We consider the construction of a special family of Runge–Kutta (RK) collocation methods based on intra-step nodal points of Chebyshev–Gauss–Lobatto type, with A-stability and stiffly accurate characteristics. This feature with its inherent implicitness makes them suitable for solving stiff initial-value problems. In fact, the two simplest cases consist in the well-known trapezoidal rule and th...

The flux reconstruction (FR) approach allows various well-known high-order schemes, such as collocation based nodal discontinuous Galerkin (DG) methods and spectral difference (SD) methods, to be cast within a single unifying framework. Recently, the authors identified a new class of FR schemes for 1D conservation laws, which are simple to implement, efficient and guaranteed to be linearly stab...

We investigate the long time behavior of the following efficient second order in time scheme for the 2D Navier-Stokes equation in a periodic box: 3ω − 4ω + ω 2k +∇(2ψ − ψ) · ∇(2ω − ω)− ν∆ω = f, −∆ψ = ω. The scheme is a combination of a 2nd order in time backward-differentiation (BDF) and a special explicit Adams-Bashforth treatment of the advection term. Therefore only a linear constant coeffic...

Abstract. We use generalized Gaussian quadratures for exponentials to develop a new ODE solver. Nodes and weights of these quadratures are computed for a given bandlimit c and user selected accuracy ǫ, so that they integrate functions e, for all |b| ≤ c, with accuracy ǫ. Nodes of these quadratures do not concentrate excessively near the end points of an interval as those of the standard, polyno...

in this paper, we introduce the new class of implicit l-stable generalized hybrid methods forthe numerical solution of first order initial value problems. we generalize the hybrid methodswith utilize ynv directly in the right hand side of classical hybrid methods. the numericalexperimentation showed that our method is considerably more efficient compared to wellknown methods used for the numer...

Abstract This paper describes an experiment that attempts to compare a range of existing collocation extraction techniques as well as the implementation of a new technique based on tests for lexical substitutability. After a description of the experiment details, the techniques are discussed with particular emphasis on any adaptations that are required in order to evaluate it in the way propose...

The present article is concerned with the numerical solution of Benjamin-BonaMahony-Burgers (BBM-Burger) equation by quartic B-spline collocation method. The method is based on quartic B-spline basis functions for space integration, and Crank-Nicolson formulation for time integration. Numerical examples considered by different researchers are discussed to illustrate the efficiency, robustness a...

a new two-step implicit linear obrechkoff twelfth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives is constructed in this paper. the purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial schrodinger equation and related problems. this algorithm belongs in the category of the multist...

We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) for the two-point boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of Tk). Super-geometric convergent rate is established for a special class of solutions.

We present optimal error estimates for spectral Petrov–Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov–Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates.