In this paper we propose and analyze a backward differentiation formula (BDF) type numerical scheme for the Cahn-Hilliard equation with third order temporal accuracy. The Fourier pseudo-spectral method is used to discretize space. The surface diffusion and the nonlinear chemical potential terms are treated implicitly, while the expansive term is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont regularization term, in the form of $−A_0\Delta t^2\Delta_N (\phi^{n+1}−\phi^n),$ is added in the numerical scheme. In particular, the energy stability is carefully derived in a modified version, so that a uniform bound for the original energy functional is available, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform-in-time $L^6_N$ bound of the numerical solution is obtained. And also, the optimal rate convergence analysis and error estimate are provided, in the $L^∞_{∆t} (0, T ;L^2 _N) ∩ L^2_{∆t} (0,T; H^2_h)$ norm, with the help of the $L^6_N$ bound for the numerical solution. A few numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence.

Symmetric and symplectic methods are classical notions in the theory of numerical methods for solving ordinary differential equations. They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space. This article is mainly concerned with the symmetric-adjoint and symplectic-adjoint Runge-Kutta methods as well as their applications. It is a continuation and an extension of the study in [14], where the authors introduced the notion of symplectic-adjoint method of a Runge-Kutta method and provided a simple way to construct symplectic partitioned Runge-Kutta methods via the symplectic-adjoint method. In this paper, we provide a more comprehensive and systematic study on the properties of the symmetric-adjoint and symplectic-adjoint Runge-Kutta methods. These properties reveal some intrinsic connections among some classical Runge-Kutta methods. Moreover, those properties can be used to significantly simplify the order conditions and hence can be applied to the construction of high-order Runge-Kutta methods. As a specific and illustrating application, we construct a novel class of explicit Runge-Kutta methods of stage 6 and order 5. Finally, with the help of symplectic-adjoint method, we thereby obtain a new simple proof of the nonexistence of explicit Runge-Kutta method with stage 5 and order 5.

We propose several immersed interface hybridized difference methods (IHDMs), combined with the Crank-Nicolson time-stepping scheme, for parabolic interface problems. The IHDM is the same as the hybrid difference method away from the interface cells, but the finite difference operators on the interface cells are modified to maintain the same accuracy throughout the entire domain. For the modification process, we consider virtual extensions of two sub-solutions in the interface cells in such a way that they satisfy certain jump equations between them. We propose several different sets of jump equations and their resulting discrete methods for one- and two-dimensional problems. Some numerical results are presented to demonstrate the accuracy and robustness of the proposed methods.

In this paper, we study the numerical solution of the time-fractional telegraph equation on the unbounded domain. We first introduce the artificial boundaries $Γ_±$ to get a finite computational domain. On the artificial boundaries $Γ_±$, we use the Laplace transform to construct the exact artificial boundary conditions (ABCs) to reduce the original problem to an initial-boundary value problem on a bounded domain. In addition, we propose a finite difference scheme based on the $\mathcal{L}1$_$2$ formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem. In order to reduce the effect of unsmoothness of the solution at the initial moment, we use a fine mesh and low-order interpolation to discretize the solution near $t = 0$. Finally, some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results.

We consider the computation of a nonlocal Helmholtz equation by using perfectly matched layer (PML). We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form. After that, we give stability estimates of some weighted-average values of the nonlocal Helmholtz solution and prove that (i) the weighted-average value of the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the weighted-average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain. Particularly for a typical kernel function $\gamma_1(s) =\frac{1}{2}e^{−|s|}$ , we obtain the Green’s function of the nonlocal Helmholtz equation, and use the Green’s function to further prove that (i) the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the nonlocal Helmholtz solution itself decays exponentially outside some domain. Based on our theoretical analysis, the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems. Finally, numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.

In this paper, we propose new algorithms for multiplicative noise removal based on the Aubert-Aujol (AA) model. By introducing a constraint from the forward model with an auxiliary variable for the noise, the NEMA (short for Noise Estimate based Multiplicative noise removal by alternating direction method of multipliers (ADMM)) is firstly given. To further reduce the computational cost, an additional proximal term is considered for the subproblem with regard to the original variable, the NEMA$_f$ (short for a variant of NEMA with fully splitting form) is further proposed. We conduct numerous experiments to show the convergence and performance of the proposed algorithms. Namely, the restoration results by the proposed algorithms are better in terms of SNRs for image deblurring than other compared methods including two popular algorithms for AA model and three algorithms of its convex variants.

In this paper, we propose a novel PDE-based model for the multi-phase segmentation problem by using a complex version of Cahn-Hilliard equations. Specifically, we modify the original complex system of Cahn-Hilliard equations by adding the mean curvature term and the fitting term to the evolution of its real part, which helps to render a piecewisely constant function at the steady state. By applying the K-means method to this function, one could achieve the desired multiphase segmentation. To solve the proposed system of equations, a semi-implicit finite difference scheme is employed. Numerical experiments are presented to demonstrate the feasibility of the proposed model and compare our model with other related ones.

The Kaczmarz algorithm is a common iterative method for solving linear systems. As an effective variant of Kaczmarz algorithm, the greedy Kaczmarz algorithm utilizes the greedy selection strategy. The two-subspace projection method performs an optimal intermediate projection in each iteration. In this paper, we introduce a new greedy Kaczmarz method, which give full play to the advantages of the two improved Kaczmarz algorithms, so that the generated iterative sequence can exponentially converge to the optimal solution. The theoretical analysis reveals that our algorithm has a smaller convergence factor than the greedy Kaczmarz method. Experimental results confirm that our new algorithm is more effective than the greedy Kaczmarz method for coherent systems and the two-subspace projection method for appropriate scale systems.

The discontinuous Galerkin method by divergence-free patch reconstruction is proposed for Stokes eigenvalue problems. It utilizes the mixed finite element framework. The patch reconstruction technique constructs two categories of approximation spaces. Namely, the local divergence-free space is employed to discretize the velocity space, and the pressure space is approximated by standard reconstruction space simultaneously. Benefit from the divergence-free constraint; the identical element patch serves two approximation spaces while using the element pair $\mathbb{P}^{m+1}/ \mathbb{P}^m$. The optimal error estimate is derived under the inf-sup condition framework. Numerical examples are carried out to validate the inf-sup test and the theoretical results.

We propose a variant modified skew-normal splitting iterative method to solve a class of large sparse non-Hermitian positive definite linear systems. Applying the preconditioning technique we also construct the preconditioned version of the proposed method. Theoretical analysis shows that the proposed method is unconditionally convergent even when the real part and the imaginary part of the coefficient matrix are non-symmetric. Meanwhile, when the real part and the imaginary part of the coefficient matrix are symmetric positive definite, we prove that the preconditioned variant modified skew-normal splitting iterative method will also unconditionally converge. Numerical experiments are presented to illustrate the efficiency of the proposed method and show better performance of it when compared with some other methods.

It is known that every tensor has an associated semi-symmetric tensor. The purpose of this paper is to investigate the shared properties of a tensor and its semi-symmetric form. In particular, a corresponding semi-symmetric tensor has smaller Frobenius norm under some conditions and can be used to get smaller bounds for eigenvalues and solutions of dynamical systems and tensor complementarity problems. In addition, every tensor has the same eigenvalues as its corresponding semi-symmetric form, also a corresponding semi-symmetric tensor inherits properties like being circulant, Toeplitz, $Z$-tensor, $M$-tensor, $H$-tensor and some others. Also, there are a two-way connection for properties like being positive definite, $P$-tensor, semi-positive, primitive and several others.