In this study, a novel fast-implicit iteration scheme called the alternating cell direction implicit (ACDI) method is combined with the approximate factorization scheme. This application aims to offer a mathematically well-defined version of the ACDI method and to increase the accuracy of the iteration scheme used for the numerical solutions of partial differential equations. The ACDI method is a fast-implicit method that can be used for unstructured grids. The use of fast implicit iteration methods with unstructured grids is not common in the literature. The new ACDI method has been applied to the unsteady diffusion equation to determine its convergence and time-dependent solution ability and character. The numerical tests are conducted for different grid types, such as structured, unstructured quadrilateral, and hybrid polygonal grids. Second, the ACDI was applied to the unsteady advection-diffusion equation to understand the time-dependent and progression capabilities of the presented method. Third, a full potential equation solution is created to understand the complex flow solving ability of the presented method. The results of the numerical study are compared with other fast implicit methods, such as the point Gauss–Seidel (PGS) and line Gauss–Seidel (LGS) methods and the fourth-order Runge-Kutta (RK4) method, which is an explicit scheme, and the Laasonen method, which is a fully implicit scheme. The study increased the abilities of the ACDI method. Due to the new ACDI method, the approximate factorization method, which is used only in structural grids that are known to be advantageous, can be applied to any mesh structure.

In this study, the polynomial expansion method (PEM) and the polynomial method of particular solutions (PMPS) are applied to solve a class of linear elliptic partial differential equations (PDEs) in two dimensions with constant coefficients. In the solution procedure, the sought solution is approximated by the Pascal polynomials and their particular solutions for the PEM and PMPS, respectively. The multiple-scale technique is applied to improve the conditioning of the resulted linear equations and the accuracy of numerical results for both of the PEM and PMPS. Some mathematical statements are provided to demonstrate the equivalence of the PEM and PMPS bases as they are both bases of a certain polynomial vector space. Then, some numerical experiments were conducted to validate the implementation of the PEM and PMPS. Numerical results demonstrated that the PEM is more accurate and well-conditioned than the PMPS and the multiple-scale technique is essential in these polynomial methods.

A new algorithm, called symmetric inertial alternating direction method of multipliers (SIADMM), is designed for separable convex optimization problems with linear constraints in this paper. The convergence rate of the SIADMM is proved to be $\mathcal{O}(1/ \sqrt{k})$. Two kinds of elliptic equation constrained optimization problems, the unconstrained cases as well as the box-constrained cases of the distributed control and the Robin boundary control, are analyzed theoretically and solved numerically. First, the existence and uniqueness of the solutions to these problems are proved. Second, these continuous optimization problems are transformed into discrete optimization problems by the finite element method, and then the discrete optimization problems are solved by the proposed SIADMM. Numerical experiments with different problems are investigated to demonstrate the efficiency of the SIADMM. And the numerical performance of the SIADMM is better than the performance of the ADMM. Moreover, the numerical results show that the convergence rate of the SIADMM tends to be faster than $\mathcal{O}(1/ \sqrt{k})$ in calculation process.

Precise calculation of the trajectory of store separation is critical in assessing whether the store can be released safely. Store ejection is the initial stage of the releasing process and any uncertainty introduced at this stage will propagate through the whole trajectory. In this work, the impact of the uncertainties in ejector modeling on the simulation of a generic store separation is investigated by using a Monte-Carlo-based approach. To reduce the extremely large computation cost resulted from the direct use CFD in Monte Carlo simulation, the CFD solutions are represented by a time-dependent Kriging model, which is constructed at each time step by using the samples from the URANS simulations. The stochastic outputs, including the distribution of probability density function, expected value and 95% confidence interval of store separation trajectory, are obtained by the Monte Carlo simulations. The sensitivity analysis is also carried out by using the Monte-Carlo-based method to determine the most significant variables in ejector modeling, which affect the output uncertainty. Our results show that ejector modeling is one of the main uncertainty sources of store separation simulation and the approximation in ejector modeling can cause a significant deviation, especially in the angular displacement.

