Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes–Darcy problem

We propose and study a combination of two second-order implicit–explicit (IMEX) methods for the coupled Stokes–Darcy system that governs flows in karst aquifers. The first is a second-order explicit two-step MacCormack scheme and the second is a second-order implicit Crank–Nicolson method. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. This combination so called the MacCormack rapid solver method is very efficient (faster, at least of second order accuracy in time and space) and can be easily implemented using legacy codes. Under time step limitation of the form $\Delta t\le Ch$ (where $h,$$\Delta t$ are mesh size and time step, respectively, and $C$ is a physical parameter) we prove both long time stability and the rate of convergence of the method. Some numerical experiments are presented and discussed.     

An iteration method for solving the linear system Ax=b

In this paper, based on a convergence splitting of the matrix $A$, we present an inner–outer iteration method for solving the linear system $Ax=b$. We analyze the overall convergence of this method without any other restriction on its parameters. Moreover, we give the accelerated inner–outer iteration method, and discuss how to apply the inner–outer iterations as a preconditioner for the Krylov subspace methods. The inner–outer iteration method can also be used for the solution of $AXB=C$. Finally, several numerical examples are given to validate the performance of our proposed algorithms.     

Trigonometric transform splitting methods for real symmetric Toeplitz systems

In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix $A$. Theoretical analyses show that if the generating function $f$ of the $n\times n$ Toeplitz matrix $A$ is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large $n$. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved.Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix $A$ is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.     

Orthogonal spline collocation method for the fourth-order diffusion system

The fourth-order diffusion systems depict the wave and photon propagation in intense laser beams and play an important role in the phase separation in binary mixture. In this paper, by using orthogonal spline collocation (OSC) method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of fourth-order fractional reaction–diffusion equations. For the original unknown $u$ and auxiliary variable $v=\Delta u$, the full-discrete unconditional stabilities based on a priori analysis are derived by virtue of properties of OSC. Moreover, the convergence rates in $L^2$-norm for unknown $u$ are strictly investigated. At the same time, the optimal error estimates in $H^1$-norm for unknown $u$ and in $L^2$-norm for variable $v$, are also derived, respectively. For further verifying the theoretical analysis, some numerical examples are provided.     

A spatially second order alternating direction implicit (ADI) method for solving three dimensional parabolic interface problems

A new matched alternating direction implicit (ADI) method is proposed in this paper for solving three-dimensional (3D) parabolic interface problems with discontinuous jumps and complex interfaces. This scheme inherits the merits of its ancestor for two-dimensional problems, while possesses several novel features, such as a non-orthogonal local coordinate system for decoupling the jump conditions, two-side estimation of tangential derivatives at an interface point, and a new Douglas–Rachford ADI formulation that minimizes the number of perturbation terms, to attack more challenging 3D problems. In time discretization, this new ADI method is found to be of first order and stable in various experiments. In space discretization, the matched ADI method achieves the second order accuracy based on simple Cartesian grids for various irregularly-shaped surfaces and spatial–temporal dependent jumps. Computationally, the matched ADI method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3D parabolic equations, in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom $N$, i.e., $O\left(N\right)$. Furthermore, unlike iterative methods, the ADI method is an exact or non-iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed $N$. Therefore, the proposed matched ADI method provides a very promising tool for solving 3D parabolic interface problems.     

A simple weak formulation for solving two-dimensional diffusion equation with local reaction on the interface

In this paper, we propose a numerical method for solving two-dimensional diffusion equation with nonhomogeneous jump condition and nonlinear flux jump condition located at the interface. We use finite element method coupled with Newton’s method to deal with the jump conditions and to linearize the system. It is easy to implement. The grid used here is body-fitting grids based on the idea of semi-Cartesian grid. Numerical experiments show that this method is nearly second order accurate in the $L^{\infty }$ norm.     

A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size $m$ and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order $m$. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order $m$, but no derivation for the queue length was provided in the general case.This paper introduces an order $m^2$ phase-type representation $(\kappa ,K)$ for the queue length distribution in the general case and proves that the order $m^2$ of the distribution cannot be further reduced in general. A matrix geometric representation $(\kappa ,K,\nu )$ is also established for the number of type $\tau ?\{1,\dots ,m\}$ customers in the system, where a customer is of type $\tau$ if the phase in which it completes service belongs to $\tau$. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute $(\kappa ,K,\nu )$. When the arrivals have a Markovian structure, the numerical procedure is reduced to solving a Quasi–Birth–Death (for the discrete time case) or fluid queue (for the continuous time case).Finally, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order $m$ phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the $R$ matrix of a Quasi–Birth–Death Markov chain.     

