In this paper, we obtain the first error estimate in $L^2$-norm for the classical two-grid method, then design an improved two-grid method by adding one more correction on the coarse space to the classical two-gird method. Furthermore, we also present the error estimates in both $L^2$-norm and $H^1$-norm for the improved two-grid method. Especially, the $L^2$ error estimate of the improved two-grid method is one order higher than that of the classical two-grid. At last, we confirm and illustrate the theoretical result by some numerical experiments.

We consider a distributed optimal control problem governed by an elliptic PDE, and propose an embedded discontinuous Galerkin (EDG) method to approximate the solution. We derive optimal a priori error estimates for the state, dual state, and the optimal control, and suboptimal estimates for the fluxes. We present numerical experiments to confirm our theoretical results.

In this paper, using modified couple stress theory, a new cylindrical shell element is introduced. Here, using modified couple stress theory in place of classical continuum theory, and using shell model in place of beam model which has been used in previous research, vibrational behavior and buckling behavior of nanotubes is investigated via the finite element method. In addition, the new cylindrical shell element is defined, the mass-stiffness matrix is developed, and the use of the super element and size-dependent finite element formulation together with shell element is extended to more precisely account for nanotube vibration and buckling. The new cylindrical shell element has been developed based on super element's shape function. In special cases, in order to investigate the application of the equations developed, the cylindrical nano-shell bending displacement is studied using modified couple stress cylindrical shell element and the results are validated using the analytical method. In addition, the effects of parameters such as length scale parameter, length, and thickness on cylindrical shell displacement are investigated. The results have indicated using the super element considerably reduces the amount of computations and analysis time in comparison with the use of molecular dynamic simulation.

New stochastic Runge-Kutta (SRK) methods for solving the Itô and Stratonovich stochastic differential equations (SDEs) with small noises are introduced. These SRK methods contain some multiple stochastic integrals simulated easily and have high global mean-square error accuracy. To simplify the calculation process, the stochastic rooted tree analysis is developed to estimate the local error and the global mean-square error estimate for a general class of SRK methods is given. Various improved SRK methods for the Itô or Stratonovich SDEs with non-commutative, commutative, diagonal, scalar, additive or colored small noises are proposed in turn. Finally, the proposed new SRK methods are examined by four test equations and all of the numerical results show the high efficiency of our methods.

In this paper we show a fourth-order compact split-step finite difference method to solve the two- and three-dimensional nonlinear Schrödinger equations. The conservation properties and stability are analyzed for the proposed scheme. Numerical results show that the method can provide accurate and stable solutions for the nonlinear Schrödinger equation.

In this paper, we propose a new approach for selecting suitable shape parameters of radial basis functions (RBFs) in the context of the localized method of approximated particular solutions. Traditionally, there are no direct connections on choosing good shape parameters and choosing interior and boundary nodes using the local collocation methods. As a result, the approximations of derivative functions are less accurate and the stability is also an issue. One of the focuses of this study is to select the interior and boundary nodes in a special way so that they are correlated. Furthermore, a test differential equation with known exact solution is selected and a good shape parameter for the given differential equation can be selected through a good shape parameter for the test differential equation. Three numerical examples, including a Poison's equation and an eigenvalue problem, are tested. Uniformly distributed node arrangement is compared with the proposed cross knot distribution with Dirichlet boundary conditions and mixed boundary conditions. The numerical results show some potentials for the proposed node arrangements and shape parameter selections.

This study presents an alternative meshless boundary collocation approach, the singular boundary method (SBM), in conjunction with the image concept for simulating SH wave scattering by the canyon topography. First the image concept is implemented to extend the original semi-infinite domain problems to infinite domain problems. Then the SBM has been used to solve the infinite domain problems. In the SBM it employs the singular fundamental solutions as basis functions and introduces the concept of source intensity factor to regularize the singularities of the related fundamental solutions, which avoids singular numerical integrals in the boundary element method. To demonstrate the effectiveness of the proposed approach for SH wave scattering by the canyon topography, several benchmark examples are considered. The present results are compared with the analytical solutions, the null-field boundary integral equation method.

