For a single small time step, this approach is much easier to implement and more efficient than the implicit method; at the same time, the stress is return-mapped effectively onto the subsequent yield surface by enforcing the consistency condition, and the problem of drift in the explicit forward Euler method is overcome.
Various types of algorithms have been developed for this problem, including the explicit forward Euler method, implicit backward Euler method, and semi-implicit method, which differ from one another in their specific (iterative) updating schemes for (15).
The stability region before the third iteration is quite similar to the stability region of the backward Euler method, while the stability region after the third iteration has a similar shape of the stability region of the forward Euler method
. The change means that the dominant control is transferred between the components of the PDC method.
This method can seem as a modified forward Euler method
(For this purpose one may use the forward Euler method).
), we may use these methods to approximate the exact solution of (4.1) (at least with [[mu].sub.1] = 0) with global mean square convergence order [r.sub.g] [greater than or equal to] 0.5 on any finite time interval [0,T] (Note that local mean and mean square consistency (A4) and (A5) with D = [R.sup.2] with rates [r.sub.0] [greater than or equal to] and [r.sub.2] [greater than or equal to] are shown by means of the forward Euler method under the presence of multiplicative white noise, provided that E = [[parallel][X.sup.0][parallel].sup.4.sub.2] + E [[parallel][Y.sub.0][parallel].sup.4.sub.2] < +[infinity] and [[DELTA].sub.min] > 0).