----20'+=- S"()=uAr?△t2△xThen Laxar(d")= O ---discretized form of 1-D transientmodel equation by forward time and central space scheme -FTCS3.Truncationerror（T.E.截断误差）of thediscretizedequationT.E.is the difference between differential and differenceoperators（微分算子与差分算子的差）T.E.= LxA(d")- L(d)(1) Definition -6/86
6/86 （1）Definition － T.E. ( ) ( ) , n n L L x t i i = − 3. Truncation error (T.E.截断误差) of the discretized equation T.E. is the difference between differential and difference operators (微分算子与差分算子的差). 1 1 1 1 1 , 2 2 ( ) 2 n n n n n n n n n i i i i i i i L u S x t i i t x x + + − + − − − − + = + − − Then , ( ) 0 n L x t i = -discretized form of 1-D transient model equation by forward time and central space scheme －FTCS
(2) Analysis- Expandingy'l, at pin (in)by Taylor series (with respect to both space and time)substituting the series into the discretized equation andrearranging into the form of two operators;For1-D model equation discretized by FTCS we havefollowing results:n+1adad-Φ=_ -20" +Φ"00ouAx?△tat2△xaxadL()"Lar,Ar()S3in = O(△t,Ax2)axT.E.How to get this result? First discussing the transient term7/86
7/86 （2）Analysis－Expanding at point（i,n) by Taylor series（with respect to both space and time）， substituting the series into the discretized equation and rearranging into the form of two operators; 1 1 , n n i i + For1-D model equation discretized by FTCS we have following results: 1 1 1 1 1 2 2 2 2 , 2 { 2 } ( , ) n n n n n n n i i i i i i i n i i n u S u t x x t x S O t x x + − − − + + − + − + − − − + − − = How to get this result? First discussing the transient term T E. . , ( ) n L x t i ( )n L i
Transient+d"d-term of FDp△tXtformn+1Bn1 as8i.e.+. = O(△t)△tdt2 otSecond, for the convection term of FD form:adO(△x3)Axdn, -d'x22axou2△x2△xad1Λx32Ox2△xadad2axpuOxRAX8/86
8/86 i.e. 1 , ( ) n n i i i n t t + − − 2 2 , , 2 ( ) ( ) . 2! n i n i n i n i t t t t t + + + Transient − term of FD form Second, for the convection term of FD form: 2 2 1 . 2 t t = + = O t ( ) 2 2 3 i 2 1 1 2 2 3 i 2 3 1 ( ) 2 = [ 2 2 1 ( ( )) 2 ] 2 2 ( ) 2 n n n i i n x x O x x x u u x x x x O x x x x x O x x u x + − + + + − − + + − + = 1 i n i n t + − = 3 2 ( ) = 2 u x O x x x + 2 = +O( ) u x x
dn-0dd=O(△r2)Thus:puou2△xOXThen for苏-dn+d'ddiffusionO(△r)Ar2dr2term :Assuming that the source term does not introduceany truncationerror, then:The T.E. of FTCS scheme for 1-D model equation:O(At,Axr?)Its mathematical meaning is:Existing two positive constants, Ky, K2, when△t →0,Ax →0 the difference between the twooperators will be less than (K,At +K,Ax)9/86
9/86 Its mathematical meaning is： Existing two positive constants，K1，K2，when → → t x 0, 0 the difference between the two 2 1 2 (K K ) + t x Thus: 1 1 2 , ( ) ( ) 2 n n i i u u O x i n x x + − − − = Assuming that the source term does not introduce any truncation error, then: The T.E. of FTCS scheme for 1-D model equation: 2 O t x ( , ) operators will be less than . Then for diffusion term : 1 2 1 2 2 2 - ( ) i n n n i i d O x x dx + − + − =
4.Consistence（相容性）ofdiscretizedequationsIf the T.E. of a discretized equation approaches zerowhen △t →0.△x >0 then:the discretized equation is said to be in consistence withthepartial differential equation (PDE)When T.E. is in the form of O(△t",Ar")(n, m > O)the discretized equations possess(具有) consistence;However when T.E. contains △t / x only when the timestep approaches zero much faster than space step , theconsistence can be guaranteed (保证).7.1.2 Discretization error and convergence10/86
10/86 4. Consistence (相容性) of discretized equations If the T.E. of a discretized equation approaches zero when then: → → t x 0, 0 the discretized equation is said to be in consistence with the partial differential equation (PDE). When T.E. is in the form of ( , )( , 0) n m O t x n m the discretized equations possess(具有) consistence； However when T.E. contains only when the time step approaches zero much faster than space step , the consistence can be guaranteed (保证). t x /