《显式隐式格式》PPT课件.ppt
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1、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)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(当U0时)蛙跳格式(跳背格式)时间、空间均为中央差显式、三时间层差分格式时间、空间均为二阶精度隐式差分格式 没有明显的计算方程来计算出初边值问题 n迎风格式(二时间层)迎风格式(二时间层)n蛙跳格式(三时间层)蛙跳格式(三时间层)则则所有所有时时刻能解刻能解已知已知 还还需要另一需要另一时间时间层层的数据。的数据。相容性 Consistency n要求差分系统和微分系统相协调。如果有这一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,该差分格式绝
2、不能模拟我们研条件不被满足,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,我们才有必要详细的研究差如果它被满足了,我们才有必要详细的研究差分格式。分格式。n相容性条件:主要是要求在小的时间步长和小相容性条件: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,差分方程应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。n相容性条件的英文表述:相容性条件的英文表述:nThe consistency:nwhen nthe FDE concides with PDE.收敛性convergenc
3、e 设设差分方程的解差分方程的解为为微分方程的解微分方程的解为为如果如果时,则则称:差分方程的解称:差分方程的解收收敛敛到到.差分算子收差分算子收敛敛到微分算子叫相容,到微分算子叫相容,收敛到而收敛是指而收敛是指稳定性稳定性 n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,当当n无限增大时,任意时刻的数值是否有界的无限增大时,任意时刻的数值是否有界的问题。假如数值解是稳定有界的,则相应的数问题。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。n计算稳定性的分析方法:计算稳定性的分析方法:n 冯纽曼方法(冯纽曼方法(Von-Neum
4、ann方法,又称方法,又称谐谐波分析法波分析法):通过测试差分格式近似解一个谐):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。拉克斯(拉克斯(LAX)等价定理)等价定理n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。n或者或者:如果差分格式是相容的如果差分格式是相容的,那么差分格式稳定等价于收敛。那么差分格式稳定等价于收敛。n或者或者:如果差分格式是相容的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那么
5、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。n其英文是其英文是:LAX theorem:LAX theoremn Given a properly posed linear initial value problem,Given a properly posed linear initial value problem,and a finite difference scheme that satisfies the and a finite difference scheme that satisfies the consistency condition,then the sta
6、bility of the FDE is consistency condition,then the stability of the FDE is the necessary and sufficient condition for convergence.the necessary and sufficient condition for convergence.CFL判据n对于迎风差分格式对于迎风差分格式n称为线性稳定性判据称为线性稳定性判据,又称又称CFL判据判据(Courant,Friedrichs和和Lewyt),中文常说成库朗判据。中文常说成库朗判据。nCFL conditio
7、n is a necessary condition for stability,but not sufficient.五、数值天气预报的概念和历五、数值天气预报的概念和历史回顾史回顾 nnumerical weather prediction uses numerical 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
8、 area for a certain time in the future.nIn 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.An initial-and boundary-value problemnIn order to do so,the meteorological vari
9、ables need to be prescribed at this initial time,which are called initial conditions.Mathematically,this corresponds to solve an initial-value problem.nDue to practical limitations,such as computing power,numerical methods,etc.,we are forced to make the numerical integration for predicting weather s
10、ystems 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.nThus,mathematically,numerical weather pred
11、iction is equivalent to solving an initial-and boundary-value problem.An initial-and boundary-value problemThe accuracy of the numerical weather predictionndepends on the accuracies of the initial conditions and boundary conditions.n The more accurate these conditions,the more accurate the predicted
12、 weather systems and processes.nthe 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.nwe do not have enough observed data over the oceans a
13、nd polar regions.nSome unconventional data,such as those retrieved from radar and satellite observations,have been used to help supply the data in data-void regions.nImprovement of global numerical weather predition model is also important in improving the accuracy of the regional numerical weather
14、prediction model since the former are often used to provide the initial and boundary conditions for the latter.The inaccuracy of numerical weather prediction nthe numerical approximation of the partial differential equations governing atmospheric motions on the discrete points of a model domainnand
15、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.Theaccuracyofanumericalmethodncan be improved by adopting a higher-order approximation of the partial differential equa
16、tions used in the numerical weather prediction models,as well as using a more accurate,but stable approximation methods.nThese require an increase of computing power as well as better understanding of numerical approximation methods.nThe accuracy of subgrid-scale parameterizations can be improved by
17、 a better understanding of the weather phenomena and processes as well as reducing the grid interval of a numerical weather prediction model.AnotherchallengeofnumericalweatherpredictionnAnother challenge of numerical weather prediction is whether the weather systems are predictable or not.n If they
18、are intrinsically unpredictable,then the improvements in more accurate initial and boundary conditions,numerical methods,and subgrid-scale parameterizations of a numerical weather prediction will have its limitations.nThe weather systems are considered to have limited predictability.The early histor
19、y of Numerical Weather PredictionnThe roots of numerical weather prediction can be traced back to the work of Vilhelm Bjerknes,a Norwegian physicist who has been called the father of modern meteorology.nIn 1904,he published a paper suggesting that it would be possible to forecast the weather by solv
20、ing a system of nonlinear partial differential equations.Vilhelm Bjerknes was a professor of applied mechanics and mathematical physics at the University of Stockholm Lewis Fry Richardson 1920nA British mathematician named Lewis Fry Richardson spent three years developing Bjerkness techniques and pr
21、ocedures to solve these equations.nRichardson computed a prediction for the change in pressure at a single point over a six-hour period.nThe calculation took him six weeks,and the prediction turned out to be completely unrealistic,but his efforts were a glimpse into the future of weather forecasting
22、.forecast factorynRichardson foresaw a“forecast factory,”where he calculated that 64,000 human“computers,”each responsible for a small part of the globe,would be needed to keep“pace with the weather”in order to predict weather conditions.后来,有些小进展nThe first meteorological radiosonde,a balloon carryin
23、g instruments to measure atmospheric temperature,pressure,humidity,and winds,was launched in the United States in 1937.nCommunication technology grew to allow hundreds of meteorological observations to be collected from around the globe.nMost importantly,by the end of World War II,the first electron
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