It is well known that nonlinear integro-differential equations play vital role in modeling of many physical processes, such as nano-hydrodynamics, drop wise condensation, oceanography, earthquake and wind ripple in desert. Inspired and motivated by these facts, we use the variation of parameters method for solving system of nonlinear Volterra integro-differential equations. The proposed technique is applied without any discretization, perturbation, transformation, restrictive assumptions and is free from Adomian’s polynomials. Several examples are given to verify the reliability and efficiency of the proposed technique.

This paper at first shows the details of finite volume-based lattice Boltzmann method (FV-LBM) for simulation of compressible flows with shock waves. In the FV-LBM, the normal convective flux at the interface of a cell is evaluated by using one-dimensional compressible lattice Boltzmann model, while the tangential flux is calculated using the same way as used in the conventional Euler solvers. The paper then presents a platform to construct one-dimensional compressible lattice Boltzmann model for its use in FV-LBM. The platform is formed from the conservation forms of moments. Under the platform, both the equilibrium distribution functions and lattice velocities can be determined, and therefore, non-free parameter model can be developed. The paper particularly presents three typical non-free parameter models, D1Q3, D1Q4 and D1Q5. The performances of these three models for simulation of compressible flows are investigated by a brief analysis and their application to solve some one-dimensional and two-dimensional test problems. Numerical results showed that D1Q3 model costs the least computation time and D1Q4 and D1Q5 models have the wider application range of Mach number. From the results, it seems that D1Q4 model could be the best choice for the FV-LBM simulation of hypersonic flows.

We construct a finite volume element method based on the constrained nonconforming rotated Q1-constant element (CNRQ1-P0) for the Stokes problem. Two meshes are needed, which are the primal mesh and the dual mesh. We approximate the velocity by CNRQ1 elements and the pressure by piecewise constants. The errors for the velocity in the H1 norm and for the pressure in the L2 norm are O(h) and the error for the velocity in the L2 norm is O(h2). Numerical experiments are presented to support our theoretical results.

In this paper, we apply the collocation methods to a class of Volterra integral functional equations with multiple proportional delays (VIFEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solutions for this type of equations, and then analyze the convergence orders of the collocation solutions and give corresponding error estimates. The numerical results verify our theoretical analysis.

In this paper, the collocation methods are used to solve the boundary integral equations of the first kind on the polygon. By means of Sidi’s periodic transformation and domain decomposition, the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers (i = 1,...,d), which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations. Numerical experiments are carried out to show that the methods are very efficient.

Here considered is the problem of transient electromagnetic scattering from overfilled cavities embedded in an impedance ground plane. An artificial boundary condition is introduced on a semicircle enclosing the cavity that couples the fields from the infinite exterior domain to those fields inside. A Green’s function solution is obtained for the exterior domain, while the interior problem is solved using finite element method. Well-posedness of the associated variational formulation is achieved and convergence and stability of the numerical scheme confirmed. Numerical experiments show the accuracy and robustness of the method.

An acceleration scheme based on stationary iterative methods is presented for solving linear system of equations. Unlike Chebyshev semi-iterative method which requires accurate estimation of the bounds for iterative matrix eigenvalues, we use a wide range of Chebyshev-like polynomials for the accelerating process without estimating the bounds of the iterative matrix. A detailed error analysis is presented and convergence rates are obtained. Numerical experiments are carried out and comparisons with classical Jacobi and Chebyshev semi-iterative methods are provided.

Characteristic boundary conditions that are capable of handling general fluid mixtures flow at all flow speeds are developed. The formulation is based on fundamental thermodynamics theories incorporated into an efficient preconditioning scheme in a unified manner. Local one-dimensional inviscid (LODI) relations compatible to the preconditioning system are proposed to obtain information carried by incoming characteristic waves at boundaries accurately. The approach has been validated against a variety of sample problems at a broad range of fluid states and flow speeds. Both acoustic waves and hydrodynamic flow features can pass through the boundaries of computational domain transparently without any un-physical reflection or spurious distortion. The approach can be reliably applied to fluid flows at extensive thermodynamic states and flow speeds in numerical simulations. Moreover, the use of the boundary condition shows to improve the computational efficiency.

