But in this paper, we are going to construct a new semi-implicit hybrid method (NSIHM) of four stages and fifth order using the algebraic order conditions given in Coleman ; the derivation also incorporates the simplifying conditions as well as the technique of minimization of the error constant.
Here, we derive the four-stage fifth-order NSIHM based on the order conditions, simplifying conditions, and minimization of the error constant [C.
10), and [rho], [tau], [lambda] is the error constant value.
The error constant value when the [rho] and [tau] has low value, non-homogeneous characteristic has low value.
The reasonable error constant value was experimentally set to 0.
It is worth pointing out that the y-intercept equals b = log K, where K is the absolute value of the error constant.
where M is the order of the error in the numerical method and K is the absolute value of the error constant.
Note that the absolute value of the error constant is equal to K = [e.
4 are numerically determined to minimize an error constant
that measures Hamiltonian (total energy) truncation errors.
Furthermore, we shall establish a new formula for calculating the asymptotic error constant of the new methods.
The 2m convergence order method formed by the above scheme will require m functions and one derivative evaluation per iterative step and the asymptotic error constant AEC (2m) of this scheme is given in .
The results were the same; some of them, including relative errors and the error constants
in the quadrature rules, are shown in Table I.