DYNAMIC TIME STEP ESTIMATES FOR TRANSIENT FIELD PROBLEMS

Rabi  H. MOHTAR

Parabolic equations govern a variety of time dependent problems in science and engineering. Space integration of the field equation produces a system of ordinary differential equations. A common problem during the numerical solution of these equations is specifying a time step that is small enough for accurate and stable results and still reasonable for economic computations. This is a challenging mathematical problem that is yet to be solved.

This study presents an experimental approach to estimate the time step that integrates the field equation within 5% of the exact solution. Time step estimates were determined and evaluated for one‑dimensional and two‑dimensional linear square elements. A section is also included on the effect of shape and size of two‑dimensional quadrilateral elements on numerical stability.

Time step estimates were determined for four numerical schemes for one‑dimensional problems and three finite element, and three finite difference schemes for two‑dimensional problems. comparisons between finite element and finite differences as well as correlation with the Froude number were conducted. The time step estimates are functions of grid size and the smallest eigenvalue λ₁. For some problem an initial mesh is required to evaluate λ₁.

The results indicates that the finite element and finite difference methods are identical in one‑dimensional problems and have similar time step estimates. These two methods had similar accuracy two‑dimensional problems. The central difference schemes were superior to the other schemes as far as the flexibility in allowing a larger time step while maintaining good accuracy. Backward difference and forward difference schemes were very close in their accuracy; the difference between the two schemes was attributed to the stability requirement of the forward difference scheme.

The dynamic time step estimate for the central difference method was: λ₁ Δ N ^{l. 18} = 1.13 for the one‑dimensional problems, and λ₁ Δ N ^{O. 55} = 1.6 for two‑dimensional problems, where N is the number of nodes in the domain.

The Froude number equivalence is defined as the Froude number that gives a time step equal to the one computed using the presented regression equations. For the range of problems presented in this work and for the central difference scheme the Froude number equivalence ranges from 0.5 to 2.7 in the one‑dimensional problems and 0.46 to 9.13 in two‑dimensional problems.