Table3-2ofTextbookEffect of grid fineness精确解64816324区间数0.047 30.048 00.047 50.047 3Φr=10.050 20.058 20.135 00.13640.13530.1350中=20.140 40.155 20.367 90.368 30.36790.369 7Φ1=30.375 20.394 4Solution of 32 intervals (区间)may be regarded asgrid-independent!TT[Ex.3-3 of Textbook]IInstability of0<x<1ax?explicit schemeatt≤0, T =2x, 0≤ x≤0.5:T=2(1-x),0.5≤x≤1Boundary condition: t > O, T(O,t) = T(1,t) = 016/86
16/86 Solution of 32 intervals (区间) may be regarded as grid-independent! [Ex. 3-3 of Textbook] Instability of explicit scheme 2 2 , 0 1 T T x t x = t T x x T x x = = − 0, 2 , 0 0.5; 2(1 ),0.5 1 Boundary condition: 0, (0, ) (1, ) 0 t T t T t = = Table3-2 of Textbook Effect of grid fineness
Numerical solutions are conducted forSolution :at=0.48, and 0.52△x?ata\t0.480.52Numerical solutionAr?Ax精确解TlT=0.5aAt= 0.48Ar1=0.0520.50.50.4t=0.1040.4=0.0480.30.3t=0.096t=0.208= 0.1920.20.20.10.10.00.00.10.20.30.40.51.60.70.80.91.00.10.30.50.70.80.90.21.61.0(b)InitialfieldAnalyticalNumerical solutions are converged ,but oscillating .This is because ofthephysicallymeaningless coefficientsofthealgebraiceguations.17/86
17/86 Solution： Numerical solutions are conducted for 2 0.48, and 0.52 a t x = Initial field 2 0.48 a t x = 2 0.52 a t x = Analytical Numerical solution Numerical solutions are converged ,but oscillating .This is because of the physically meaningless coefficients of the algebraic equations
7.2 von Neumann Method for AnalyzingStabilityofInitialProblems7.2.1Propagation of errorvectorwithtime7.2.2DiscreteFourierexpansion7.2.3Basic idea of vonNeumannanalysis7.2.4 Examples of von Neumann analysis7.2.5Discussion on vonNeumannanalysis18/86
18/86 7.2 von Neumann Method for Analyzing Stability of Initial Problems 7.2.1 Propagation of error vector with time 7.2.2 Discrete Fourier expansion 7.2.3 Basic idea of von Neumann analysis 7.2.4 Examples of von Neumann analysis 7.2.5 Discussion on von Neumann analysis
7.2vonNeumannMethod forAnalyzingStabilityofInitialProblems7.2.1Propagation（传递）oferrorvectorwithtimeMatrix expression of discretized equationsa?TOT0<x<Latax(i,n)T(x,O)= F(x)nAT(O,t)= f(t), T(L,t) = fz(t)ixn+Tn-2T"+T"i=1,2,,3.....(1-1)111(a)=F(x), i=0,12,3... Initial conditionn = f(n△t), Tn = f(nAt), n =1,2,Boundarycondition19/86
19/86 7.2 von Neumann Method for Analyzing Stability of Initial Problems 7.2.1 Propagation (传递) of error vector with time 1 Matrix expression of discretized equations 2 2 , 0 T T a x L t x = 1 2 ( ,0) ( ) (0, ) ( ), ( , ) ( ) T x F x T t f t T L t f t = = = 1 1 1 2 2 , ,2,3, 1 ( 1 . . . ) n n n n n T T T T T i i i i i i t x a I + − − + + − = = − 0 ( ), ,1,2,3. 0 T F x i i i = = I 0 1 2 ( ), ( ), ,2,. 1 . n n T f n t T f n t n = = = I t n t i x Initial condition Boundary condition (a)
atSetr,thedifferenceeqs.canbeexpressed asArTn+l = T"(1- 2r)+r(T" +T"), i= 1,2,..(I -1)For a fixed time level n, the above egs. can be re-written for each inner point as follows:Tn+l = T"(1- 2r)+ r(T" + T’)i=1,T++ =T(1-2r)+r(T" +T)i=2,Tg++=T(1-2r)+r(T'+T))i=3,Trl = T2(1 - 2r)+ r(T2+ + Tr3)i=I-2,Tnt = Tn(1 - 2r)+ r(Tn + Tr2)i=I-1.20/86
20/86 the difference eqs. can be expressed as 1 1 1 (1 2 ) ( ), 1,2,.( 1) n n n n T T r r T T i I i i i i + = − + + = − + − Set 2 , a t r x = For a fixed time level n, the above eqs. can be rewritten for each inner point as follows: 1 0 1 1 2 1, (1 2 ) ) ( n n n n i T T r r T T + = = − + + 1 2 2 1 3 2, (1 2 ) ( ) n n n n i T T r r T T + = = − + + 1 3 3 2 4 3, (1 2 ) ( ) n n n n i T T r r T T + = = − + + 1 2 2 1 3 2, (1 2 ) ( ) n n n n I I I I i T T r I r T T + = = − + + − − − − − 1 1 1 2 1, (1 2 ( ) ) n n n I I n I I i T T r I r T T + = = − + + − − − −