1.Discretizationerror(离散误差）Op" =d(i,n)-d'Analytical solution of FDEAnalytical solution of PDE2. Factors affecting discretization errorT.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 higher schemeschemes are needed!11/86
11/86 ( , ) 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 scheme schemes are needed!
3.Convergence（收敛性)ofthediscretizedequationsWhen △t →0,△x →0 if pn →0 then it is said:the discretized equationspossessconvergenceProving convergence mathematically for a specific scheme is not easy.It should be noted :that above descriptions of consistence andconvergence are only qualitatively（定性地），not in the strictmathematical sense. But enough for engineering students.（Quantitatively----定量地）7.1.3Round-offerror(舍入误差）andstabilityofinitialproblems7"=" -Φ1. Round-off errorY-- actual solution from computer we can obtain12/86
12/86 When → → t x 0, 0 if 0 n i → Proving convergence mathematically for a specific scheme 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
2.Factoraffecting round-offerrorLength of computer word: Numerical solution method3.Totalerrorofnumerical solutionsd(i,n)-Φ, = (i,n)-d + -= p" +e"For most engineering problems, generally P,ispredominant（占优)4. Stability of initial problemsThe solution procedureof aninitial problem is ofmarching (步进) type; if errors introduced at any timelevel are enlarged (放大)in the subsequent（随后的)simulation such that the solutions become infinite （无限），this schemeis called unstable（不稳定）；Otherwise the scheme for the initial problem is stable.13/86
13/86 ( , ) n i i n − = ( , ) n n i n i i i n − + − = n n i i + 3. Total error 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
Stabilityis aninherent（固有的）characterofascheme,no matter what kind of error is introduced.7.1.4 Example[Example3-1ofTextbook|EffectofT.E.and gridnumber234d'.dp2 = 0 ,d(0)= 0; d(4) =1dx+dxdα=h=1aFind: Values of nodes 2, 3 and 4dp中Solution: 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 used14/86
14/86 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
2xsh34The analytical1solution&=h=1TTable 3-1 in the textbookdsp4p2p3d格式100.36790.04730.1350精确解(i=2,3,4)10.394 400.155 20.0582二阶格式(i=3)100.391.80.134 80.0505四阶格式The fourth order scheme is only adopted at Node3, while solution accuracy is greatly improved15/86
15/86 2 4 8 x x e e e e − − = − The analytical solution The fourth order scheme is only adopted at Node 3, while solution accuracy is greatly improved Table 3-1 in the textbook