Weighting Factors for Single-Step Trapezoidal Method

[+] Author and Article Information
Fuzhang Zhao

Department of Mechanical Engineering and Mechanics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104Fuzhang.Zhao@drexel.edu

J. Heat Transfer 128(4), 409-412 (Nov 07, 2005) (4 pages) doi:10.1115/1.2175155 History: Received June 15, 2005; Revised November 07, 2005

The uses of a weighting factor along with a time step in a single-step trapezoidal method to solve a first-order parabolic system have been systematically studied. The weighting factors are used in two main types: constants and variables. The most commonly used constant weighting factors can be defined by the ratio of the Fibonacci sequence. Among them, the optimal weighting factor is 0.618, resulting in a balance between the overall accuracy and efficiency. With the finite element formulation, the space and time dimensions can be discretized separately. For the time discretization only, there exists a zero-error dimensionless time step if a weighting factor is within the range of 0.5–1.0. By taking advantage of the zero-error condition, the weighting factor can be correlated with a time step. The influence of spatial dimensions is lumped into a nonzero eigenvalue of the system. Through validity tests of two benchmark linear problems, the variable weighting factor for a single-step trapezoidal method is shown to be accurate, efficient, and stable. The relevant features have been captured.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

DAE as a function of DTS with different θ

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Figure 2

Temperature comparison for example 1

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Figure 3

Error as a function of position for example 2

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Figure 4

Error comparison as a function of time step and methods




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