Dynamic time step estimates for one-dimensional linear transient field problems
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abstract
Space and time integration of the parabolic time-dependent field equation produce a system of algebraic equations. A common problem during the numerical solution of these equations is determining a time step small enough for accurate and stable results yet large enough for economic computations. This study presents an experimental approach to defining the time step that integrates the linear one-dimensional field equation within 5% of the exact solution for four time stepping schemes; forward, central, and backward differences and Galerkin schemes. The dynamic time step estimates are functions of grid size and the smallest eigenvalue, 1. For a particular problem, a preliminary calculation is required to evaluate 1. The dynamic time step estimates were successfully tested for various problems. Evaluation results indicate that the central difference scheme is superior to the other three schemes as far as the flexibility in allowing a larger time step while maintaining accuracy of the numerical solution. Backward difference and forward difference schemes were very similar in their accuracy. The slight discrepancy between these two schemes is attributed to the numerical stability encountered by the forward difference scheme. The presented dynamic time step equations can be used in numerical software as a pre-priori, automatic, user independent, time step estimate.
