OOF2: The Manual
SS22 (SS22) — Zienkowicz and Taylor's SS22 algorithm for solving equations with second order time derivatives.
- First moment of the weight function for time averages. Type: A real number in the range [0, 1].
- Second moment of the weight function for time averages. Type: A real number in the range [0, 1].
SS22 is a TimeStepper that implements the SS22
algorithm from Zienkiewicz and Taylor. Like CrankNicolson, it is a second order
method, so its error is of order on each step. Unlike CrankNicolson and the other time
steppers in OOF2,
SS22 integrates second
order time derivatives directly, without resorting to equation (6.141).
SS22 works by expanding the field values
Taylor series in time
are vectors of
Field values and their time derivatives at time
is the time within the time step, and
is unknown. This expression is inserted into the differential
equation being solved (e.g, equation
(2.9)) and integrated over the
duration of the time step with a weighting
function . plays the role of a
finite element shape function in the time coordinate. The result is
a matrix equation which can be solved for , which in turn gives
Field values and time derivatives at the end of the step, via
The exact form of the weighting function turns out to be
irrelevant. All that is needed are its first and second moments,
which must be provided as the parameters
theta2. The k-th moment is defined by
Using small values near 0 will make
more like an explicit method, and values near 1 will make it more
like an implicit method. Values near 0.5 seem to work well.
SS22 can be used on first order equations,
such as the heat equation, although it's most advantageous for
second order equations, like the force balance equation.
 O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Fifth Edition, Volume 1, page 511.