OOF2: The Manual

`SS22 (SS22)`
	6.5.2. Subclasses

Name

SS22 (SS22) — Zienkowicz and Taylor's SS22 algorithm for solving equations with second order time derivatives.

Synopsis

SS22(theta1,theta2)

Details

Base class: TimeStepper
Parameters:

theta1

First moment of the weight function for time averages. Type: A real number in the range [0, 1].

theta2

Second moment of the weight function for time averages. Type: A real number in the range [0, 1].

Description

SS22 is a TimeStepper that implements the SS22 algorithm from Zienkiewicz and Taylor^[34]. Like CrankNicolson, it is a second order method, so its error is of order $(\Delta t)^3$ 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 $\phi$ in a Taylor series in time

$\phi = \phi_n + \tau\dot\phi_n + \frac12 \tau^2\alpha_n$

(6.161)

where $\phi_n$ and $\dot\phi_n$ are vectors of Field values and their time derivatives at time $t_n$ , $\tau = t-t_n$ is the time within the time step, and $\alpha_n$ is unknown. This expression is inserted into the differential equation being solved (e.g, equation (2.9)) and integrated over the duration $\Delta t$ of the time step with a weighting function $W(\tau)$ . $W(\tau)$ plays the role of a finite element shape function in the time coordinate. The result is a matrix equation which can be solved for $\alpha_n$ , which in turn gives the Field values and time derivatives at the end of the step, via equation (6.161).

The exact form of the weighting function $W(\tau)$ turns out to be irrelevant. All that is needed are its first and second moments, which must be provided as the parameters theta1 and theta2. The k-th moment is defined by

$\theta_k = \frac1{\Delta t^k} \frac{\int_0^{\Delta t} W(\tau) \tau^k\,d\tau} {\int_0^{\Delta t} W(\tau)\,d\tau}$

(6.162)

Using small values near 0 will make SS22 act 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.

^[34] O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Fifth Edition, Volume 1, page 511.


`Rodrigues (Rodrigues)`		`Other Mesh (ScalarOtherMeshInit)`