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Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2020-05-13 , DOI: 10.1007/s10915-020-01222-z
Lai Wang , Meilin Yu

We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock–Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for the high-order flux reconstruction/correction procedure via reconstruction solution of the unsteady Navier–Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method solver is employed with a low storage element-Jacobi preconditioner to solve the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.



中文翻译:

使用高阶通量重构公式比较非稳态流动的ROW,ESDIRK和BDF2

我们对无雅可比线性无隐式Rosenbrock-Wanner(ROW)方法,显式第一阶段,单对角隐式Runge-Kutta(ESDIRK)方法和高阶二阶后向微分公式(BDF2)进行了比较研究。通过非稳态Navier–Stokes方程的重建解进行高阶通量重建/校正过程。伪瞬态连续用于求解ESDIRK的每个阶段(不包括第一阶段)和BDF2的每个步骤的非线性方程。重新启动的广义最小残差方法求解器的无Jacobian实现与低存储元素-Jacobi预条件器一起使用,以求解ROW的每个阶段以及ESDIRK和BDF2的每个伪时间迭代的线性系统。几个数值实验 包括层流和湍流模拟,都进行了比较。我们观察到,多级ROW2和ESDIRK2比多步BDF2更有效,高阶隐式时间积分器比低阶BDF2更有效。通常,当使用元素-雅可比预调节器时,尤其是对于有边界的流,ESDIRK方法比ROW方法允许更大的物理时间步长,用于非恒定流模拟。细化时间步长后,ROW方法比ESDIRK方法更有效。当使用元素-雅可比预处理器时,尤其是对于有边界的流,ESDIRK方法比ROW方法允许更大的物理时间步长,用于非恒定流模拟。细化时间步长后,ROW方法比ESDIRK方法更有效。当使用元素-雅可比预处理器时,尤其是对于有边界的流而言,ESDIRK方法在非稳态流模拟中比ROW方法具有更大的物理时间步长;细化时间步长后,ROW方法比ESDIRK方法更有效。

更新日期:2020-05-13
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