第二章显式隐式格式

numerical weather prediction uses numerical
An initial- and boundaryvalue problem
In order to do so, the meteorological variables need to be prescribed at this initial time, which are called initial conditions. Mathematically, this corresponds to solve an initial-value problem. Due to practical limitations, such as computing power, numerical methods, etc., we are forced to make the numerical integration for predicting weather systems in a finite area. In order to do so, it is necessary to specify the meteorological variables at the boundaries, which include upper, lower, and lateral boundaries, of the domain of interest. Mathematically, this corresponds to solve a boundary-value problem. Thus, mathematically, numerical weather prediction is equivalent to solving an initial- and boundary- value problem.
Fi
n +1
− Fi 2∆t
2
n −1
+U
2
F
n i +1
−F 2∆x
n i −1
=0
O ∆x , ∆t
(
)
显式、三时间层差分格式 时间、空间均为二阶精度
隐式差分格式
Fi
n +1
− Fi F −F +U ∆t 2∆x
n
n +1 i +1
n +1 i −1
=0
没有明显的计算方程来计算出 Fi n+1
An initial- and boundaryvalue problem
The accuracy of the numerical weather prediction
depends on the accuracies of the initial conditions and boundary conditions. The more accurate these conditions, the more accurate the predicted weather systems and processes. the lack of sufficient and accurate initial conditions, as well as more accurate and sufficient boundary conditions and appropriate ways in implementing them at the lateral boundaries of a finite domain of interest. we do not have enough observed data over the oceans and polar regions. Some unconventional data, such as those retrieved from radar and satellite observations, have been used to help supply the data in data-void regions. Improvement of global numerical weather predition model is also important in improving the accuracy of the regional numerical weather prediction model since the former are often used to provide the initial and boundary conditions for the latter.
∆t n n = Fi − U ( Fi − Fi −1 ) ∆x
n
∆t ∆x
已知
F0 (t ), F0 ( x)
还需要另一时间 层的数据。 层的数据。
则所有时刻能解
相容性 Consistency
要求差分系统和微分系统相协调。 要求差分系统和微分系统相协调。如果有这一 条件不被满足， 条件不被满足，该差分格式绝不能模拟我们研 究的初值问题。我们可以说这个条件是基本的， 究的初值问题。我们可以说这个条件是基本的， 如果它被满足了， 如果它被满足了，我们才有必要详细的研究差 分格式。 分格式。 相容性条件： 相容性条件：主要是要求在小的时间步长和小 的空间格距趋于零的极限条件下， 的空间格距趋于零的极限条件下，差分方程应 等同于微分方程。 等同于微分方程。 相容性条件的英文表述： 相容性条件的英文表述： The consistency: when ∆x → 0, ∆t → 0 the FDE concides with PDE.
The accuracy of a numerical method
can be improved by adopting a higher-order approximation of the partial differential equations used in the numerical weather prediction models, as well as using a more accurate, but stable approximation methods. These require an increase of computing power as well as better understanding of numerical approximation methods. The accuracy of subgrid-scale parameterizations can be improved by a better understanding of the weather phenomena and processes as well as reducing the grid interval of a numerical weather prediction model.
CFL判据
对于迎风差分格式 0 ≤ µ ≤ 1 称为线性稳定性判据,又称 又称CFL判据 称为线性稳定性判据 又称 判据 (Courant, Friedrichs和Lewyt)， 和 ， 中文常说成库朗判据。 中文常说成库朗判据。 CFL condition is a necessary condition for stability, but not sufficient.
初边值问题
迎风格式（二时间层） 迎风格式（二时间层）
Fi
n +1
∂F ∂F +U =0 ∂t ∂x F ( x,0) = F((x0)) F (0, t ) = F0 (t )
蛙跳格式（三时间层） 蛙跳格式（三时间层）
n n Fi n +1 = Fi n −1 − U ( Fi +1 − Fi −1 )
Fi n
收敛到 F ( xi , t n )
稳定性
差分近似的稳定性是指对于任意给定的初值， 差分近似的稳定性是指对于任意给定的初值， 无限增大时， 当n无限增大时，任意时刻的数值是否有界的 无限增大时 问题。假如数值解是稳定有界的， 问题。假如数值解是稳定有界的，则相应的数 值格式称为稳定的格式。 值格式称为稳定的格式。 计算稳定性的分析方法： 计算稳定性的分析方法： 冯纽曼方法（ 方法， 冯纽曼方法（Von-Neumann方法，又称谐 方法 又称谐 波分析法）： ）：通过测试差分格式近似解一个谐 波分析法）：通过测试差分格式近似解一个谐 波分量的稳定性，研究差分格式的稳定性。 波分量的稳定性，研究差分格式的稳定性。
4、时间的显隐，时间层数
显式差分格式和隐式差分格式
迎风格式（时间向前差，空间 向后差）
Fi
n +1
− Fi Fi − F +U ∆t ∆x
n n
n i −1
=0
（当U>0时）
O(∆x, ∆t )
数学上不喜欢用这种格式,精 度不高,但物理上它的物理意 义明确
蛙跳格式（跳背格式） 时间、空间均为中央差
拉克斯（ 拉克斯（LAX）等价定理 ）
对于一个适定的初值问题和它的一个具有相容性的差分格式， 对于一个适定的初值问题和它的一个具有相容性的差分格式，收 敛性的充分必要条件是其稳定性。 敛性的充分必要条件是其稳定性。 或者:如果差分格式是相容的,那么差分格式稳定等价于收敛。 或者:如果差分格式是相容的,那么差分格式稳定等价于收敛。 或者:如果差分格式是相容的, 或者:如果差分格式是相容的,那么差分解收敛的充要条件是差分 格式是稳定的。 格式是稳定的。 其英文是:LAX 其英文是:LAX theorem Given a properly posed linear initial value problem, and a finite difference scheme that satisfies the consistency condition, then the stability of the FDE is the necessary and sufficient condition for convergence.
收敛性convergence
设差分方程的解为 Fi n 微分方程的解为
F ( xi , t n )
ε in = Fi n − F ( xi , t n )
如果
.
∆t → 0, ∆x → 0
时，ε i
n
→0
Biblioteka Baidu则称： 则称：差分方程的解 Fi n
收敛到 F ( xi , t n )
差分算子收敛到微分算子叫相容， 差分算子收敛到微分算子叫相容， 而收敛是指
五、数值天气预报的概念和历 史回顾
methods to approximate a set of partially differential equations on discrete grid points in a finite area to predict the weather systems and processes in a finite area for a certain time in the future. In order to numerically integrate the partial differential equations, which govern the atmospheric motions and processes, with time, one needs to start the integration at certain time.
The inaccuracy of numerical weather prediction
the numerical approximation of the partial differential equations governing atmospheric motions on the discrete points of a model domain and the representation of the weather phenomena and processes occurred within grid points of a numerical model, i.e. the parameterization of subgrid-scale weather phenomena and processes.