*********************** Time Step Parameter *********************** :: Time step parameter = ----------------------- Description / Usage ----------------------- This card allows the user to vary the time integration scheme. The usual settings are: .. tabularcolumns:: |l|L| ======================= ======================================================================== 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 :math:`\Delta t` is used the Trapezoid rule can exhibit oscillations. If such a large :math:`\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 :math:`\theta`, the solution at time step :math:`n` as :math:`y^n`, and the PDE to be solved as .. math:: \frac{\partial y}{\partial t} = g \left( y \right) then the time integration method takes the form .. math:: \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 .. math:: \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 :math:`\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).