# 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 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).