Thermodynamics HW Solutions 496

Thermodynamics HW Solutions 496 - Chapter 5 Numerical...

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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 Tx t t t t t t mi (, ) (, ) ) ) += + + + ΔΔ Δ 1 2 2 2 2 L The finite difference formulation of the time derivative at the same nodal point is expressed as t t Tx t t TT t m i m i ) +− = + Δ 1 or t t t ) ) +≅ + 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 series expansion is proportional to , and thus the local discretization error is also proportional to . () Δ t 2 Δ t 2 The global discretization error is proportional to the step size to Δ t itself since, at the worst case,
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