NumericalHeatTransfer（数值传热学）Chapter7MathematicalandPhysicalCharacteristicsofDiscretizedEquations(Chapter3ofTextbook)1/86
1/86 Numerical Heat Transfer (数值传热学) Chapter 7 Mathematical and Physical Characteristics of Discretized Equations (Chapter 3 of Textbook)
Contents7.1Consistence,Convergence andStabilityofDiscretizedEquations7.2vonNeumannMethodforAnalysingStabilityofInitialProblems7.3ConservationofDiscretizedEquations7.4TransportivePropertyofDiscretizedEquations7.5Sign-preservationPrincipleforAnalyzingConvectiveStability2/86
2/86 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
7.1 Consistence,Convergence and Stability ofDiscretizedEquations7.1.1Truncationerrorandconsistence(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 Examples3/86
3/86 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）
7.1 Consistence, Convergence and Stabilityof Discretized Equations7.1.1Truncationerrorandconsistence（相容性）1. 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 inthe solutionprocedure,denotedbvIt is assumed that Taylor expansion can be applied to thedaccurate numerical solutions2.Differentialvs.differenceoperators（微分与差分算子）4/86
4/86 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 (微分与差分算子) 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
(1）Differentialoperator（微分算子）：Implementing(执行)some differential （微分）and/or arithmetic(算术) operations on function Φ(i, n) at a point (i,n):a?dadadL(d)inS)ataxOxThen L(Φ)i.n = O --1-D transient model diffu-convec. equation.（2）Differenceoperator(差分算子）：Implementing some difference（差分）and/or arithmetic operationson function , at point (i,n)5/86
5/86 Then , ( ) 0 L i n = -1-D transient model diffu-convec. 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 = + − −