Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods
Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods
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DOI：
10.1007/s10915-021-01635-4
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发表时间：
2021-09
影响因子：
2.5
通讯作者：
B. Yeager;E. Kubatko;Dylan Wood
中科院分区：
文献类型：
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作者：
B. Yeager;E. Kubatko;Dylan Wood
不连续伽辽金有限元空间离散化通常用于线法方法中，该方法具有常微分方程求解器以及时向前推进解。显式强稳定性保持时间步长是一种流行的选择，因为它们可证明保持前向欧拉方法应用于不连续Galerkin半离散方程的非线性稳定性，受到时间步长约束。在强保稳定性保证非线性稳定性的同时，组合格式的线性稳定性还必须满足一个单独的条件。在这项工作中，我们评估的线性稳定性的不连续Galerkin空间离散与一组强稳定性保持多步Runge-Kutta方法。我们发现，在所有情况下，线性稳定性的约束比强稳定性保持的约束更严格。对于每一个订单，从多步龙格-库塔方法的集合中，我们选择一个最佳的时间步进，需要最少的不连续Galerkin算子的评价。所有的方法都进行了测试，在应用程序中的线性和非线性偏微分方程的收敛性，并发现所有的方法收敛在这两种情况下使用的最大稳定的时间步长确定本文中发现的稳定性约束。
Discontinuous Galerkin finite element spatial discretizations are often used in a method-of-lines approach with an ordinary differential equation solver to step the solution forward in time. Explicit strong-stability-preserving time steppers are a popular choice because they provably preserve the nonlinear stability properties of the forward Euler method applied to discontinuous Galerkin semi-discretized equations, subject to a time step constraint. While nonlinear stability is guaranteed by strong-stability-preservation, a separate condition for linear stability of the combined scheme must also be satisfied. In this work, we assess the linear stability properties of discontinuous Galerkin spatial discretizations with a set of strong-stability-preserving multistep Runge–Kutta methods. We find that, in all cases, the constraint for linear stability is more strict than that for strong-stability-preservation. For each order, from the set of multistep Runge–Kutta methods, we select an optimal time stepper that requires the fewest evaluations of the discontinuous Galerkin operator. All methods are tested for convergence in application to both a linear and a nonlinear partial differential equation, and all methods are found to converge in both cases using the maximum stable time step as determined by the stability constraints found in this paper.