Abstract: In this paper, we introduce a method for improving
the embedded Runge-Kutta-Fehlberg4(5) method. At each integration
step, the proposed method is comprised of two equations for the
solution and the error, respectively. These solution and error are
obtained by solving an initial value problem whose solution has the
information of the error at each integration step. The constructed algorithm
controls both the error and the time step size simultaneously and
possesses a good performance in the computational cost compared to
the original method. For the assessment of the effectiveness, EULR
problem is numerically solved.
Abstract: A block backward differentiation formula of uniform
order eight is proposed for solving first order stiff initial value
problems (IVPs). The conventional 8-step Backward Differentiation
Formula (BDF) and additional methods are obtained from the same
continuous scheme and assembled into a block matrix equation which
is applied to provide the solutions of IVPs on non-overlapping
intervals. The stability analysis of the method indicates that the
method is L0-stable. Numerical results obtained using the proposed
new block form show that it is attractive for solutions of stiff problems
and compares favourably with existing ones.
Abstract: The implicit block methods based on the backward
differentiation formulae (BDF) for the solution of stiff initial value
problems (IVPs) using variable step size is derived. We construct a
variable step size block methods which will store all the coefficients
of the method with a simplified strategy in controlling the step size
with the intention of optimizing the performance in terms of
precision and computation time. The strategy involves constant,
halving or increasing the step size by 1.9 times the previous step size.
Decision of changing the step size is determined by the local
truncation error (LTE). Numerical results are provided to support the
enhancement of method applied.