论∂ζ/∂t+u∂ζ/∂x=0方程的所谓“先向前差、后中央差”差分格式的稳定性和收敛性

ON THE STABILITY AND CONVERGENCE OF THE SO-CALLED “FIRST-FORWARD-THEN-CENTERED DIFFERENCE ANALOGUE” FOR THE EQUATION OF THE FORM ∂ζ/∂t+u∂ζ/∂x=0

中文摘要:

      针对∂ζ/∂_t+u∂ζ/∂_x=0方程,本文首先讨论了在一般情形下所谓"先向前差、后中央差"差分格式的稳定性,并给出了稳定的条件。其结果和一般只用中央差分的情形有所不同。利用三种差分格式进行比较,发现目前广泛应用的用向前差计算第一个时间步长,以后全用中央差的格式是最差的;当λ(=uΔt/Δs)→1-0时,计算并不稳定。如果用另两种格式中任一种代替向前差,则可以避免这个困难。其中又以"逆2-3格式"较好。

英文摘要:

      By the so-called "first-forward-then-centered difference analogue" for the equation of the form ∂ζ/∂_t+u∂ζ/∂x=0 in conventional operation,in meteorology,is meant an analogue,in which the space derivatives are replaced by centered differences,but the time derivative is replaced by a forward difference at the first time step and then by a centered difference.For the sake of investigating the computational stability and convergence of this analogue under two arbitrary initial conditions an analogue with centered differences in both space and time is discussed.The results show that the analogue is stable when and only when the conditions:λ(=uΔt/Δ_s)<1 and 1/√1-λ² being fiaite,are exactly satisfied,where u the velocity of the basic current;Δt andΔs are time step and grid length respectively.However even in a stable case,the difference solution so obtained can not converge to the corresponding exact solution,unless the value of vorticity at the end of the first time step,i.e.ζ⁽¹⁾,is convergent.For the convenience of comparison,three difference analogues used to compute ζ⁽¹⁾ are investigated.It is found that the conventional analogue is the worst one of them and when λ→1-0.it is computationally unstable.If any one of the other two up-wind analogues is used in this case it is computationally stable.