热流科学与工程西步文源大堂G教育部重点实验室7+d"dn+1 -d" -2 +- - S"?OuAxr?2△x△tThen Lar.r(d") = O ---discretized form of 1-D transientmodel equation: Forward time and central space -FTCS3.Truncationerror（T.E.截断误差）ofthediscretizedequationT.E.isthe difference between differential anddifferenceoperators（微分算子与差分算子的差）T.E.= Lx(d")- L(Φ)(1) Definition -ΦCFD-NHT-EHT6/40CENTER
6/40 （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: Forward time and central space －FTCS
热流科学与工程西步文源大堂E教育邻重点实验室(2) Analysis-Expanding d!l, t point (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+1adadon一*-2"+ΦoAr?Atat2△xaxaL()"Lar,Ar(d')Bin =O(t,Ax2)axT.E.How to get this result? First discussing the transient term中CFD-NHT-EHT7/40CENTER
7/40 （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
热流科学与工程西步文源大堂G教育部重点实验室Transientd'77Cterm of FDp△tXtformon+1r1 asi.e.O+. = O(△t)△tdi2 otSecond, for the convection term of FD form:adO(△x3Axo"ax2-02Oxou2△x2△xap10(Ax3))2Ox2△xadad21axpuOxRAXCFD-NHT-EHTG8/40CENTER
8/40 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
热流科学与工程西步文源大堂E教育部重点实验室dn0ad一0(△x)Thus:puou2△xdxThen ford'$?+0diffusionO(Ax2)Ar2dr?term :Assuming that the source term does not introduceany truncationerror,thenThe T.E. of FTCS scheme for 1-D model equationO(△t,Ar)Its mathematical meaning is:Existing two positive constants, Kl, K2, when△t → O, △x → O the difference between the twooperators will be less than (K,At + K,Ax°)ΦCFD-NHT-EHT9/40CENTER
9/40 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
热流科学与工程西步文源大堂E教育部重点实验室4.Consistence（相容性）ofdiscretizedequationsIf the T.E. of 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",Ax")(n,m > O)the discretized equations possess(其有)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 convergenceCFD-NHT-EHT中10/40CENTER
10/40 4. Consistence (相容性) of discretized equations If the T.E. of 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 /