The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1,0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

We consider the nonlinear and ill-posed inverse problem where the Robin coefficient in the Laplace equation is to be estimated using the measured data from the accessible part of the boundary. Two regularisation methods are considered — viz. L2 and H1 regularisation. The regularised problem is transformed to a nonlinear least squares problem; and a suitable regularisation parameter is chosen via the normalised cumulative periodogram (NCP) curve of the residual vector under the assumption of white noise, where information on the noise level is not required. Numerical results show that the proposed method is efficient and competitive.

We discuss an efficient preconditioner and iterative numerical method to solve large complex linear algebraic systems of the form (W + iT)u = c, where W and T are symmetric matrices, and at least one of them is nonsingular. When the real part W is dominantly stronger or weaker than the imaginary part T, we propose a block multiplicative (BM) preconditioner or its variant (VBM), respectively. The BM and VBM preconditioned iteration methods are shown to be parameter-free, in terms of eigenvalue distributions of the preconditioned matrix. Furthermore, when the relationship between W and T is obscure, we propose a new preconditioned BM method (PBM) to overcome this difficulty. Both convergent properties of these new iteration methods and spectral properties of the corresponding preconditioned matrices are discussed. The optimal value of iteration parameter for the PBM method is determined. Numerical experiments involving the Helmholtz equation and some other applications show the effectiveness and robustness of the proposed preconditioners and corresponding iterative methods.

Incompressible flows with zero Reynolds number can be modeled by the Stokes equations. When numerically solving the Stokes flow in stream-vorticity formulation with high-order accuracy, it will be important to solve both the stream function and velocity components with the high-order accuracy simultaneously. In this work, we will develop a fifth-order spectral/combined compact difference (CCD) method for the Stokes equation in stream-vorticity formulation on the polar geometries, including a unit disk and an annular domain. We first use the truncated Fourier series to derive a coupled system of singular ordinary differential equations for the Fourier coefficients, then use a shifted grid to handle the coordinate singularity without pole condition. More importantly, a three-point CCD scheme is developed to solve the obtained system of differential equations. Numerical results are presented to show that the proposed spectral/CCD method can obtain all physical quantities in the Stokes flow, including the stream function and vorticity function as well as all velocity components, with fifth-order accuracy, which is much more accurate and efficient than low-order methods in the literature.

In error estimates of various numerical approaches for solving decoupled forward backward stochastic differential equations (FBSDEs), the rate of convergence for one variable is usually less than for the other. Under slightly strengthened smoothness assumptions, we show that the fully discrete Euler scheme admits a first-order rate of convergence for both variables.

In this paper, we propose a new SSOR-like method with four parameters to solve the augmented system. And we analyze the convergence of the method and get the optimal convergence factor under suitable conditions. It is proved that the optimal convergence factor is the same as the GMPSD method [M.A. Louka and N.M. Missirlis, A comparison of the extrapolated successive overrelaxation and the preconditioned simultaneous displacement methods for augmented systems, Numer. Math. 131(2015) 517-540] with five parameters under the same assumption.

Model and parameter uncertainties are common whenever some parametric model is selected to value a derivative instrument. Combining the Monte Carlo method with the Smolyak interpolation algorithm, we propose an accurate efficient numerical procedure to quantify the uncertainty embedded in complex derivatives. Except for the value function being sufficiently smooth with respect to the model parameters, there are no requirements on the payoff or candidate models. Numerical tests carried out quantify the uncertainty of Bermudan put options and down-and-out put options under the Heston model, with each model parameter specified in an interval.

Simplified models like the shallow water equations (SWE) are commonly adopted for describing a wide range of free surface flow problems, like flows in rivers, lakes, estuaries, or coastal areas. In the literature, numerical methods for the SWE are mostly mesh-based. However, this macroscopic approach is unable to accurately represent the complexity of flows near coastlines, where waves nearly break. This fact prompted the idea of coupling the mesh-based SWE model with a meshless particle method for solving the Euler equations. In a previous paper, a method to couple the staggered scheme SWE and the smoothed particle hydrodynamics (SPH) Euler equations was developed and discussed. In this article, this coupled model is used for simulating solitary wave run-up on a sloping beach. The results show strong agreement with the experimental data of Synolakis. Simulations of wave overtopping over a seawall were also performed.

There has been a lot of study on the SOR-like methods for solving the augmented system of linear equations since the outstanding work of Golub, Wu and Yuan (BIT 41(2001)71-85) was presented fifteen years ago. Based on the SOR-like methods, we establish a class of accelerated SOR-like methods for large sparse augmented linear systems by making use of optimization technique, which will find the optimal relaxation parameter ω by optimization models. We demonstrate the convergence theory of the new methods under suitable restrictions. The numerical examples show these methods are effective.

The signal transduction pathway is the important process of communication of the cells. It is the dynamical interaction between the ligand-receptor complexes and an inhibitor protein in second messenger synthesis. The signaling molecules are detected and bounded by receptors, typically G-Protein receptors, across the cell membrane and that in turns alerts intracellular molecules to stimulate a response or a desired consequence in the target cells. In this research, we consider a model of the signal transduction process consisting of a system of three differential equations which involve the dynamic interaction between an inhibitor protein and the ligand-receptor complexes in the second messenger synthesis. We will incorporate a delay τ in the time needed before the signal amplification process can take effect on the production of the ligand-receptor complex. We investigate persistence and stability of the system. It is shown that the system allows positive solutions and the positive equilibrium is locally asymptotically stable under suitable conditions on the system parameters.

