热流科学与工程西步文源大学G教育部重点实验室Y1.Discretizationerror（离散误差）p" =d(i,n)-Φ'Analytical solution of FDEAnalytical solution of PDE2.Factors affecting discretization error T.E.: The higher the order , the smaller the value of p,(1)for the same grid system,(2) Grid step: For the same order of accuracy, a finer gridsystem leads to less numerical error.For conventional engineering simulation, usually2nd order for diffusion term and 2nd or 3rd order for convection termare used. For direct simulation of turbulent flow much higherschemes are needed!Φ-NHT-EHHFD.11/40CENTER
11/40 ( , ) n n i i i n Analytical solution of PDE Analytical solution of FDE 2. Factors affecting discretization error 1. Discretization error(离散误差) n i （2）Grid step：For the same order of accuracy, a finer grid system leads to less numerical error. （1）T.E.：The higher the order ，the smaller the value of for the same grid system; n i For conventional engineering simulation, usually: 2 nd order for diffusion term and 2 nd or 3rd order for convection term are used. For direct simulation of turbulent flow much higher schemes are needed!
热流科学与工程亚步文源大堂G教育部重点实验空3.Convergence（收敛性）ofthediscretizedequationsWhen △t →0,Ax →>0 if p" →0 then it is said:thediscretizedequationspossess convergenceProving convergence is not easy.It should be noted :that above descriptions of consistenceand convergenceareonlyqualitatively（定性地），not inthestrict mathematical sense. But enough for engineeringstudents.(Quantitatively----定量地)7.1.3Round-offerror(舍入误差）andstabilityofinitialproblems"=n -Φ1. Round-off error-- actual solution from computer we can obtain中CFD-NHT-EHT12/40CENTER
12/40 When t x 0, 0 if 0 n i Proving convergence is not easy. the discretized equations possess convergence. 3. Convergence (收敛性) of the discretized equations then it is said: It should be noted :that above descriptions of consistence and convergence are only qualitatively (定性地）, not in the strict mathematical sense. But enough for engineering students.（Quantitatively-定量地） 1. Round-off error n i n n n i i i n i - actual solution from computer we can obtain 7.1.3 Round-off error(舍入误差) and stability of initial problems
热流科学与工程西步文源大堂E教育部重点实验室2.Factoraffectinground-offerrorLength of computer word: Numerical solution method3.Errorsofnumericalsolutionsd(i,n)-Φ, = d(i,n)-d" ±d" -Φ= p" +s"For most engineering problems, generally Pispredominant（占优）4.Stability of initial problemsThe solutionprocedureof aninitial problem is ofmarching (步进) type; if errors introduced at any timet(随后的）level are enlarged (放大) in the subsequentsimulation such that the solutions become infinite (无限），this scheme is called unstable（不稳定）；Otherwise the scheme for the initial problem is stable.ΦCFD-NHT-EHT13/40CENTER
13/40 ( , ) n i i n ( , ) n n i n i i i n n n i i 3. Errors of numerical solutions For most engineering problems, generally n i is predominant (占优). Length of computer word; Numerical solution method 2. Factor affecting round-off error 4. Stability of initial problems The solution procedure of an initial problem is of marching (步进) type; if errors introduced at any time level are enlarged (放大) in the subsequent （随后的） simulation such that the solutions become infinite（无 限）, this scheme is called unstable（不稳定）； Otherwise the scheme for the initial problem is stable
热流科学与工程西步文源大堂G教育部重点实验室Stabilityis aninherent（固有的）characterofascheme, no matter what kind of error is introduced.7.1.4Example[Example3-1 ofTextbook|Effect ofT.E.and gridnumber23451d'p,dp2b = 0,d(0) = 0; d(4)= 1dxdxdα=h=1aFind: Values of nodes 2, 3 and 4d0Solution: By FDM: replacingbyFDExp.dx?' dxFirst way: for all three points 2nd order scheme isadopted, then the FD Eqs can be established ;Second way: for Node 3 fourth order scheme isadopted: Nodes 2 and 4----second order scheme is usedΦCFD-NHT-EH14/40CENTER
14/40 7.1.4 Example [Example 3-1 of Textbook ] Effect of T.E. and grid number 2 2 2 0 (0) 0; (4) 1 d d dx dx ， First way: for all three points 2nd order scheme is adopted，then the FD Eqs can be established ; Second way: for Node 3 fourth order scheme is adopted; Nodes 2 and 4-second order scheme is used. Solution：By FDM：replacing by FD Exp. 2 2 , d d dx dx Find: Values of nodes 2, 3 and 4. Stability is an inherent (固有的）character of a scheme，no matter what kind of error is introduced