3672057777 - Lecture Notes ME4906, Mechanical Engineering,...

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Lecture Notes ME4906, Mechanical Engineering, PolyU 2008/2009 5.3.2 Implicit Scheme It has been shown that numerical instability occurs due to the explicit scheme. An implicit scheme is constructed as follows to avoid such instability of finite difference solution for parabolic PDEs. Instead of using the space derivative, 22 / Tx , of the past time step, l , for which the values are known, we choose to use the current time step, l+ 1. The scheme gives () ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 / ,1 , 1 ,12 ,1 1 , 1, 1 1 2 , 1 1 , , κ =∆∆ +− − +− ++ + = ∆∆ → − +++ + += ct x Til Ti l l tx cT i l c T i l cT i l T i l which is an equation of three unknown variable on the left-hand side. Applying this to all interior points from i =2 to i = N x -1 for the level of l+ 1, one has a set of linear equations for all variables on this level. To simplify the notation, we omit indices l +1 and add a superscript (past) to indicate the time level l which is already found. The set of equations is written as follows, P N 1 23 2 4 3 34 5 4 21 1 12 xx x x known past past past
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3672057777 - Lecture Notes ME4906, Mechanical Engineering,...

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