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Heat Chap05-096

# Heat Chap05-096 - Chapter 5 Numerical Methods in Heat...

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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 difference between a numerical solution and the exact solution (the error) is primarily due to two sources: The discretization error (also called the truncation or formulation error) which is caused by the approximations used in the formulation of the numerical method, 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 derivatives by differences in each step, or replacing the actual temperature distribution between two adjacent nodes by a straight line segment. The difference between the two solutions at each time step is called the local discretization error . The total discretization error at any step is called the global or accumulated discretization error . The local and global discretization errors are identical for the first time step. 5-98C Yes, the global (accumulated) discretization error be less than the local error during a step. The global discretization error usually increases with increasing number of steps, but the opposite may occur when the solution function changes direction frequently, giving rise to local discretization errors of opposite signs which tend to cancel each other. 5-99C The Taylor series expansion of the temperature at a specified nodal point m about time t i is + + + = + 2 2 2 ) , ( 2 1 ) , ( ) , ( ) , ( 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 formulation 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 ( , ) ( , ) ( , ) 2245 + - = - + 1 or T x t t T x t t T x t t m i m i m i ( , ) ( , ) ( , ) + 2245 + which resembles the Taylor series expansion terminated after the first two terms. 5-98

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Chapter 5 Numerical Methods in Heat Conduction 5-100C The Taylor series expansion of the temperature at a specified nodal point m about time t i is T x t t T x t t T x t t t T x t t m i m i m i m i ( , ) ( , ) ( , ) ( , ) + = + + + 1 2 2 2 2 The finite difference formulation 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 ( , ) ( , ) ( , ) 2245 + - = - + 1 or T x t t T x t t T x t t m i m i m i ( , ) ( , ) ( , ) + 2245 + which resembles the Taylor series expansion terminated after the first two terms. Therefore, the 3rd and following terms in the Taylor series expansion represent the error involved in the finite difference approximation. For a sufficiently small time step, these terms decay rapidly as the order of derivative increases, and their contributions become smaller and smaller. The first term neglected in the Taylor
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Heat Chap05-096 - Chapter 5 Numerical Methods in Heat...

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