Chapter 5 Numerical Methods in Heat Conduction Special Topic: Controlling the Numerical Error 5-96C The results obtained using a numerical method differ from the exact results obtained analytically because the results obtained by a numerical method are approximate. The diff erence betwe en a numerical solution and the exact sol ution (the error) is primarily due to two s ources: The discretization error (also called the truncation or formulation error) which is caused by the approximations used in the formulation of the numer ica l met hod, and the round-off error which is caused by the computers' representing a number by using a limited number of significant digits and continuously rounding (or chopping) off the digits it cannot retain. 5-97C The discretization error (also called the truncation or formulation error) is due to replacing the deriv ative s by diff eren ces in each step , or repla cing the actua l temp erature distributi on betw een two adjacent nodes by a straight line segment. The difference between the two solutions at each time step is cal led th e local discretization error . The tot al dis cre ti ati on error at any ste p is calle d the global or accumulated discretization error . The local and global discretiation errors are identical for the first time step. 5-98C !es, the global (accumulated) discretiation error be less than the local error during a step. The global discretiation error usually increases with increasing number of step s, but the opposite may occur when the solu tion function changes direc tion fre" uentl y , giving rise to loca l disc retia tion error s of opposite signs which tend to cancel each other. 5-99C The Ta ylor series expansion of the temperature at a specified nodal point m about time t i is + ∆ + ∆ + = ∆ + # # # ) , ( # $ ) , ( ) , ( ) , ( t t x T t t t x T t t x T t t x T i m i m i m i m ∂ ∂ ∂ ∂ The finite difference fo rmulation of the time derivative at the same nodal point is expressed as ∂ ∂ T x t t T x t t T x t t T T t m i m i m i m i m i ( , ) ( , ) ( , ) ≅ + − = − + ∆ ∆ ∆ $ or Tx t t Tx t t T x t t m i m i m i ( , ) ( , ) ( , ) + ≅ + ∆ ∆ ∂ ∂ which resembles the Ta ylor series expansion terminated after the first two terms. %&
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