A fast algorithm for solving the space–time fractional diffusion equation

In this paper, we propose a fast algorithm for efficient and accurate solution of the space–time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian ${(?\Delta )}_s^{\alpha /2}$ results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix–vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method.     

On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations

In this article, a finite difference scheme for coupled nonlinear Schrödinger equations is studied. The existence of the difference solution is proved by Brouwer fixed point theorem. With the aid of the fact that the difference solution satisfies two conservation laws, the finite difference solution is proved to be bounded in the discrete $L_{\infty }$ norm. Then, the difference solution is shown to be unique and second order convergent in the discrete $L_{\infty }$ norm. Finally, a convergent iterative algorithm is presented.     

Superconvergence of local discontinuous Galerkin methods for one-dimensional convection–diffusion equations

In this paper, we study the convergence behavior of the local discontinuous Galerkin (LDG) methods when applied to one-dimensional time dependent convection–diffusion equations. We show that the LDG solution will be superconvergent towards a particular projection of the exact solution, if this projection is carefully chosen based on the convection and diffusion fluxes. The order is observed to be at least $k+2$ when piecewise $P^k$ polynomials are used. Moreover, the numerical traces for the solution are also superconvergent, sometimes, of higher-order. This is a continuation of our previous work [Cheng Y, Shu C-W. Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J Comput Phys 2008;227:9612–27], in which superconvergence of DG schemes for convection equations is discussed.     

Spectral-element discontinuous Galerkin lattice Boltzmann simulation of flow past two cylinders in tandem with an exponential time integrator

In this paper, a spectral-element discontinuous Galerkin (SEDG) lattice Boltzmann discretization and an exponential time-marching scheme are used to study the flow field past two circular cylinders in tandem arrangement. The basic idea is to discretize the streaming step of the lattice Boltzmann equation by using the SEDG method to get a system of ordinary differential equations (ODEs) whose exact solutions are expressed by using a large matrix exponential. The approximate solution of the resulting ODEs are obtained from a projection method based on a Krylov subspace approximation. This approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension. The exponential time integration scheme is useful especially when computations are carried out at high Courant–Friedrichs–Lewy ($CFL$) numbers, where most explicit time-marching schemes are inaccurate. Simulations of flow were carried out for a circular cylinder at $Re=20$ and for two circular cylinders in tandem at $Re=40$ and a spacing of $2.5D$, where $D$ is the diameter of the cylinders. We compare our results with those from a fourth-order Runge–Kutta scheme that is restricted by the $CFL$ number. In addition, important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme. These results are compared with results from our simulation using the RK scheme and with existing benchmark results.     

Two-grid method for two-dimensional nonlinear Schrödinger equation by mixed finite element method

A conservative two-grid mixed finite element scheme is presented for two-dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse-grid solution as the initial guess. Moreover, error estimates are conducted for the two-grid method. It is shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy $H=O\left(h^{\frac{1}{2}}\right)$ in the two-grid method. The numerical results show that this method is very effective.     

Traveling waves for epidemic models with nonlocal dispersal in time and space periodic habitats

This paper deals with the spatial spreading speed and traveling wave solutions of a general epidemic model with nonlocal dispersal in time and space periodic habitats. It should be mentioned that the existence of spreading speed and traveling wave solutions of nonlocal dispersal cooperative system in space–time periodic habitats have been established previously. In this paper, we further show that the epidemic system has a spreading speed $c^1\left(\xi \right)$ and for any $c>c^1\left(\xi \right)$, there exist a unique, continuous space–time periodic traveling wave solution $({\Phi }_1(x?ct\xi ,t,ct\xi ),{\Phi }_2(x?ct\xi ,t,ct\xi ))$ of epidemic model in the direction of $\xi$ with speed $c$, and there is no such solution for $c<c^1\left(\xi \right)$.     

A stabilized finite volume method for the evolutionary Stokes–Darcy system

In this paper, we propose a stabilized fully discrete finite volume method based on two local Gauss integrals for a non-stationary Stokes–Darcy problem. This stabilized method is free of stabilized parameters and uses the lowest equal-order finite element triples $P1\text{–}P1\text{–}P1$ for approximating the velocity, pressure and hydraulic head of the Stokes–Darcy model. Under a modest time step restriction in relation to physical parameters, we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements. Finally, a series of numerical experiments are provided to demonstrate the validity of the theoretical results.