一维平流方程迎风格式的最优时空步长

Optimal steps in space and time for 1D advection equation with upwind difference scheme

在考虑舍入误差影响的情况下,研究一维平流方程迎风格式最优时空步长的选取.首先,分析每一时间层产生的离散误差和舍入误差,以及2种误差向高时间层传播的累积,得到数值解总误差的理论上界;然后推导出最优时间步长和最优空间步长的理论公式,进而得到2种不同机器精度下最优时间步长之比满足的一个仅与机器精度有关的普适关系;最后通过数值算例验证了结论的可靠性.

The optimal steps in space and time are studied in numerical solution of one-dimensional advection equation with upwind difference scheme when round off error is taken into account.Firstly,discretization error and round off errors in the nu-merical computation are analyzed theoretically,as well as the accumulation of the two kinds of errors propagating from each lower time layer to the highest time layer.The theoretical approximate formula for total error bound is derived.Then the theoreti-cal formulae for determining optimal steps in space and time are obtained,and a universal relation between two optimal time steps under any two different machine precision is established,which only relates to the two involved machine precision.Final-ly,the reliability of the conclusion are confirmed by numerical examples.