The superconvergence of a two-dimensional time-independent nonlinear Schrödinger equation are analyzed with the rectangular Lagrange type finite element of order $k$. Firstly, the error estimate and superclose property are given in $H^1$-norm with order $\mathcal{O}(h^{k+1})$ between the finite element solution $u_h$ and the interpolation function $u_I$ by use of the elliptic projection operator. Then, the global superconvergence is obtained by the interpolation post-processing technique. In addition, some numerical examples with the order $k = 1$ and $k = 2$ are provided to demonstrate the theoretical analysis.

The aim of this paper is to present the backward substitution method for solving a class of fractional dual-phase-lag models of heat transfer. The proposed method is based on the Fourier series expansion along the spatial coordinate over the orthonormal basis formed by the eigenfunctions of the corresponding Sturm-Liouville problem. This Fourier expansion of the solution transforms the original fractional partial differential equation into a sequence of multi-term fractional ordinary differential equations. These fractional equations are solved by the use of the backward substitution method. The numerical examples with temperature-jump boundary condition and parameters of the tissue confirm the high accuracy and efficiency of the proposed numerical scheme.

A cell-centered Lagrangian scheme is developed for the numerical simulation of wave propagations in one dimensional (1D) elastic-plastic flow. The classical elastic-plastic material model initially proposed by Wilkins is adopted. The linear elastic model (Hooke’s Law), perfectly plastic model and von Mises yield criterion are used to describe the constitutive relationship of elastic-plastic solid. The second-order extension of this scheme is achieved by a linear reconstruction method. Various numerical tests are simulated to check the capability of this scheme in capturing nonlinear elastic-plastic waves. Compared with the well-developed operator splitting method used in simulating elastic-plastic flow, this scheme is more accurate due to the consideration of a list of 64 different types of the nonlinear elastic-plastic waves when constructing the elastic-perfectly plastic Riemann solver. The numerical simulations of typical examples show competitive results.

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography. The governing equations are reformulated as a nonlinear system of conservation laws with differential source forces and reaction terms. Coupling between the flow layers is accounted for in the system using a set of exchange relations. The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservative finite element method whose approximate solutions are discontinuous across the inter-element boundaries. The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-Lobatto-Legendre nodes for the quadrature used in the approximation of source terms. The method can also be viewed as a high-order version of upwind finite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standard finite element methods fail. To deal with the source terms we also implement a high-order splitting operator for the time integration. The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surface flows over both flat and non-flat beds. The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method. The proposed method is also applied to solve a recirculation flow problem in the Strait of Gibraltar.

The localized method of fundamental solutions (LMFS) is a relatively new meshless boundary collocation method. In the LMFS, the global MFS approximation which is expensive to evaluate is replaced by local MFS formulation defined in a set of overlapping subdomains. The LMFS algorithm therefore converts differential equations into sparse rather than dense matrices which are much cheaper to calculate. This paper makes the first attempt to apply the LMFS, in conjunction with a domain-decomposition technique, for the numerical solution of steady-state heat conduction problems in two-dimensional (2D) anisotropic layered materials. Here, the layered material is decomposed into several subdomains along the layer-layer interfaces, and in each of the subdomains, the solution is approximated by using the LMFS expansion. On the subdomain interface, compatibility of temperatures and heat fluxes are imposed. Preliminary numerical experiments illustrate that the proposed domain-decomposition LMFS algorithm is accurate, stable and computationally efficient for the numerical solution of large-scale multi-layered materials.

This present paper proposes aerodynamic forces and entropy generation characteristics on the flow past two-dimensional airfoil at low Reynolds number by multiple-relaxation-time lattice Boltzmann method to clarify the flow loss mechanism. The block mesh refinement was adopted in which a higher accuracy was needed in parts of the domain characterized by complex flow. The interpolated bounce-back method was used to treat the irregular curve. This numerical method can effectively solve the complex flow field simulation problems with reasonable accuracy and reliability by simulating flow around plate and airfoil. Based on second law of thermodynamics, an expression of entropy generation rate for arbitrary control volume was derived theoretically which could accurately quantify the local irreversible loss of the flow field at any position. After that, a comprehensive numerical study was conducted to analyze relationship of entropy generation and drag force by taking NACA0012 airfoil as the research object. For unsteady condition, entropy generation rate and the drag force are not linearly related any more. Losses due to steady effects mainly consider the irreversibility in the boundary layer and wake while the unsteady effects come from the interaction between the main separation vortex and the trailing shedding vortex.