《数值传热学》研究生课程教学资源（课件讲稿）Chapter 7 Mathematical and Physical Characteristics of Discretized Equations（1/2，7.1-7.2）

热流科学与工程西步文源大学G教育部重点实验室Numerical HeatTransfer数值传热学）Chapter7MathematicalandPhysical CharacteristicsofDiscretizedEguations(Chapter3ofTextbook）QInstructorTao,Wen-QuanKeyLaboratoryofThermo-FluidScience&EngineeringInt.JointResearchLaboratoryofThermalScience&EngineeringXi'an Jiaotong UniversityInnovativeHarborofWestChina,Xian2022-Dec.-01CFD-NHT-EHTΦ1/40CENTER

1/40 Instructor Tao, Wen-Quan Key Laboratory of Thermo-Fluid Science & Engineering Int. Joint Research Laboratory of Thermal Science & Engineering Xi’an Jiaotong University Innovative Harbor of West China, Xian 2022-Dec.-01 Numerical Heat Transfer (数值传热学) Chapter 7 Mathematical and Physical Characteristics of Discretized Equations (Chapter 3 of Textbook)

热流科学与工程西步文源大堂G教育部重点实验室Contents7.1Consistence,Convergenceand StabilityofDiscretized Equations7.2vonNeumannMethodforAnalysingStabilityofInitialProblems7.3ConservationofDiscretized Equations7.4TransportivePropertyofDiscretizedEquations7.5Sign-preservation PrincipleforAnalyzingConvectiveStabilityCFD-NHT-EHTΦ2/40CENTER

2/40 7.1 Consistence, Convergence and Stability of Discretized Equations 7.3 Conservation of Discretized Equations Contents 7.4 Transportive Property of Discretized Equations 7.5 Sign-preservation Principle for Analyzing Convective Stability 7.2 von Neumann Method for Analysing Stability of Initial Problems

热流科学与工程西步文通大堂E教育部重点实验室7.1 Consistence,Convergence and Stability ofDiscretizedEquations7.1.1Truncation errorand consistence(相容性）and7.1.2Discretizationerror（离散误差）convergence(收敛性）（舍入误差）7.1.3 Round off errorand stability稳定性）ofinitialproblems（初值问题）7.1.4 ExamplesCFD-NHT-EHTΦ3/40CENTER

3/40 7.1 Consistence, Convergence and Stability of Discretized Equations 7.1.1 Truncation error and consistence(相容性） 7.1.3 Round off error （舍入误差）and stability （稳定性） of initial problems（初值问题） 7.1.4 Examples 7.1.2 Discretization error（离散误差） and convergence(收敛性）

热流科学与工程西步文通大堂G教育部重点实验室7.1 Consistence, Convergence and StabilityofDiscretizedEquationse（相容性）7.1.1Truncationerrorandconsistencel.Accurate (analytical) solution of the discretizedequations（离散方程的精确解It refers to the numerical solution without any round.off(舍入)error introduced in the solution proceduredenoted by ΦIt is assumed that Taylor expansion can be applied tothe accurate numerical solutions d’;2.Differential vs.difference operators (算子)4/40中CFD-NHT-EHTCENTER

4/40 7.1 Consistence, Convergence and Stability of Discretized Equations 7.1.1 Truncation error and consistence (相容性) 1. Accurate (analytical) solution of the discretized equations (离散方程的精确解) 2. Differential vs. difference operators (算子) n i  ； It is assumed that Taylor expansion can be applied to the accurate numerical solutions It refers to the numerical solution without any round￾off (舍入) error introduced in the solution procedure, denoted by . n i

热流科学与工程西步文源大堂E教育部重点实验室(1）Differentialoperator（微分算子）：Implementing(执行）somedifferential（微分）and/orarithmetic(算术) operations on function d(i, n) at a point (i,n):a?dadadL(d)inS)iouataxaxThen L(Φ)in = O --1-D transient model equation.（2）Differenceoperator（差分算子）：Implementing some difference（差分）and/or arithmetic operationson function Φ, at point (i,n)中CFD-NHT-EHT5/40CENTER

5/40 Then , ( ) 0 L  i n  -1-D transient model equation. （2）Difference operator(差分算子) ： n i Implementing some difference (差分) and/or arithmetic operations on function at point (i,n) （1）Differential operator (微分算子)： Implementing(执行) some differential (微分) and/or arithmetic(算 术) operations on function at a point ( ( , ) i n i,n): 2 , , 2 ( ) ( ) L u S i n i n t x x                 

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 /