We consider a splitting method for the numerical solution of the regularized Kobayashi-Warren-Carter (KWC) system which describes the growth of single crystal particles of different orientations in two spatial dimensions. The KWC model is a system of two nonlinear parabolic PDEs representing gradient flows associated with a free energy in two variables. Based on an implicit time discretization by the backward Euler method, we suggest a splitting method and prove the existence as well as the energy stability of a solution. The discretization in space is taken care of by Lagrangian finite elements with respect to a geometrically conforming, shape regular, simplicial triangulation of the computational domain and requires the successive solution of two individual discrete elliptic problems. Viewing the time as a parameter, the fully discrete equations represent a parameter dependent nonlinear system which is solved by a predictor corrector continuation strategy with an adaptive choice of the time step size. Numerical results illustrate the performance of the splitting method.

This paper describes an application of weighted Laguerre polynomial functions to produce an unconditionally stable and accurate finite-difference scheme for the numerical solution of transient diffusion and convection-diffusion problems. The unconditionally stability of Laguerre-FDM (L-FDM) is guaranteed by expanding the time dependency of the unknown potential as a series of orthogonal functions in the domain (0, ∞), avoiding thus any time integration scheme. The L-FDM is a marching-on-in-degree scheme instead of traditional marching-on-in-time methods. For the two heat-transfer problems, we demonstrated the accuracy, numerical stability and computational efficiency of the proposed L-FDM by comparing its results against closed-form analytical solutions and numerical results obtained from classical finite-difference schemes as, for instance, the Alternating Direction Implicit (ADI).

A class of nonlinear complementarity problems are first reformulated into a series of equivalent implicit fixed-point equations in this paper. Then we establish a modulus-based synchronous multisplitting iteration method based on the fixed-point equation. Moreover, several kinds of special choices of the iteration methods including multisplitting relaxation methods such as extrapolated Jacobi, Gauss-Seidel, successive overrelaxation (SOR), and accelerated overrelaxation (AOR) of the modulus type are presented. Convergence theorems for these iteration methods are proven when the coefficient matrix $A$ is an $H_+$-matrix. Numerical results are also provided to confirm the efficiency of these methods in actual implementations.

This paper develops and analyses numerical approximation for linear-quadratic optimal control problems governed by elliptic interface equations. We adopt variational discretization concept to discretize optimal control problems, and apply an interface-unfitted finite element method due to [A. Hansbo and P. Hansbo. An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg., 191(47-48): 5537-5552, 2002] to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around interface are enriched into standard conforming finite element space. Optimal error estimates in both $L$² norm and a mesh-dependent norm are derived for the optimal state, co-state and control under different regularity assumptions. Numerical results verify the theoretical results.

Starting from the definition of an $n\times n$ $g$-Toeplitz matrix, $\!T_{n,g}\!\!\left(\!u\!\right)\!=\!\left[\!\widehat{u}_{rgs}\!\right]_{\!r,s=0}^{n-1}\!,$ where $g$ is a given nonnegative parameter, $\{\widehat{u}_{k}\}$ is the sequence of Fourier coefficients of the Lebesgue integrable function $u$ defined over the domain $\mathbb{T}=(-\pi,\pi]$, we consider the product of $g$-Toeplitz sequences of matrices $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\},$ which extends the product of Toeplitz structures $\{T_{n}(f_{1})T_{n}(f_{2})\},$ in the case where the symbols $f_{1},f_{2}\in L^{\infty}(\mathbb{T}).$ Under suitable assumptions, the spectral distribution in the eigenvalues sequence is completely characterized for the products of $g$-Toeplitz structures. Specifically, for $g\geq2$ our result shows that the sequences $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\}$ are clustered to zero. This extends the well-known result, which concerns the classical case (that is, $g=1$) of products of Toeplitz matrices. Finally, a large set of numerical examples confirming the theoretic analysis is presented and discussed.

In this paper, we consider a stochastic parabolic Volterra equation driven by the infinite dimensional fractional Brownian motion with Hurst parameter $H$ ∈ [$\frac{1}{2}$,1). We apply the piecewise constant, discontinuous Galerkin method to discretize this equation in the temporal direction. Based on the explicit form of the scalar resolvent function and the refined estimates for the Mittag-Leffler function, we derive sharp mean-square regularity results for the mild solution. The sharp regularity results enable us to obtain the optimal error bound of the time discretization. These theoretical findings are finally accompanied by several numerical examples.

