The piecewise variational iteration method (VIM) for solving Riccati differential equations (RDEs) provides a solution as a sequence of iterates. Therefore, its application to RDEs leads to the calculation of terms that are not needed and more time is consumed in repeated calculations for series solutions. In order to overcome these shortcomings, we propose an easy-to-use piecewise-truncated VI...

In the present work, exponential integrators for time integration of semilinear problems are studied. These integrators, as there name suggests, use the exponential and often functions which are closely related to the exponential function inside the numerical method. Three main classes of exponential integrators, exponential linear multistep (multivalue), exponential Runge–Kutta (multistage) an...

In this paper, Homotopy perturbation method (HPM) has been applied to investigate the effect of magnetic field on Cu-water nanofluid flow in non-parallel walls. The validity of HPM solutions were verified by comparing with numerical results obtained using a fourth order Runge–Kutta method. Effects of active parameters on flow have been presented graphically. The results show that velocity i...

In this work we consider symplectic Runge Kutta Nyström (SRKN) methods with three stages. We construct a fourth order SRKN with constant coefficients and a trigonometrically fitted SRKN method. We apply the new methods on the two-dimentional harmonic oscillator, the Stiefel-Bettis problem and on the computation of the eigenvalues of the Schrödinger equation.

A B-spline finite element method is used to solve the equal width equation numerically. This approach involves a collocation method using quintic B-splines at the knot points as element shape. Time integration of the resulting system of ordinary differential equations is effected using the fourth order Runge–Kutta method, instead of the finite difference method, the resulting system of ordinary...

Galerkin fully discrete approximations for parabolic equations with time-dependent coefficients are analyzed. The schemes are based on implicit Runge-Kutta methods, and are coupled with preconditioned iterative methods to approximately solve the resulting systems of linear equations. It is shown that for certain classes of Runge-Kutta methods, the fully discrete equations exhibit parallel featu...

Despite the popularity of high-order explicit Runge–Kutta (ERK) methods for integrating semi-discrete systems of equations, ERK methods suffer from severe stability-based time step restrictions for very stiff problems. We implement a discontinuous Galerkin finite element method (DGFEM) along with recently introduced high-order implicit–explicit Runge–Kutta (IMEX-RK) schemes to overcome geometry...

In this study the centre manifold is applied for reduction and limit cycle calculation of a highly nonlinear structural aeroelastic wing. The limit cycle is arisen from structural nonlinearity due to the large deflection of the wing. Results obtained by different orders of centre manifolds are compared with those obtained by time marching method (fourth-order Runge-Kutta method). These comparis...

We introduce a novel local time-stepping technique for marching-in-time algorithms. The technique is denoted as Causal-Path Local Time-Stepping (CPLTS) and it is applied for two time integration techniques: fourth order low–storage explicit Runge–Kutta (LSERK4) and second order Leapfrog (LF2). The CPLTS method is applied to evolve Maxwell’s curl equations using a Discontinuous Galerkin (DG) sch...

In this paper, we consider exponential Runge-Kutta methods for the numerical pricing of options. The methods are shown to be an alternative to other existing procedures for the numerical valuation of jump -diffusion models. We show that exponential Runge-Kutta methods give unconditional second order accuracy for European call options under Merton's jump -diffusion model with constant coefficien...