优化初始值问题的时间谱解法
Optimizing time-spectral solution of initial-value problems
AI总结：
本文比较了时间谱通用加权残差法（GWRM）与显式Lax-Wendroff和隐式Crank-Nicolson方法在求解初始值问题中的效率与精度，发现GWRM在高精度下更高效，尤其在处理双时间尺度问题时，CPU时间和内存需求显著优于有限差分法。
摘要：
时间谱解法通常被认为效率低下，其时间域扩展相较于有限差分法导致大量数值运算和高内存需求。本文通过引入时间与空间子域算法结合稀疏矩阵方法，显著提升了性能。本文将最近发展的时间谱通用加权残差法（GWRM）的精度和效率与显式Lax-Wendroff方法和隐式Crank-Nicolson方法进行比较。采用三个初始值PDE作为模型问题：1D Burger方程、受迫1D波动方程和14个线性化理想磁流体力学（MHD）方程耦合系统。发现GWRM在高精度下比时间步进方法更高效。对于双时间尺度受迫波动方程的时间平均解，GWRM在CPU时间和内存需求上比有限差分法高出一个数量级。MHD方程在时间与空间子域数量增加时，CPU时间和内存使用具有良好的可扩展性。
英文摘要：
Time-spectral solution of ordinary and partial differential equations is often regarded as an inefficient approach. The associated extension of the time domain, as compared to finite difference methods, is believed to result in uncomfortably many numerical operations and high memory requirements. It is shown in this work that performance is substantially enhanced by the introduction of algorithms for temporal and spatial subdomains in combination with sparse matrix methods. The accuracy and efficiency of the recently developed time spectral, generalized weighted residual method (GWRM) is compared to that of the explicit Lax-Wendroff method and the implicit Crank-Nicolson method. Three initial-value PDEs are employed as model problems; the 1D Burger equation, a forced 1D wave equation and a coupled system of 14 linearized ideal magnetohydrodynamic (MHD) equations. It is found that the GWRM is more efficient than the time-stepping methods at high accuracies. For time-averaged solution of the two-time-scales, forced wave equation GWRM performance exceeds the finite difference methods by an order of magnitude both in terms of CPU time and memory requirement. Favourable scaling of CPU time and memory usage with the number of temporal and spatial subdomains is demonstrated for the MHD equations.
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