Sentence examples for midpoint rule and from inspiring English sources

The method is based on extrapolation of the two-step explicit midpoint rule and incorporates adaptive time step and order selection.

A programme, 'simkine' is developed using a semi-implicit extrapolation method (SIEM) which uses the implicit midpoint rule and extrapolation to simulate complex mechanisms based on the kinetics of homogeneous chemical systems.

We propose a two-step iterative method that first solves the evolution equation by the implicit midpoint rule and then solves the Helmholtz equation using a three-point sixth-order compact scheme.

The particle equations of motion and the Maxwell's equations are differenced implicitly in time by the midpoint rule and solved concurrently by a Jacobian-free Newton Krylov (JFNK) solver.

The applicability of the proposed estimator to several but selected existing time integration algorithms including the well known schemes like the Newmark method, HHT- α, Classical midpoint rule and in addition, new algorithms and designs as well is demonstrated with single and multi-degree of freedom, linear and nonlinear dynamical problems.

Furthermore, in [11], Xu et al. used contractions to regularize the semi-implicit midpoint rule (1.3) and presented the following viscosity implicit midpoint rule for nonexpansive mappings: x_{n+1}=alpha_{n}Q(x_{n})+ 1- alpha_{n})T biggl(frac{x_{n}+x_{n+1}}{2} biggr),quad nge0, (1.4) where (alpha_{n}in 0,1)) and Q is a contraction.

The error functionals,, and for the midpoint rule, trapezoidal quadrature and Simpson's rule are, respectively, (46).

Error in calculation of the value of I m at which the lower Hopf bifurcation occurs using the midpoint rule (red stars) and the Gaussian quadrature (blue circles).

Error in the calculation of the value of I m at which the upper Hopf bifurcation occurs using the midpoint rule (red stars) and the Gaussian quadrature (blue circles).

Error in the calculated period of the synchronised oscillators as a function of the number of neurons simulated (N) for the midpoint rule (red stars) and the Gaussian quadrature (blue circles).

The numerical formula is derived by integration over two steps, where the integrals of terms on the right-hand side of (1) are approximated via the trapezoidal rule and the midpoint rule, respectively.