The Ambrosio-Tortorelli functional is a phase-field approximation of the Mumford-Shah functional that has been widely used for image segmentation. It has the advantages of being easy to implement, maintaining the segmentation ability, and Γ-converging to the Mumford-Shah functional as the regularization parameter goes to zero. However, it has been observed in actual computation that the segmentation ability of the Ambrosio-Tortorelli functional varies significantly with different values of the parameter and it even fails to Γ-converge to the original functional for some cases. In this paper we present an asymptotic analysis on the gradient flow equation of the Ambrosio-Tortorelli functional and show that the functional can have different segmentation behaviors for small but finite values of the regularization parameter and eventually loses its segmentation ability as the parameter goes to zero when the input image is treatd as a continuous function. This is consistent with the existing observation as well as the numerical examples presented in this work. A selection strategy for the regularization parameter and a scaling procedure for the solution are devised based on the analysis. Numerical results show that they lead to good segmentation of the Ambrosio-Tortorelli functional for real images.

In this paper, we propose the Uzawa splitting iteration method for solving a class of singular saddle point problems. The semi-convergence of the Uzawa splitting iteration method is carefully analyzed, which shows that the iteration sequence generated by this method converges to a solution of the singular saddle point problems under certain conditions. Moreover, the characteristics of the eigenvalues and eigenvectors of the iteration matrix of the proposed method are studied. The theoretical results are supported by the numerical experiments, which implies that Uzawa splitting iteration method is effective and feasible for solving singular saddle point problems.

In this paper, based on the recursive algorithm of the non-tensor-product-typed bivariate divided differences, the bivariate polynomial interpolation is reviewed firstly. And several numerical examples show that the bivariate polynomials change as the order of the ortho-triples, although the interpolating node collection is invariant. Moreover, the error estimation of the bivariate interpolation is derived in several cases of special distributions of the interpolating nodes. Meanwhile, the high order bivariate divided differences are represented as the values of high order partial derivatives. Furthermore, the operation count approximates $\mathcal{O}(n^2)$ in the computation of the interpolating polynomials presented, including the operations of addition/substraction, multiplication, and division, while the operation count approximates $\mathcal{O}(n^3)$ based on radial basis functions for sufficiently large $n$.

We consider an approximation of second-order hyperbolic interface problems by partially penalized immersed finite element methods. In order to penalize the discontinuity of IFE functions, we add some stabilization terms at interface edges. Some semi-discrete and fully discrete schemes are presented and analyzed. We prove that the approximate solutions have optimal convergence rate in an energy norm. Numerical results not only validate the theoretical error estimates, but also indicate that our methods have smaller point-wise error over interface elements than classical IFE methods.

Multi-dimensional coupled Burgers' equations are important nonlinear partial differential equations arising in fluid mechanics. Developing high-order and efficient numerical methods for solving the Burgers' equation is essential in many real applications since exact solutions can not be obtained generally. In this paper, we seek a high-order accurate and efficient numerical method for solving multi-dimensional coupled Burgers' equations. A linearized combined compact difference (CCD) method together with alternating direction implicit (ADI) method is proposed. The CCD-ADI method is sixth-order accuracy in space variable and second-order accuracy in time variable. The resulting linear system at each ADI computation step corresponds to a block-tridiagonal system which can be effectively solved by the block Thomas algorithm. Fourier analysis shows that the method is unconditionally stable. Numerical experiments including both two-dimensional and three-dimensional problems are conducted to demonstrate the accuracy and efficiency of the method.

In this paper, a high order compact difference scheme has been developed for second order wave equations with piecewise discontinuous coefficients. The idea presented here can be used to solve a wide variety of hyperbolic models for nonhomogenous inner structures. Thermal wave model of bio-heat transfer in a multilayered skin structure with different thermal and physical properties is investigated subject to constant, linear, exponential and sinusoidal heating. The success of the new procedure is demonstrated by solving test problem as well as by application to the triple layered skin structure.

In this paper, a Legendre Galerkin Chebyshev collocation method for the Burgers-like equations with fractional nonlinear term and diffusion term is developed. This method is based on the Legendre-Galerkin variational form, but the nonlinear term and the right-hand term are treated by Chebyshev-Gauss interpolation. Rigorous stability and convergence analysis are developed. Numerical examples are shown to demonstrate the accuracy, stability and effectiveness of this method.

In this paper, we study the convergence and stability properties of explicit exponential general linear methods for delay differential equations. We prove that, under some assumptions, for delay differential equations in Banach spaces, these numerical methods converge essentially with the order min {$P, Q+1$}, where $P$ and $Q$ denote the order and stage order of the methods for ordinary differential equations, respectively. By using an interpolation procedure for the delay term, we analyze the linear and nonlinear stability of exponential general linear methods for two classes of delay differential equations. The sufficient conditions on the stability of exponential general linear methods for the testing delay differential equations are provided. Several numerical experiments are given to demonstrate the conclusions.

In this article, we present a new method for selecting a base that corresponds to the modified left shift operator in operational calculus. The method is illustrated by Emden-Fowler equation and differential equations with variable coefficients. The method, combined with Pade approximant, is also applied to solve a differential-difference equation which was solved by the Adomian decomposition method. Since the new method does not involve integrals, it is more efficient than the one in the literature.

This paper presents a superconvergence analysis of the finite volume method for Stokes problems using the $P_1$ – $P_1$ velocity-pressure element pair. Based on some superclose estimates on the interpolation function, we derive a superconvergence result of rate $\mathcal{O}(h^{\frac{3}{2}})$ for the post-processed velocity approximation in the $H^1$-norm and for the directly computed pressure approximation in the $L_2$-norm, respectively. Numerical experiments are provided to illustrate the theoretical analysis.