A high-order spatiotemporal precision-matching Taylor–Li scheme for time-dependent problems
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Based on the Taylor series method and Li’s spatial differential method, a high-order hybrid Taylor–Li scheme is proposed. The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with third-order accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme’s computation speed. A more rapid and high-precision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers’ equation.
Key wordsTaylor–Li scheme high-order scheme Burgers’ equation
结合Taylor级数法和Li空间微分方案的优点, 实现了高阶时间积分的Taylor-Li算法格式, 并且进行了多组数值试验.线性平流方程的试验结果表明对于高斯函数型的初值, 高阶精度算法可以取得非常好的计算效果.计算非线性无粘Burgers方程时, 高阶精度算法能否获得好的计算结果, 除了受初始场形式的影响, 还与计算的目标时刻有关.当目标时刻解的各阶导数连续时, 高阶算法效果非常好; 研究发现时间积分方案的阶数大于3之后, 对应的最优空间差分精度阶数可以比6阶提高很多, 这可以解释以前某些研究中6阶以上空间差分格式对结果无改进的现象, 是由于没有使用足够高精度的时间积分方案引起的.提出了使用递归微分的方式来提高Taylor-Li算法的计算速度的技术, 所实现的快速高精度差分格式的比直接计算高阶空间微分项的方案速度显著提升, 配合多精度计算的工具库, 实现了平流方程超高阶的时间积分-空间差分格式.
关键词Taylor-Li格式 高阶算法 Burgers方程
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The author gratefully acknowledges Prof. Jianping LI for his suggestions and discussions. The work is supported by the National Natural Sciences Foundation of China (Grant Nos. 41375112 and 41530426) and the Chinese Academy of Sciences Key Technology Talent Program.
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