西安交通大学：《数值传热学》课程教学资源（课件讲稿）第四章 扩散与对流方程的离散格式 4.5 Discussion on false diffusion 4.6 Methods for overcoming or alleviating effects of false diffusion 4.7 Discretization of multi-dimensional problem and B.C. treatment

adadAt aduAx a?p+O(△x2,△t) (1)22axatataxwhere the transient 2nd derivative can be re-written as follows:aadadaaa"sapaaap1u214ar?Ot?ataxataxaxataxat△tabad△tSubstituting into Equation (1))122axaasadad+O(Ax?,△t2))(ataxThus at the sense of 2nd-order accuracy abovediscretized equation simulates a convective-diffusiveprocess，rather than an advectionprocess（平流，纯对流）6/55

6/55 2 , 2 2 2 , , , 2 2 ) ) ) ( , ) 1 2 2 ) i n i n i n i n t u x u O x t t x x t          = − − + +        （） where the transient 2nd derivative can be re-written as follows: 2 2 ( ) t t t      =    ( ) u t x     − =   u ( ) x t   −   2 2 2 u u u ( ) x x x      − − =     Substituting into Equation (1): 2 2 2 , , 2 , ) ) [ (1 )]( ) 2 ( , ) i n i n i n u x u t u t x t x O x x        = − + − +        Thus at the sense of 2nd -order accuracy above discretized equation simulates a convective-diffusive process , rather than an advection process（平流，纯对流）. 2 2 , 2 ) i n t t     2 2 2 ; 2 t u x    = 

uAtOnly whenOthiserrordisappears（消失）Axutis called Courant number， in memory of (纪念)aAxGerman mathematician Courant.apaduAtuAx+O(Ax2,△t)u-at2axAxdxRemark: We only study the false diffusion at the sense of2nd-order accuracy; i.e., if inspecting(审视) at the 2nd-orderaccuracytheabovediscretizedequationactuallysimulatesaconvection-diffusion process.For most engineering problems2nd-order accuracy solutions are satisfied.7/55

7/55 Only when 1 0 this error disappears（消失）. u t x  − =  u t x   is called Courant number，in memory of （纪念） a German mathematician Courant. 2 2 2 , , 2 , ) ) [ (1 )]( ) 2 ( , ) i n i n i n u x u t u t x t x O x x        = − + − +        Remark：We only study the false diffusion at the sense of 2 nd -order accuracy；i.e., if inspecting(审视) at the 2nd-order accuracy the above discretized equation actually simulates a convection-diffusion process. For most engineering problems 2 nd -order accuracy solutions are satisfied

2.Extendedmeaning（扩展的意义(现存）literatures almost allIn most existingnumerical errors are called false diffusion, which includes:(1) 1st-order accuracy schemes of the 1st order derivatives(original meaning);(2)Oblique intersection(倾斜交叉)of flowdirection with gridlines;(3) The effects of non-constant source term which are notconsideredinthediscretized schemes4.5.2Examplescausedby1st-orderaccuracyschemes1.1-D steady convection-diffusionproblemWhenconvectiontermisdiscretizedbyFUD,diffusion(偏离)term by CD, numerical solutions will severely deviatefrom analytical solutions:8/55

8/55 2. Extended meaning (扩展的意义) In most existing （现存）literatures almost all numerical errors are called false diffusion，which includes： (1) 1 st -order accuracy schemes of the 1st order derivatives (original meaning)； (2) Oblique intersection(倾斜交叉) of flow direction with grid lines； (3) The effects of non-constant source term which are not considered in the discretized schemes. 4.5.2 Examples caused by 1st-order accuracy schemes 1. 1-D steady convection-diffusion problem When convection term is discretized by FUD，diffusion term by CD, numerical solutions will severely deviate （偏离） from analytical solutions:

1.0OFUDP.=20,PA=4$-中oFUD: Physically-CDP-00exactplausible solution0.5FUD: severeerrorCD: oscillatingsolution0.50.51.0r/L2. 1-D unsteady advection problem (Noye,1976) (20241008)adop0 ≤ x ≤1, u = 0.1 (Linear problem) Φ(0,t) = Φ(1,t) = 0-uataxIn the range of x e[O,O.1] initial distribution is antriangle, others are zero. The two derivatives are discretized by9/55

9/55 2. 1-D unsteady advection problem (Noye,1976)（20241008） u ,   (0, ) (1, ) 0 t t = = t x     = −   0 1, 0.1(Linear problem)   = x u triangle，others are zero. CD: oscillating solution FUD: Physically plausible solution FUD: severe error In the range of x[0,0.1] initial distribution is an FUD CD The two derivatives are discretized by

the 1st -order accuracy schemes. The results are as follows虹t=4Caused by falset=41.0FI c=1.0diffusion of the1storderaccuracyu=0.1scheme=0dC=0.81.of.Initialcondition00.20.40.60.81.0r1a.u=0.10.5t=80叫0.20.40.60.81.071.0Ac=1.0Au=0.1Caused byfalse0.5diffusion of theist order accuracyc=0.8scheme00.80.20.40.61.0r10/55

10/55 Initial condition t=4 t=8 Caused by false diffusion of the 1 st order accuracy scheme Caused by false diffusion of the 1 st order accuracy scheme the 1st –order accuracy schemes. The results are as follows