implicit Euler method

You can select the implicit Euler method as the corrector in the time-integration procedure. Ansys Polyflow automatically uses the explicit forward Euler formula (Equation 29–11) as the predictor.

Substituting into Equation 29–12, the following set of nonlinear algebraic equations corresponding to the implicit Euler method is obtained:

(29–14)

The implicit Euler method is a first-order scheme, that is, the time-integration accuracy is O . It does not cause oscillatory behavior, no matter how large the time-step size is.

The implicit Euler method is the default method.

Galerkin method

With this method, the corrector is obtained by setting in Equation 29–12, which yields

(29–15)

The explicit forward Euler formula (Equation 29–11) is used, as with the implicit Euler method.

The Galerkin method is more accurate than the implicit Euler method. However, it can generate oscillatory errors if the time step is large, though these are not as troublesome as with the Crank-Nicolson method. The Galerkin method is a compromise between the implicit Euler and Crank-Nicolson methods.

Crank-Nicolson method

This method is also called the modified Euler method or the trapezoid rule. It is a second-order method with an accuracy of . It is advisable to use a predictor of the same accuracy, so the following Adams-Bashforth explicit formula is used in Ansys Polyflow:

(29–16)

The corrector corresponding to the Crank-Nicolson method has been obtained by setting in Equation 29–12, which yields

(29–17)

The Adams-Bashforth explicit formula (Equation 29–16) requires the knowledge of the time-derivative of at two previous time steps: and . It cannot be used until the completion of the first three time steps. Therefore, the time step adjustment as a function of the local truncation errors can only begin at the 4th step. For this reason, the initial time step should not be too large, since there is no error control during the first three time steps.

The Crank-Nicolson method is the most accurate of the three methods available in Ansys Polyflow. A drawback of this method is the generation of oscillatory errors when the time step is large. To avoid this, you should set the tolerance (that is, ) quite small.