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If we consider a sequence of warped product length spaces, what conditions on the sequence of warping functions imply compactness of the sequence of distance functions? In particular, we want to know when a subsequence converges to a well-defined metric space on the same manifold with the same topology. What conditions on the sequence of warping functions imply Lipschitz bounds for the sequence of distance functions and/or the limiting distance function? In this paper we give answers to both of these questions as well as many examples which elucidate the theorems and show that our hypotheses are necessary.

We study a uniform-in-time convergence from the discrete-time (in short, discrete) Cucker–Smale (CS) model to the continuous-time CS model, which is valid for the whole time interval, as time-step tends to zero. Classical theory yields the convergence results which are valid only in any finite-time interval. Our uniform convergence estimate relies on two quantitative estimates “asymptotic flocking estimate” and “uniform${\ell }_2$-stability estimate with respect to initial data”. In the previous literature, most studies on the CS flocking have been devoted to the continuous-time model with general communication weights, whereas flocking estimates have been done for the discrete-time model with special network topologies such as the complete network with algebraically decaying communication weights and rooted leaderships. For the discrete CS model with a regular and algebraically decaying communication weight, asymptotic flocking estimate has been extensively studied in the previous literature. In contrast, for a general decaying communication weight, corresponding flocking dynamics has not been addressed in the literature due to the difficulty of extending the Lyapunov functional approach to the discrete model. In this paper, we present asymptotic flocking estimate for the discrete model using the Lyapunov functional approach. Moreover, we present a uniform ${\ell }_2$-stability estimate of the solution for the discrete CS model with respect to initial data. We combine asymptotic flocking estimate and uniform stability to derive a uniform-in-time convergence from the discrete CS model to the continuous CS model, as time-step tends to zero.

In this article, a singularly perturbed reaction-diffusion problem with Robin boundary conditions, is considered. In general, the solution of this problem possesses boundary layers at both the ends of the domain. To solve this problem, we propose a numerical scheme, involving the cubic spline scheme for boundary conditions and the classical central difference scheme for the differential equation (DE) at the interior points. The grid is generated by the equidistribution of a positive monitor function. It has been proved that classical forward–backward approximation for mixed type boundary conditions, gives first-order convergence, whereas our proposed cubic spline scheme provides second-order accuracy independent of the perturbation parameter. Numerical experiments have been provided to validate the theoretical results.

In this paper, a coupled system of $m(\ge 2)$ second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.

The aim of this paper is to construct and analyze a fitted mesh finite difference method (FMFDM) for a class of time-dependent singularly perturbed convection–diffusion–reaction problems with a turning point. While such problems have been widely studied in the case of boundary layers, little has been achieved for interior layer problems. In addition, most of works have considered the coefficient functions as the functions of space only. In this work, we study problems where the coefficient functions are dependent on both space and time variables, and their solutions exhibit an interior layer. We derive the solution bounds and their derivatives. Then we use the classical implicit Euler method to discretize the time variable with a constant step-size. This method consists of two upwind finite difference schemes defined on an appropriate non-uniform mesh of Shishkin type. This mesh is fine near the turning point and coarse elsewhere. We prove that the method is almost first-order uniformly convergent with respect to the perturbation parameter. To improve the accuracy of the proposed method, we apply Richardson extrapolation technique. Numerical simulations are provided to confirm the theoretical findings.

In this paper, we provide numerical approaches for singularly perturbed convection–diffusion problem with a discontinuous source term. As a result of the discontinuity in the source term, the problem highlights an exponential-type boundary layer and a weak interior layer. To handle both the layers at the same time, we design a nonsymmetric discontinuous Galerkin finite element technique with interior penalties (NIPG). The problem is solved using the standard Shishkin mesh. In the DG norm, uniform error estimates are derived for the problem. To verify the theoretical estimates, numerical tests are carried out.

In this paper, the modified Burgers’ equation is considered. These kinds of problems come from the field of sonic boom and explosions theories. At big Reynolds’ number there is a boundary layer in the right side of the domain. From numerical point of view, the major difficulty in dealing with this type of problem is that the smooth initial data can give rise to solution varying regions i.e. boundary layer regions. To tackle this situation, we propose a numerical method on nonuniform mesh of Shishkin type, which works well at high as well as low Reynolds number. The proposed method comprises of Euler implicit scheme and hybrid scheme in time and space direction, respectively. First, we discretize the continuous problem in temporal direction by Euler implicit method, which yields a set of ode’s at each time level. The resulting set of differential equations are approximated by a hybrid scheme on Shishkin mesh i.e. upwind in regular region (nonboundary layer region) and central difference in boundary layer regions. The convergence of proposed method has been shown parameter uniform. Some numerical experiments have been carried out to corroborate the theoretical results.

The present paper deals with the approximation properties of generalization of Lupaş–Păltănea’s operators preserving exponential functions. We obtain moments using the concept of moment generating functions and establish a Voronovskaya type theorem, uniform convergence estimate and also an asymptotic formula in quantitative sense. In the end we present comparative study through graphical representation and propose an open problem.

This paper presents a numerical scheme based on a nonstandard finite difference method for solving the singularly perturbed time-dependent two-parameter problem. This model is a common occurrence in the theory of chemical flow reactors and boundary layers controlled by fluid suction. There are several numerical methods in the literature for solving singularly perturbed two-parameter problems with temporal dependence, but most of them are either of low order or else dependent on parameter. Here, we have constructed a numerical method based on Crank–Nicolson in time and by nonstandard finite difference (NSFD) in space. Consequently, the derived method is found to be higher-order accurate in space and time and also shows parameter uniformity. An extensive convergence analysis has been performed to demonstrate the method’s uniform convergence and the numerical answers back up the theoretical findings of the study.

In this paper, a second-order singularly perturbed initial value problem is considered. A hybrid scheme which is a combination of a cubic spline and a modified midpoint upwind scheme is proposed on various types of layer-adapted meshes. The error bounds are established for the numerical solution and for the scaled numerical derivative in the discrete maximum norm. It is observed that the numerical solution and the scaled numerical derivative are of second-order convergence on the layer-adapted meshes irrespective of the perturbation parameter. To show the performance of the proposed method, it is applied on few test examples which are in agreement with the theoretical results. Furthermore, existing results are also compared to show the robustness of the proposed scheme.

No abstract received.

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a quasilinear Sobolev or pseudo-parabolic equation with initial jump. We have derived a method based on using finite elements with piecewise linear functions in space and with exponential functions in time and appropriate quadrature formulae with remainder term in integral form. In the initial layer, we introduce a special non-uniform mesh which is constructed by using estimates of derivatives of the exact solution and the analysis of the local truncation error. For the time integration we use the implicit rule. The fully discrete scheme is shown to be convergent of order 2 in space and of order one in time, uniformly in the singular perturbation parameter. Numerical results supporting the theory are presented.

We consider families of life distributions with the first three moments belonging to small neighborhoods of respective powers of a positive number. For various shapes of the neighborhoods, we determine exact convergence rates of their Prokhorov radii to zero. This provides a refined evaluation of the effect of specific moment convergence on the weak one.

Let G be a finite smooth Jordan domain of the complex plane ℂ, z₀ ∈ G, and let