We construct modulus-based synchronous multisplitting iteration methods to solve a large implicit complementarity problem on parallel multiprocessor systems, and prove their convergence. Numerical results confirm our theoretical analysis and show that these new methods are efficient.

An adaptive finite element method is adopted to simulate the steady state coupled Schrödinger equations with a small parameter. We use damped Newton iteration to solve the nonlinear algebraic system. When the solution domain is elliptic, our numerical results with Dirichlet or Neumann boundary conditions are consistent with previous theoretical results. For the dumbbell and circular ring domains with Dirichlet boundary conditions, we obtain some new results that may be compared with future theoretical analysis.

In this paper we consider new perturbation bounds analysis of a kind of generalized saddle point systems. We provide perturbation upper bounds for the solutions of generalized saddle point systems, which extend the corresponding results in [W.-W. Xu, W. Li, New perturbation analysis for generalized saddle point systems, Calcolo., 46(2009), pp. 25-36] to more general cases.

In this paper, we consider a two-point boundary value problem with Caputo fractional derivative, where the second order derivative of the exact solution is unbounded. Based on the equivalent form of the main equation, a finite difference scheme is derived. The L∞ convergence of the difference system is discussed rigorously. The convergence rate in general improves previous results. Numerical examples are provided to demonstrate the theoretical results.

We investigate the stability and periodic orbits of a predator-prey model with harvesting. The model has a biologically-meaningful interior, an attractor undergoing damped oscillations, and can become destabilised to produce periodic orbits via a Hopf bifurcation. Some sufficient conditions for the existence of the Hopf bifurcation are established, and a stability analysis for the periodic solutions using a Lyapunov function is presented. Finally, some computer simulations illustrate our theoretical results.

We provide some computable error estimates in solving a nonsymmetric eigenvalue problem by general conforming finite element methods on general meshes. Based on the complementary method, we first give computable error estimates for both the original eigenfunctions and the corresponding adjoint eigenfunctions, and then we introduce a generalised Rayleigh quotient to deduce a computable error estimate for the eigenvalue approximations. Some numerical examples are presented to illustrate our theoretical results.

Mean curvature-based image registration model firstly proposed by Chumchob-Chen-Brito (2011) offered a better regularizer technique for both smooth and nonsmooth deformation fields. However, it is extremely challenging to solve efficiently this model and the existing methods are slow or become efficient only with strong assumptions on the smoothing parameter β. In this paper, we take a different solution approach. Firstly, we discretize the joint energy functional, following an idea of relaxed fixed point is implemented and combine with Gauss-Newton scheme with Armijo's Linear Search for solving the discretized mean curvature model and further to combine with a multilevel method to achieve fast convergence. Numerical experiments not only confirm that our proposed method is efficient and stable, but also it can give more satisfying registration results according to image quality.

A two-layer non-hydrostatic numerical model is proposed to simulate the formation of undular bores by tsunami wave fission. These phenomena could not be produced by a hydrostatic model. Here, we derived a modified Shallow Water Equations with involving hydrodynamic pressure using two layer approach. Staggered finite volume method with predictor corrector step is applied to solve the equation numerically. Numerical dispersion relation is derived from our model to confirm the exact linear dispersion relation for dispersive waves. To illustrate the performance of our non-hydrostatic scheme in case of linear wave dispersion and non-linearity, four test cases of free surface flows are provided. The first test case is standing wave in a closed basin, which test the ability of the numerical scheme in simulating dispersive wave motion with the correct frequency. The second test case is the solitary wave propagation as the examination of owing balance between dispersion and nonlinearity. Regular wave propagation over a submerged bar test by Beji is simulated to show that our non-hydrostatic scheme described well the shoaling process as well as the linear dispersion compared with the experimental data. The last test case is the undular bore propagation.

Multi-step modified Newton-HSS (MMN-HSS) methods, which are variants of inexact Newton methods, have been shown to be competitive for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices. Previously, we established these MMN-HSS methods under Lipschitz conditions, and we now present a semilocal convergence theorem assuming the nonlinear operator satisfies milder Hölder continuity conditions. Some numerical examples demonstrate our theoretical analysis.

The pricing model for American lookback options can be characterised as a two-dimensional free boundary problem. The main challenge in this problem is the free boundary, which is also the main concern for financial investors. We use a standard technique to reduce the pricing model to a one-dimensional linear complementarity problem on a bounded domain and obtain a corresponding variational inequality. The inequality is discretised by finite differences and finite elements in the temporal and spatial directions, respectively. By enforcing inequality constraints related to the options using Lagrange multipliers, the discretised variational inequality is reformulated as a set of semi-smooth equations, which are solved by a primal-dual active set method. One of the major advantages of our algorithm is that we can obtain the option values and the free boundary simultaneously, and numerical simulations show that our approach is as efficient as some other methods.