In order to improve the resolution power of the hybrid cell-edge and cell-node dissipative compact schemes (HDCS), a series of multiderivative combined dissipative compact schemes (MDCS) have been proposed in this paper. The design concept of the HDCS has been followed to develop the MDCS satisfying the geometric conservation law (GCL) and possessing inherent dissipation. Evaluations of multiple derivatives are included in the MDCS for the purpose of increasing the scheme resolution. The performance of the MDCS is evaluated by theoretical analysis and numerical tests. The multiple derivatives demonstrate their capability in significantly improving resolution power of the MDCS. A MDCS can achieve much higher resolution power than a HDCS with the same order of accuracy. Based on the solutions of the transition and turbulence decay in three-dimensional Taylor-Green vortex, a fifth-order MDCS with three tri-diagonal operators is recommended. This MDCS has better performance than the fifth-order HDCS in resolving multiple-scales turbulent structures.

The multiderivative combined dissipative compact scheme (MDCS) is extended to implement applications on complex curvilinear meshes. According to our previous evaluation on the scheme, a fifth-order MDCS, which has coexistent superior resolution power and relatively high efficiency, is chosen to present the performance of the MDCS. The capability of the fifth-order MDCS is evaluated by increasingly complex meshes in three typical tests: acoustic scattering from two cylinders, flow over a rod-airfoil configuration and flow over a landing gear model. On the curvilinear meshes, high resolution power possessed by the representative fifth-order MDCS is demonstrated for resolving acoustic wave. Moreover, the MDCS presents promising capability in simulating multiple scales in turbulent flow.

The multi-physical transport phenomena through the different size geometries are studied by developing a general local grid refinement approach for lattice Boltzmann methods. Revisiting the method of local fine patches on the coarse grid, through the Chapman-Enskog expansion, the multi-physicochemical source terms such as ion electro-migration, heat source, electric body force, and free net electric charge density can be rigorously incorporated to the rescaling relations of the distribution functions, which interchange between fine and coarse grids. We propose two general local refinement approaches for lattice Boltzmann for momentum and advection-diffusion equations with source terms. To evaluate our algorithm, (i) a body-force driven Poiseuille flow in a channel; (ii) an electro-osmotic flow in which the coupled Poisson-Nernst-Planck with Navier-Stokes equations for overlapped and non-overlapped electric double layers; (iii) a symmetric and asymmetric 1D and 2D heat conduction with heat generation in a flat plate; and (iv) an electric potential distribution near a charged surface, are modeled numerically. Good agreements with the available analytical solutions demonstrated the robustness of the proposed algorithm for diffusion or advection-diffusion equations, which may be coupled or decoupled. The present model may broaden the applications of local grid refinement for modeling complex transport phenomena, such as multi-physicochemical transport phenomena in different size geometries.

Considering a fractional integro-differential equation involving a general form of Hilfer fractional derivative with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation, we also prove the existence, uniqueness of solutions and Ulam-Hyers stability of this problem by employing a variety of tools of fractional calculus including Banach fixed point theorem. An example is provided to illustrate our main results.

In this paper, we propose a weak Galerkin finite element method (WG) for solving the stationary incompressible Stokes equation in two or three dimensional space. The weak Galerkin finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. However, since additional variables are introduced, the computational cost is much higher. Our new method can significantly reduce the computational cost  while maintaining the accuracy. Optimal error orders are established for the weak Galerkin finite element approximations in various norms. Some numerical results are presented to demonstrate the efficiency of the method.

In this report, a partitioned time stepping algorithm for Navier Stokes/Darcy model is analyzed. This method requires only solving one, uncoupled Navier Stokes and Darcy problems in two different sub-domains respectively per time step. On the interface, the simplified Beavers-Joseph-Saffman conditions are imposed with an additional assumption ${\bf u}\cdot {\bf n}_f>0$ (not hold for general case but still in many situation, such as the gentle river). Under a modest time step restriction of the form $\Delta  t\leq C$, where $C=C$ (physical parameters), we prove stability of the method and get the error estimates. Numerical tests illustrate the validity of the theoretical results.

