A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator
A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator
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通过直接构造近似时间演化算子来解决波传播问题的高阶时间并行方案
DOI：
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发表时间：
2016
期刊：
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通讯作者：
B. Wingate
中科院分区：
文献类型：
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作者：
T. Haut;T. Babb;P. Martinsson;B. Wingate
我们的手稿展示了一种有效求解经典波动方程、浅水方程以及更一般的 ∂u/∂t=Lu∂u/∂t=Lu 形式的方程的技术，其中 LL 是斜厄米微分算子。这个想法是针对相对较大的时间步长 ττ 显式构造时间演化算子 exp(τL)exp(τL) 的近似值。利用最近开发的通过有理函数逼近振荡标量函数的技术以及用于计算离散微分算子函数的加速算法。所提出的方法的主要优点包括：即使对于大时间步长也具有稳定性，在许多特征波长上及时并行化的可能性以及在需要长时间模拟的情况下比现有方法大幅加速。给出了涉及非均匀介质中二维旋转浅水方程和二维波动方程的数值示例，并将该方法与四阶龙格-库塔（RK4）方法以及切比雪夫多项式的使用进行了比较。新方法在长时间间隔内实现了高精度，并且速度比 RK4 和切比雪夫多项式的使用快几个数量级。
Our manuscript demonstrates a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u/∂t=Lu∂u/∂t=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(τL)exp(τL) for a relatively large time-step ττ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.