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 thirdorder
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 highprecision
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.

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• A High-Order Spatiotemporal Precision-Matching Taylor–Li Scheme for Time-Dependent Problems