Time Step Parameter

Time step parameter = <float>

Description / Usage

This card allows the user to vary the time integration scheme. The usual settings are:

0.0	Backward Euler method (1st order in time)
0.5	Trapezoid rule (2nd order in time)

Examples

This is a sample card that sets the time integration scheme to Trapezoidal rule:

Time step parameter = 0.5

Technical Discussion

One should usually use the Trapezoid rule. When a large time step \(\Delta t\) is used the Trapezoid rule can exhibit oscillations. If such a large \(\Delta t\) is required then the Backward Euler method can be used (it will damp oscillations), albeit at a cost of accuracy.

If we designate the time step parameter as \(\theta\), the solution at time step \(n\) as \(y^n\), and the PDE to be solved as

\[\frac{\partial y}{\partial t} = g \left( y \right)\]

then the time integration method takes the form

\[\frac{y^{n+1} - y^n}{\Delta t} = \frac{2 \theta}{1 + 2 \theta} \dot{y}^n + \frac{1}{1 + 2 \theta} g \left( y^{n+1} \right)\]

where

\[\dot{y}^{n+1} = \frac{1 + 2 \theta}{\Delta t} \left( y^{n+1} - y^n \right) - 2 \theta \dot{y}^n = g \left( y^{n+1} \right).\]

Note that there is no choice of finite \(\theta\) that will yield a Forward Euler method. See Gartling (1987) for more information.

FAQs

For porous flow problems with mass lumping, you should always choose backward Euler method.

References

SAND86-1816: NACHOS 2: A Finite Element Computer Program for Incompressible Flow Problems - Part 2 - User’s Manual, Gartling, David K. (September, 1987).