An Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for solving convection and convection-diffusion equations
An Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) method for solving convection and convection-diffusion equations
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DOI：
10.1016/j.jcp.2022.111589
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发表时间：
2022-04
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影响因子：
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通讯作者：
Joseph Nakao;Jiajie Chen;Jing-Mei Qiu
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文献类型：
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作者：
Joseph Nakao;Jiajie Chen;Jing-Mei Qiu
提出了一种新的欧拉-拉格朗日-库塔有限体积法，用于数值求解对流和对流扩散方程。欧拉-拉格朗日方法和半拉格朗日方法越来越受欢迎，主要是因为它们能够允许大的时间步长。我们所提出的方案是在由Rankine-Hugoniot跳跃条件确定的特征的近似划分的时空区域上积分PDE，然后将时间积分形式重写为时间微分形式，以便通过线法方法应用Runge-Kutta(RK)方法。该格式可以看作是标准Runge-Kutta有限体积(RK-FV)格式的推广，在Runge-Kutta有限体积(RK-FV)格式中，用零速度的近似特征来划分时空区域。高阶空间重建采用新近发展的具有自适应阶次的加权基本无振荡格式(WENO-AO)，高阶时间精度由对流方程的显式RK方法和对流扩散方程的隐式-显式RK方法实现。我们的算法通过维度分裂扩展到更高的维度。数值实验证明了该算法的健壮性、高阶精度和处理超大时间步长的能力。
We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines approach. The scheme can be viewed as a generalization of the standard Runge-Kutta finite volume (RK-FV) scheme for which the space-time region is partitioned by approximate characteristics with zero velocity. The high-order spatial reconstruction is achieved using the recently developed weighted essentially non-oscillatory schemes with adaptive order (WENO-AO); and the high-order temporal accuracy is achieved by explicit RK methods for convection equations and implicit-explicit (IMEX) RK methods for convection-diffusion equations. Our algorithm extends to higher dimensions via dimensional splitting. Numerical experiments demonstrate our algorithm's robustness, high-order accuracy, and ability to handle extra large time steps.