Volume 24, Issue 2

In the present paper, a numerical method is proposed for solving one-dimensional time dependent Burgers’ equation for various values of Reynolds number on a rectangular domain in the x - t plane. For large values of Reynolds number, a boundary layer produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a quasi-linear singularly perturbed parabolic problem. Classical numerical methods with uniform mesh are known to be inadequate to solve such problems, unless an unacceptably large number of mesh points are used. Therefore, in order to overcome this drawback associated to classical numerical methods with uniform mesh, we construct a numerical scheme that comprises of implicit Euler method to discretize in temporal direction on uniform mesh and a B-spline collocation approach to discretize the spatial variable with piecewise uniform Shishkin mesh. Quasi-linearization process is used to tackle the non-linearity and shown that quasi-linearization process converges quadratically. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. These bounds are applied in convergence analysis of the proposed method on Shishkin mesh. The method has been shown to be first order convergent in the temporal variable and almost second order accurate in the spatial variable. Higher accuracy and convergence of the method is demonstrated by numerical examples and an estimate of the error is given. Comparisons of numerical results are made with others available in the literature for both low as well as high Reynolds numbers.

In this paper some new methods for curvature approximation of circular arcs by low-degree Bézier curves are presented. Interpolation by geometrically continuous ( G 1 ) parametric polynomials is considered. All derived approximants are given explicitly and are therefore practically applicable. Moreover, obtained results indicate that G 1 biarcs with at least G 1 continuity at the junction have smaller curvature error as parametric polynomial counterparts of the same degree. It is also shown that all considered methods provide optimal asymptotic approximation order.

We present the error analysis of a particular Oldroyd-B type model with the limiting Weissenberg number going to infinity. Assuming a suitable regularity of the exact solution we study the error estimates of a standard finite element method and of a combined finite element/finite volume method. Our theoretical result shows first order convergence of the finite element method and the error of the order 𝓞( h 3/4 ) for the finite element/finite volume method. These error estimates are compared and confirmed by the numerical experiments.

In the current paper, we derive the a priori error analysis ( hp version) of the streamline diffusion DG finite element approximation for optimal distributed control problem governed by the first order linear hyperbolic equation. We present the stability of such method, obtain the a priori error upper bound for the state and the control approximation, and prove the convergence of numerical method. For the optimal control problem, these estimates are apparently not available in the literature.