Artificial bee colony (ABC) algorithm is one of the most popular swarm intelligence algorithms. Owing to its simpleness and effectiveness, it has been widely applied in many fields. Many modified versions of ABC algorithm were used to solve constrained optimization problems (COPs). This paper introduces an artificial bee colony algorithm based on multiobjective and nondominated solution replacement mechanism (MONABC) for solving COPs. This new approach presents four modifications on the foundation of the original ABC algorithm. The COPs are converted into unconstrained multiobjective optimization problems (MOPs), and the hybrid search mechanism of small population is applied in the employed bee phase. Moreover, nondominated solution replacement mechanism is devoted to updating the population. According to the dominating ability and feasibility, a new following probability formula based on the overall rank is proposed. In the scout bee phase, new archive and replacement mechanism will be constructed. To verify the performance of our approach, MONABC algorithm is tested on 24 and 18 well-known constrained problems from 2006 and 2010 IEEE Congress on Evolution Computation (CEC 2006 and 2010). The results indicate that MONABC is competitive with the state-of-the-art algorithms for solving COPs and MOPs.

In this paper, we consider a stochastic collocation method for the Maxwell equations with random inputs. We first verify the regularity of the solutions for the model problem, which depends on the random dielectric constant, random magnetic permeability and the initial and boundary data. Then the convergence of our numerical approach is proved. Further some numerical examples are presented to support the analysis.

Finite difference scheme for the variable coefficients subdiffusion equations with non-smooth solutions is constructed and analyzed. The spatial derivative is discretized on a uniform mesh, and $L$1 approximation is used for the discretization of the fractional time derivative on a possibly graded mesh. Stability of the proposed scheme is given using the discrete energy method. The numerical scheme is $\mathcal{O}$ ($N$^{−min{2−$α$,$rα$}}) accurate in time, where $α$ (0 < $α$ < 1) is the order of the fractional time derivative, $r$ is an index of the mesh partition, and it is second order accurate in space. Extension to multi-term time-fractional problems with nonhomogeneous boundary conditions is also discussed, with the stability and error estimate proved both in the discrete $l$²-norm and the $l$^∞-norm on the nonuniform temporal mesh. Numerical results are given for both the two-dimensional single and multi-term time-fractional equations.

Modulus-based matrix splitting iteration methods were recently proposed for solving implicit complementarity problems. In this paper, to solve a class of implicit complementarity problems, two-step modulus-based matrix splitting iteration methods are presented and analyzed. The convergence theorems are established when the system matrix is an $H$₊-matrix. Numerical results show that the proposed methods are efficient and can accelerate the convergence performance with less iteration steps and CPU time.

The Degasperis–Procesi equation is split into a system of a hyperbolic equation and an elliptic equation. For the hyperbolic equation, we use the high order finite difference WENO-Z scheme to approximate the nonlinear flux. For the elliptic equation, the wavelet collocation method is employed to discretize the high order derivative. Due to the combination of the WENO-Z reconstruction and the wavelet collocation, the splitting method shows an excellent ability in capturing the formation and propagation of shockpeakon solutions. The numerical simulations for different solutions of the Degasperis–Procesi equation are conducted to illustrate high accuracy and capability of the proposed method.

We propose a cubic spline rule for the calculation of the Hadamard finite-part integral on an interval. The error estimate is presented in theory, and the superconvergence result of the cubic spline rule for Hadamard finite-part integral is derived. When the singular point coincides with a prior known point, the convergence rate is one order higher than what is globally possible. Numerical experiments are given to demonstrate the efficiency of the theoretical analysis.

In this paper, we discuss the weak Galerkin (WG) finite element method for the obstacle problem and the second kind of the elliptic variational inequality. We use piecewise linear functions to approximate the exact solutions. The WG schemes for the first and the second kind of elliptic variational inequality are established and the well-posedness of the two schemes are proved. Furthermore, we can obtain the optimal order estimates in $H$¹ norm. Finally, some numerical examples are presented to confirm the theoretical analysis.

Here, reduction in acoustic noise due to insertion of thin reflecting barriers has been estimated. Accurate simulations have been performed using numerical methods with a near spectral resolving ability, neutral stability and the physical dispersion relation preserving ($DRP$) nature. Use of high accuracy schemes makes present approach useful in analysis of an acoustic field consisting of acoustic waves with a large frequency band. Present investigation has been carried out for eight geometrically different acoustic barriers and for nine discrete frequencies in the one-third octave band starting from 250 $Hz$ to 1600 $Hz$. Use of high accuracy $DRP$ scheme allows one to compute the complex acoustic field by accurately capturing reflection, diffraction and interference of acoustic waves. Comparison of numerically obtained sound pressure level ($SPL$) distributions at different heights with the available experimental results in the literature shows a good match for the case of $I$-shaped barrier. In order to quantify effects of acoustic source frequencies over a large band, we have extended our analysis for a frequency range of 100 $Hz$ to 5000 $Hz$ for the case of $I$-shaped barrier. Present analysis shows that the thick $T$-shaped barrier and grooved barrier work efficiently in the low and mid frequency range while in the higher frequency range all the barriers are equally effective due to minimal diffraction.

This paper presents an $hp$-version Chebyshev spectral collocation method for nonlinear Volterra integro-differential equations with weakly singular kernels. The $hp$-version error bound of the collocation method under the $H$¹-norm is established on an arbitrary mesh. Numerical experiments demonstrate the effectiveness of the proposed method.