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 roundoff (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