提供高阶空间离散。最后，我们展示了针对一维刚性方程组的一维和二维数值测试。我们重铸了预期的宏观行为。为了获得高阶精度，我们将IMEX时间离散与延迟校正程序结合使用，而自然RD提供高阶空间离散。最后，我们展示了针对一维刚性方程组的一维和二维数值测试。我们重铸了预期的宏观行为。为了获得高阶精度，我们将IMEX时间离散与延迟校正程序结合使用，而自然RD提供高阶空间离散。最后，我们展示了针对一维刚性方程组的一维和二维数值测试。