Numerical HeatTransfer（数值传热学）Chapter4DiscretizedSchemesofDiffusionandConvectionEquation(2)1/55
1/55 Numerical Heat Transfer (数值传热学) Chapter 4 Discretized Schemes of Diffusion and Convection Equation (2)
Chapter 4Discretized diffusion-convection eguation4.1 Two waysof discretization of convection term4.2CD andUDoftheconvectionterm4.3 Hybrid and power-law schemes4.4Characteristics of fivethree-pointschemes4.5Discussiononfalsediffusion4.6Methodsforovercomingoralleviating effectsoffalsediffusion4.7Discretizationof multi-dimensional problemandB.C.treatment2/51
2/51 4.1 Two ways of discretization of convection term 4.2 CD and UD of the convection term 4.3 Hybrid and power-law schemes Chapter 4 Discretized diffusion-convection equation 4.4 Characteristics of five three-point schemes 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
4.5Discussion on false diffusion4.5.1Meaningandreasonsoffalsediffusion(假扩散）（最初的含义）1.Original meaning2.Extendedmeaning（扩展后的含义）3.Taylorexpansionanalysis4.5.2Examplesofseverefalsediffusioncausedby1st-orderscheme4.5.3Errorscausedbyobliqueintersection（倾斜交叉）ofgridlines4.5.4Falsediffusioncausedbynon-constantsourceterm4.5.5Twofamousexamples3/55
3/55 4.5.1 Meaning and reasons of false diffusion(假扩散） 4.5.2 Examples of severe false diffusion caused by 1st-order scheme 1.Original meaning （最初的含义） 3.Taylor expansion analysis 4.5 Discussion on false diffusion 4.5.3 Errors caused by oblique intersection (倾斜交叉) of grid lines 4.5.4 False diffusion caused by non-constant source term 4.5.5 Two famous examples 2.Extended meaning（扩展后的含义）
4.5 Discussion on false diffusion4.5.1MeaningandreasonsoffalsediffusionFalsediffusion（假扩散),also called numerical viscosity（数值黏性），isanimportantnumericalcharacterofthediscretizedconvectivescheme1.Original meaningNumerical errors caused by discretized scheme with 1st orderaccuracy is called false diffusion;By Taylor expension, the eguation of such scheme at thesecond-order sense contains 2nd order derivative, thus the diffusionaction is somewhat magnified (放大)at the sense of（在...意义上)second-order accuracy, hence the numerical error is called “falsediffusion"4/55
4/55 4.5 Discussion on false diffusion 4.5.1 Meaning and reasons of false diffusion False diffusion (假扩散), also called numerical viscosity (数 值黏性)，is an important numerical character of the discretized convective scheme. 1. Original meaning Numerical errors caused by discretized scheme with 1st order accuracy is called false diffusion； By Taylor expension, the equation of such scheme at the second-order sense contains 2nd order derivative, thus the diffusion action is somewhat magnified (放大) at the sense of （在.意义上） second-order accuracy, hence the numerical error is called “false diffusion
Taking 1-D unsteady advection (平流)equationasanexampleThe two 1st-order derivatives are discretized by 1st-order accuracyschemesist-orderderivatives1st-oder scheme+ ---,-adad△t△xu>0ataxExpanding 0", at (i,n) by Taylor series, andsubstituting into the above equation:+1Mt-u5/55
5/55 Taking 1-D unsteady advection（平流） equation as an example. The two 1st -order derivatives are discretized by 1st -order accuracy schemes. u t x = − 1 st-oder scheme 1 1 n n n n i i i i u t x + − − − = − u >0 Expanding 1 1 , n n i i + − at (i,n) by Taylor series，and 2 3 2 3 , 2 3 , , 1 1 ) ) . 2 ) 6 n n i i n i n i n i t t t t t t t + + + − = , 2 3 2 3 2 3 , , 1 1 [ ) ) . ) 2 6 n n i i i n i n i n x x x x x u x x − − + − − substituting into the above equation： 1 st -order derivatives