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.

Issue Section:
Technical Briefs
Keywords:
weighting factor, time step, single-step trapezoidal method, parabolic system, eigenvalue, finite element method, parabolic equations, Fibonacci sequences, finite element analysis, eigenvalues and eigenfunctions, numerical stability, heat conduction
Topics:
Eigenvalues, Errors, Finite element analysis, Temperature, Dimensions
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Copyright © 2006
 by American Society of Mechanical Engineers
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