This study presents a modification of the central-upwind Kurganov scheme for approximating the solution of the 2D Euler equation. The prototype, extended from a 1D model, reduces substantially less dissipation than expected. The problem arises from over-restriction of some slope limiters, which keep slopes between interfaces of cells to be Total-Variation-Diminishing. This study reports the defect and presents a re-derived optimal formula. Numerical experiments highlight the significance of this formula, especially in long-time, large-scale simulations.

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson’s ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.

A bounded high order upwind scheme is presented for the modified Burgers’ equation by using the normalized-variable formulation in the finite volume framework. The characteristic line of the present scheme in the normalized-variable diagram is designed on the Hermite polynomial interpolation. In order to suppress unphysical oscillations, the present scheme respects both the TVD (total variational diminishing) constraint and CBC (convection boundedness criterion) condition. Numerical results demonstrate the present scheme possesses good robustness and high resolution for the modified Burgers’ equation.

In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures. Numerical results using these material parameters are presented to demonstrate the cloaking effect.

This paper reports an improved constitutive model for the shear creep behavior of offwhite marbles which are selected from slope and underground cavern and contain green schist’s weak structural planes. The shear creep behavior of the samples is characterized using the rheological tests. Based on the experimental measurements on mechanical properties under different normal stress conditions, an improved model is proposed to analyze the experimental results. It is demonstrated from a further discussion that such model can reflect the non-linear creep characteristics of structural planes, and especially, it is suitable for description of the viscoelastic and viscoplastic deformation behavior of structural planes.

The pullback asymptotic behavior of the solutions for 2D Nonau-tonomous G-Navier-Stokes equations is studied, and the existence of its L2-pullback attractors on some bounded domains with Dirichlet boundary conditions is investigated by using the measure of noncompactness. Then the estimation of the fractal dimensions for the 2D G-Navier-Stokes equations is given.

The stability and natural vibration of a standing tapered vertical column under its own weight are studied. Exact stability criteria are found for the pointy column and numerical stability boundaries are determined for the blunt tipped column. For vibrations we use an accurate, efficient initial value numerical method for the first three frequencies. Four kinds of columns with linear taper are considered. Both the taper and the cross section shape of the column have large influences on the vibration frequencies. It is found that gravity decreases the frequency while the degree of taper may increase or decrease frequency. Vibrations may occur in two different planes.

A hybridization of a high order WENO-Z finite difference scheme and a high order central finite difference method for computation of the two-dimensional Euler equations first presented in [B. Costa and W. S. Don, J. Comput. Appl. Math., 204(2) (2007)] is extended to three-dimensions and for parallel computation. The Hybrid scheme switches dynamically from a WENO-Z scheme to a central scheme at any grid location and time instance if the flow is sufficiently smooth and vice versa if the flow is exhibiting sharp shock-type phenomena. The smoothness of the flow is determined by a high order multi-resolution analysis. The method is tested on a benchmark sonic flow injection in supersonic cross flow. Increase of the order of the method reduces the numerical dissipation of the underlying schemes, which is shown to improve the resolution of small dynamic vortical scales. Shocks are captured sharply in an essentially non-oscillatory manner via the high order shock-capturing WENO-Z scheme. Computations of the injector flow with a WENO-Z scheme only and with the Hybrid scheme are in very close agreement. Thirty percent of grid points require a computationally expensive WENO-Z scheme for high-resolution capturing of shocks, whereas the remainder of grid points may be solved with the computationally more affordable central scheme. The computational cost of the Hybrid scheme can be up to a factor of one and a half lower as compared to computations with a WENO-Z scheme only for the sonic injector benchmark.

Linear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay. The corresponding convergence results are obtained and successfully confirmed by some numerical examples. The results obtained in this work extend the corresponding ones in literature.

New numerical methods based on collocation methods with the mechanical quadrature rules are proposed to solve some systems of singular integro-differential equations that are defined on arbitrary smooth closed contours of the complex plane. We carry out the convergence analysis in classical Hölder spaces. A numerical example is also presented.

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and L∞ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

This work is concerned with the numerical simulations for two reaction-diffusion systems, i.e., the Brusselator model and the Gray-Scott model. The numerical algorithm is based upon a moving finite element method which helps to resolve large solution gradients. High quality meshes are obtained for both the spot replication and the moving wave along boundaries by using proper monitor functions. Unlike [33], this work finds out the importance of the boundary grid redistribution which is particularly important for a class of problems for the Brusselator model. Several ways for verifying the quality of the numerical solutions are also proposed, which may be of important use for comparisons.