φ(t) = F (t, φ(t)), t ∈ [a, b], φ(a) = φa. (1.1) where φa, φ(t) ∈ C n and F : R × C → C. Let {t}n=0 be equally spaced nodes in the interval [a, b] with t0 = a, tN = b. Let {t j n} M j=0 be the Legendre-Gauss-Lobatto nodes in the subinterval [tn, tn+1] with t 0 n = tn, t M n = tn+1. Denote ∆n = t j+1 n − t j n. 1.1 Forward Euler Scheme The Picard integral equation in each [tn, tn+1] associated w...

In this paper we consider ODEs whose solutions satisfy exponential monotonic quadratic forms. We show that the quadratic preserving Gauss–Legendre–Runge–Kutta methods do not preserve this qualitative feature, while certain Lie group preserving methods, as Crouch–Grossman methods and methods based on projection techniques are suitable integrators for such a kind of differential problems. We also...

Parallel implementation issues of the explicit Verlet{type (VT) and the implicit Runge{Kutta (RK) methods for molecular dynamics integration (MD) on parallel distributed memory processors with the ring topology are described and compared. Both methods are applied to a system of N particles interacting through the Lennard{Jones potential. The RK method is carried out for the two{stage fourth{ord...

In this paper, three numerical solutions for the Kortewegde Vries Burgers’ (KdVB) equation are presented. Two of these methods are based on cardinal Chebyshev basis function with Galerkin method. Gauss-quadrature formula and El-gendi method are used to convert the problem into system of ordinary differential equations. In the third proposed method, the cardinal Legendre basis function with Gale...

This work is to provide spectral and pseudo-spectral Jacobi-Galerkin approaches for the second kind Volterra integral equation. The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. For some spectral and pseudo-spectral Jacobi-Galerkin methods, a ...

A mesh refinement method is described for solving optimal control problems using Legendre-Gauss-Radau collocation. The detects discontinuities in the solution by employing an edge detection scheme based on jump function approximations. When are identified, refined with a targeted h-refinement approach whereby discontinuity locations bracketed points. remaining smooth portions of previously deve...