If we apply the half-explicit Euler method
which was proposed in  to the test DAE (3.16), then we obtain the DAE stability function
To estimate the depth of the deep structures, the Euler method
is applied the RTE-TMI map.
[11,12] studied the Vaserstein bound for Euler method
and they proved an O([h.sup.(2/3-epsilon])]) for a one-dimensional diffusion process where h is the step-size and then they generalize the result to SDEs of any dimension with 0(h[square root of log(1/h))] bound when the coefficients are time-homogeneous.
As we all know, there are many solutions corresponding to the second-order ODEs, including the Euler method
, the improved Euler method
and the Runge-Kutta method et al.
The results provided by the presented method will be compared with those by the modified Euler method
, Newmark method, widely-used fourth-order RK method, and the exact results, respectively.
The direct application of the method to such pathological cases has been compared with the classical Euler method
, showing that singularity and ambiguity drawbacks do not affect the proposed solution.
When applying the multiplicative Euler method
to the mth subsystem, we obtain a single step at a time:
The numerical properties of impulsive differential equations attracted attentions of scholars since Ran et al.'s work in , which showed that the explicit Euler method
is stable for impulsive differential equations, while the implicit Euler method
(a) The exponential Euler method
(explicit), denoted by expEuler:
Here [theta] = 0 defines the explicit Euler method
, [theta] = 1 defines the backward Euler method
and [theta] =1/2 gives the symmetric (or trapezoidal) Euler method
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