Dynamic time step estimates for one-dimensional linear transient field problems

Space and time integration of the parabolic time-dependent field equation p roduce a system of algebraic equations. A common problem during the numeric al solution of these equations is determining a time step small enough for accurate and stable results yet large enough for economic computations. Thi s study presents an experimental approach to defining the time step that in tegrates the linear one-dimensional field equation within 5% of the exact s olution for four time stepping schemes; forward, central, and backward diff erences and Galerkin schemes. The dynamic time step estimates are functions of grid size and the smallest eigenvalue, lambda 1. For a particular probl em, a preliminary calculation is required to evaluate lambda(1) The dynamic time step estimates were successfully tested for various problems. Evaluat ion results indicate that the central difference scheme is superior to the other three schemes as far as the flexibility in allowing a larger time ste p while maintaining accuracy of the numerical solution. Backward difference and forward difference schemes were very similar in their accuracy. The sl ight discrepancy between these two schemes is attributed to the numerical s tability encountered by the forward difference scheme. The presented dynami c time step equations can be used in numerical software as a pre-priori, au tomatic, user independent, time step estimate.