In this paper, we develop a technique of adaptive time stepping and combine it with dynamic mesh adaptation to simulate unsteady Navier-Stokes flows over stationary or moving bodies. The second order Backward Differentiation Formula (BDF2) is employed for time discretization and the adaptation of time step is based on the estimation of temporal error. Via a PID (Proportional Integral Derivative) controller or a classical heuristic controller, the size of time step is determined adaptively by the estimate of temporal error and the specified tolerance. In order to eliminate the lag of the adapted mesh behind the unsteady solution, which is associated with the size of time step, a predictor-corrector scheme is adopted in the dynamic mesh adaptation. The efficiency and reliability of the present adaptive time and space discretization approach are validated by the numerical experiments for two- and three-dimensional flows. In the numerical experiments, the behaviors of different error estimators and step-size controllers have also been compared and discussed.

An adaptive continuous interior penalty finite element method (ACIPFEM) for symmetric second order linear elliptic equations is considered. Convergence and quasi-optimality of the ACIPFEM are proved. Compared with the analyses for the adaptive finite element method or the adaptive interior penalty discontinuous Galerkin method, extra work is done to overcome the difficulties caused by the additional penalty term. Numerical tests are provided to verify the theoretical results and show advantages of the ACIPFEM.

Considering primary and secondary regimes of creep, deformation of a rotating disc made of Al-SiC Functionally Graded Material (FGM) is investigated using Generalized Differential Quadrature Method (GDQ). Primary and secondary creep are described by Norton law in which creep parameters depended on volume fraction distribution of SiC reinforcement particles, temperature and particle size. All mechanical and thermal properties are functions of volume fraction percentage of SiC particles. Using equilibrium, constitutive and strain-displacement equations, displacement-based creep equation is obtained. This non-closed form equation is solved using GDQ method and a self-developed solution algorithm. Different graphs of creep strains and stresses are extracted using this presented method of creep analysis. Studies show that functionally distribution of particle content and prevailing temperature does not influence the stress fields considerably but obviously, the creep rates depend on temperature level and percentage of reinforcing particles.

This paper is concerned with the $C^0$ weak Galerkin finite element method for a fourth order linear parabolic equation. The method is based on the construction of a discrete weak Laplacian operator. The error estimates are obtained for semi-discrete weak Galerkin finite element method. Numerical results are presented to confirm the theory.

In this paper, a Legendre-collocation spectral method is developed for the second order Volterra integro-differential equation with delay. We provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both $L^{2}$-norm and $L^{\infty}$-norm. In the end, the numerical experiment is illustrated to confirm the theoretical analysis.

A dual-level method of fundamental solutions in conjunction with kernel- independent fast multipole method is proposed in this study. The competitive attributes of the method are that it inherits high accuracy of the method of fundamental solutions, yet avoids producing the resulting ill-conditioned linear system of equations. In contrast to the method of fundamental solutions, the proposed method places two sets of source nodes on the fictitious boundary and physical boundary, respectively, and then combines the fundamental solutions generated by these two sets of source nodes as the modified fundamental solutions of the Laplace equation. This strategy improves significantly the stability of the method of fundamental solutions. In addition, the method is accelerated by the kernel-independent fast multipole method, which reduces the asymptotic complexity of the method to $\mathcal{O}(N)$ from $\mathcal{O}({N}^{2})$. Numerical experiments show that the method can simulate successfully the large-scale heat conduction problems via a single laptop with up to 250000 degrees of freedom.

This paper presents a modified moving least square (MMLS) collocation method for solving wave equations. In contrast to the conventional moving least square (CMLS) method, this method modifies how discretization of computational points is done and decreases the number of base functions to simplify shape functions while solving high-dimensional problems. In addition, the proposed method maintains the independence of discretization for different dimensions, which is convenient to deal with computational domains in a simple manner while retaining a local character. The above improvement results in this approximation significantly saving calculation time while preserving accuracy of the solution. The numerical simulations show that MMLS collocation method has good stability and accuracy in analyzing high-dimensional wave propagation.

A new numerical method is proposed and investigated for solving two- dimensional Black-Scholes option pricing model. This model is represented  by Dirichlet initial-boundary value problem in a rectangular domain for a parabolic equation with advection-diffusion operator containing mixed derivatives. It is approximated by using a finite element method in spatial variables and alternating direction implicit (ADI) method in time variable. The ADI scheme is based on the semi-implicit approximation. The stability and convergence of the constructed scheme is proved rigorously. The provided computational results demonstrate the efficiency and high accuracy of the proposed method.