西安交通大学：《数值传热学》课程教学资源（课件讲稿）第三章 求解扩散方程的数值方法及其应用 3.1 1-D Heat Conduction Equation 3.2 Fully Implicit Scheme of Multi-dimensional Heat Conduction Equation 3.3 Treatments of Source Term and B.C.

3.1.5Onlyfully implicit scheme can guaranteephysically meaningful solutionIllustrated by an example1[Known] 1-D transient HC withoutTnitialsource term, uniform initialfield. Two surfaces weresuddenly cooleddowntozero.[Find]Variation of inner point0xtemperature with timeW[Solution] Discretized by Practice AFAdopting three grids: W, P, and EAxAxPhysically the variation trendshown in right fig. can be expected!16/55

16/55 3.1.5 Only fully implicit scheme can guarantee physically meaningful solution Illustrated by an example. 1-D transient HC without source term, uniform initial field. Two surfaces were suddenly cooled down to zero. [Known] Variation of inner point temperature with time [Find] [Solution] Discretized by Practice A Adopting three grids: W, P, and E. Physically the variation trend shown in right fig. can be expected!

Analyzing the 2nd time level:Substituting:T,=T=Tw=T=0;Sc=0, S,=0apT,=aLfTe +(1-f)TI+awLfTw+(1-f)TaT[a, -(1- f)ae-(1-f)aw +(1- f)8,A,(x)Ax)+8cA,(x)AxYieldsa,T,=T,[a-(1-f)ae-(1-f)aw]a-(1-f)(aw+a)ap-(1-f)(aw+ae)Tpi.e.:Toa, +f(aw +ae)apa/△xat△tpc,Ar2a.1agOap4r?Ar?pc,Ax/ △tAxr△tpc,a1-2(1- f)a\tT.Grid FourierFinally:FoT,Ar2number!aAt1+2f17/55

17/55 Analyzing the 2nd time level: 0 0 0 ; T T T T E E W W     Yields 0 0 [ (1 ) (1 ) ] P P E W P P a T T f a f a      a 0, 0 C P S S   i.e.： 0 0 0 0 (1 )( ) (1 )( ) ( ) P P W E P W E P P P W E T a f a a a f a a T a a f a a           1 , E W a a x      0 , p P c x a t     Finally: 0 2 2 1 2(1 )( ) 1 2 ( ) P P a t x a f T T t x f         0 2 2 / ( ) / E P p p a x t a t a c x t c x x               0 0 [ (1 ) ] [ (1 ) ] P P E E E W W W a T a fT f T a fT f T        0 0 [ (1 ) (1 ) (1 ) ( ) ] ( ) T f a f a f S A x x S A P P E W P P C P a          x x Substituting： 0 0 0 0 0 0 2 a t Fo x     Grid Fourier number!

T,1-2(1- f)Fo1.0To1+2fFoOnly fully implicit0.8scheme can guaranteePhysically it is0.6positive ratiorequired :全隐格式（f=1）0.4T0.2>0ToFIT0.024T12142XF0AT0.275initial=0.5)Crauk-NColson(-0.4-0.6..显式（=0)-0.801.0WEAxAx18/55

18/55 0 1 2(1 ) 1 2 P P T f Fo T fFo       Physically it is required ： 0 0 P P T T  Only fully implicit scheme can guarantee positive ratio