One of the flash analytic approximate methods of nowadays for the solution of highly nonlinear algebraic or differential equations arising from the mathematical modelling of industrial and technological applications is the homotopy analysis method (HAM). The success of the HAM is mainly due to a so-called convergence control parameter, $h$, plugged into the system externally which is missing in other competing methods. A simple algorithm to determine this parameter is introduced in this paper, besides the well-known approaches of constant $h$−level curves, the squared residual error and the recent ratio technique. Comparison of the four approaches yields nearly the same convergence control parameters with the advantage of the newly proposed approach in terms of its simplicity and less CPU time requirement. Moreover, a convergence accelerating method is suggested here based on updating the initial guess of the solution at some low-order homotopy series approximation of the solution. It appears to extend the interval of convergence control parameter. The provided examples of real life phenomena in combination with this technique demonstrate a successful improvement over the classical HAM method.

A (3+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation is considered, which can be used to describe many nonlinear phenomena in plasma physics. By virtue of binary Bell polynomials, a bilinear representation of the equation is succinctly presented. Based on its bilinear formalism, we construct soliton solutions and Riemann theta function periodic wave solutions. The relationships between the soliton solutions and the periodic wave solutions are strictly established and the asymptotic behaviors of the Riemann theta function periodic wave solutions are analyzed with a detailed proof.

This work is devoted to performing systematic sensitivity analysis of different turbulence models and various incoming wind conditions in predicting the wake flow behind a horizontal-axis wind turbine represented by an actuator disc (AD). The tested turbulence models are the standard $k−є$ model and the Reynolds Stress Model (RSM). Employing each turbulence model, the wind turbine immersed in four inflow conditions, including both uniform and non-uniform ones, is numerically studied. Simulation results are validated against Sexbierum field experimental data. Comparisons show that $k−є$ model is much more sensitive to the employed inflow conditions, with simulated wake velocity and turbulence profiles strongly differ from one condition to another; among them, a uniform TI & Length scale condition delivers the most accurate predictions. By contrast, comparisons identify that RSM is less sensitive to the inflow condition implemented, and the results under all inflow conditions are consistently in fair match with the measurements; the RSM is found to be more robust for capturing wake behavior reliably when using the AD/RANS approach.

Carbonate reservoir is one of the important reservoirs in the world. Because of the characteristics of carbonate reservoirs, horizontal well and acid fracturing have become a key technology for efficiently developing carbonate reservoir. Establishing corresponding mathematical models and analyzing transient pressure behaviors of this type of well-reservoir configuration can provide a better understanding of fluid flow patterns in formation as well as estimations of important parameters.
A mathematical model for an oil-water two-phase flow fractured horizontal well in triple media carbonate reservoir by conceptualizing vugs as spherical shapes is presented in this article. A semi-analytical solution is obtained in the Laplace domain by using source function theory, Laplace transformation and superposition principle. Analysis of transient pressure responses indicates that nine characteristic flow periods of fractured horizontal wells in triple media carbonate reservoir can be identified. Parametric analysis shows that water saturation of matrix, vug and fracture system and fracture half-length, fracture number and fracture spacing can significantly influence the transient pressure responses of fractured horizontal wells in triple media carbonate reservoir. The model presented in this article can be applied to obtain important parameters pertinent to reservoir or fracture by type curve matching and it can also provide useful information for optimizing fracture parameters.

Using a separable Buchwald representation in cylindrical coordinates, we show how under certain conditions the coupled equations of motion governing the Buchwald potentials can be decoupled and then solved using well-known techniques from the theory of PDEs. Under these conditions, we then construct three parametrized families of particular solutions to the Navier-Lamé equation in cylindrical coordinates. In this paper, we specifically construct solutions having 2π-periodic angular parts. These particular solutions can be directly applied to a fundamental set of linear elastic boundary value problems in cylindrical coordinates and are especially suited to problems involving one or more physical parameters. As an illustrative example, we consider the problem of determining the response of a solid elastic cylinder subjected to a time-harmonic surface pressure that varies sinusoidally along its axis and we demonstrate how the obtained parametric solutions can be used to efficiently construct an exact solution to this problem. We also briefly consider applications to some related forced-relaxation type problems.