The Stokeslet and stresslet kernels are commonly used in boundary element simulations and singularity methods for slow viscous flow. Evaluating the velocity induced by a collection of Stokeslets and stresslets by direct summation requires $\mathcal{O}(N^2)$ operations, where $N$ is the system size. The present work develops a treecode algorithm for 3D Stokeslets and stresslets that reduces the cost to $\mathcal{O}(N\log N)$. The particles are divided into a hierarchy of clusters and well-separated particle-cluster interactions are computed by a far-field Cartesian Taylor approximation. The terms in the approximation are contracted to promote efficient computation. Serial and parallel results display the performance of the treecode for several test cases. In particular, the method has relatively simple structure and low memory usage and this enhances parallel efficiency for large systems.

It is a common practice in industry to model the elasticity in flexible multibody dynamics, when the deformations are small, by means of a linear finite element approach and of a model condensation strategy. Taking into account the flexibility in multibody modelling may require computationally expensive numerical models to be managed. Proper shape functions are introduced in this paper to model the displacements of flexible slender beam components, without the need of any spatial discretization; a novel formulation of the flexible properties of beam-like components follows and a small size motion equation set can be obtained. Modelling aspects, from point location to constraint equations and to elastodynamic modelling, are discussed. An ideal quick return mechanism, properly actuated, is modelled as a test case to prove the effectiveness of the proposed approach.

In this article, we investigate the construction of a meshless local discrete collection method suitable for solving a class of boundary integral equations of the second kind with logarithmic singular kernels. These types of boundary integral equations can be deduced from boundary value problems of Laplace's equations with linear Robin boundary conditions. The numerical solution presented in the current paper is obtained by moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The logarithm-like singular integrals appeared in the method are computed via a particular nonuniform Gauss-Legendre quadrature rule. Since the offered scheme is based on the use of scattered points spread on the solution domain and does not require any background meshes, it can be identified as a meshless local discrete collocation (MLDC) method. We also obtain the error bound and the convergence rate of the presented method. The new technique is simple, efficient and flexible for most classes of boundary integral equations. The convergence accuracy of the new technique is examined over four integral equations on various domains and obtained results confirm the theoretical error estimates.

In this article, a quadratic finite volume method is applied to solve the nonlinear elliptic equation. Firstly, we construct a finite volume scheme for this nonlinear equation. Then, under certain assumptions, the boundedness and ellipticity of the corresponding bilinear form are obtained. Moreover, we get the optimal error estimates not only in $H^{1}$-norm but also in $L^{2}$-norm where the optimal error estimate in $L^{2}$-norm depends on the optimal dual partition. In addition, the effect of numerical integration is analyzed. To confirm the theoretical analysis, we solve the nonlinear equation by the Newton iteration method and prove the quadratic rate of convergence. The numerical results show the effectiveness of our method.

The paper proposes a general numerical framework to simplify a CAD model into a volume mesh model under reliable control of certain prescribed physical quantity that the designer is interested in. Different from previous work, the proposed approach does not assume that the candidate features have been detected and can directly generate the simplified volume mesh model. In addition, it can efficiently estimate the quantitative impact of each individual feature via solving a linear equation of small dimension less than 10. This is achieved by reformulating the problem as estimating the solution differences caused by different stiffness matrices, using the $combined$ $approximation$ approach. Performance of this approach is demonstrated via numerical 2D examples.

In this paper, we modify the weak Galerkin method introduced in [15] for Stokes equations. The modified method uses the $\mathbb{P}_k/\mathbb{P}_{k-1}$ $(k\geq1)$ discontinuous finite element combination for velocity and pressure in the interior of elements. Especially, the numerical traces ${v}_{hb}$ which are defined in the interface of the elements belong to the space $C^0(\mathcal{E}_h)$, this change leads to less degree of freedom for the resultant linear system. The stability, priori error estimates and $L^2$ error estimates for velocity are proved in this paper. In addition, we prove that the modified method also yields globally divergence-free velocity approximations and has uniform error estimates with respect to the Reynolds number. Finally, numerical results illustrate the performance of the method, support the theoretical properties of the estimator and show the efficiency of the algorithm.

A new simple and robust type of finite difference well-balanced weighted essentially non-oscillatory (WENO) schemes is designed for solving the one- and two-dimensional shallow water equations with or without source terms on structured meshes in this paper. Compared with the classical WENO schemes [5] in this field, the set of linear weights of these new WENO schemes could be chosen arbitrarily with one constraint that their summation equals one, maintain the optimal order of accuracy in smooth regions and keep essentially non-oscillatory property in non-smooth regions. For the shallow flow problems with smooth or discontinuous bed, we combine with the well-balanced procedure for balancing the flux gradients and the source terms and then these new WENO schemes with any set of linear weights will satisfy the exact C-property for still stationary solutions and maintain the other advantages of other high-order WENO schemes at the same time. Some benchmark numerical examples are performed to obtain high-order accuracy in smooth regions, keep exact C-property, sustain good convergence property for some steady-state problems and show sharp shock transitions by such new type of finite difference WENO schemes.

In this paper, a class of two-dimensional Riesz space-fractional diffusion equations (2D-RSFDE) with homogeneous Dirichlet boundary conditions is considered. In order to reduce the computational complexity, the alternating direction implicit Crank-Nicholson (ADI-CN) method is applied to reduce the two-dimensional problem into a series of independent one-dimensional problems. Based on the fact that the coefficient matrices of these one-dimensional problems are all real symmetric positive definite Toeplitz matrices, a fast method is developed for the implementation of the ADI-CN method. It is proved that the ADI-CN method is uniquely solvable, unconditionally stable and convergent with order $\mathcal{O}(\tau^{2}+h_{x}^{2}+h_{y}^{2})$ in the discrete $L_{\infty}$-norm with time step $\tau$ and mesh size $h_{x},$ $h_{y}$  in the $x$ direction and the $y$ direction, respectively. Finally, several numerical results are provided to verify the theoretical results and the efficiency of the fast method.

In $R\&D$ investment projects, the value and the flexibility of investments are investigated by real option models, while the competition among the investors are mainly analyzed by the game theory. In this paper, we propose an option game model to analyze the case in which two firms invest the same project during a period of time. The firm taking the initiative, the leader, achieves more market shares, whereas the follower gains some investment information from the leader. We apply the fitted finite volume method and the power penalty method in order to get the solution of the option game model. Several managerial results including the payoffs and the effects of parameters are also discussed. Finally, by comparing the compound American option with the European one, we find the former more suitable for the analysis of $R\&D$ investment projects.

Vendor Managed Inventory (VMI) model coordinates the supply chain, for the vendor is authorized to manage retailer's inventory and revenue-sharing contract helps to achieve the coordination. In this paper, a multi-period two-echelon supply chain coordination model, which is composed of a manufacturer and a retailer, is built. Under the assumption that the market demand rate is affected by time, sales price and the current inventory level, the supply chain profit models of centralized decision and decentralized decision based on VMI model are established respectively, with optimal decisions obtained in each model. Furthermore, the revenue sharing contract is used to synchronize the supply chain in the VMI model and in turn the analytical model in the supply chain to achieve Pareto improvement is obtained. An equation describing thin film oscillating is used to correct the profit value in VMI model to simulate the reality. Finally, a numerical illustration validating the model and analytical results showing parameters' varying effects on the profit and optimal quantity of each party, are presented. The results demonstrate that the revenue sharing contract is able to simultaneously improve the interests of parties in the multi-period VMI supply chain.