Storage requirement and computational efficiency have always been challenges for the efficient implementation of discontinuous Galerkin (DG) methods for real life applications. In this paper, a fully implicit Jacobian-Free Newton-Krylov (JFNK) method is developed in the context of DG discretizations for the three-dimensional compressible Euler and Navier-Stokes equations. Compared with the Jacobian-based methods, the Jacobian-Free approach saves the storage for the Jacobian matrix which can be of great importance for DG methods. Three types of preconditioners are investigated in which the block diagonal preconditioner requires the least storage, while the block LU-SGS and ILU0 preconditioners require more storage but are more computationally efficient. An implicit time-stepping strategy is adopted for the stability of the current solver, which is based upon a hexahedral spatial mesh and the nonlinear solver package Kinsol is used to improve the computational efficiency and robustness. Numerical results demonstrate that the preconditioned JFNK-DG solver can substantially reduce the storage requirement compared with the Jacobian based method without significantly compromising accuracy or efficiency. Furthermore, as a good compromise between efficiency and storage requirement, the ILU0 preconditioner shows the best choice of the preconditioners presented.

Fluid-structure interaction (FSI) occurs in many situations in nature and industries. Traditional descriptions of this problem in computational fluid dynamics (CFD) are mostly at macroscopic level. In this paper, an alternative mesoscopic description, based on lattice Boltzmann method (LBM), is presented for the FSI problems. The FSI are viewed as the collective behaviors of neighboring fictitious particles of the LBM deviating from their equilibrium state when solid boundaries presented in flow. To illustrate the rationality of the present idea, a forced convection over a stationary heated circular cylinder and a circular cylinder with in-line oscillation in fluids are simulated at first and the results are validated by comparing with existing numerical and experimental data in the literatures. For applications, natural convections in a square cavity with maximum three circle particles suspended inside are then carried out and the mechanisms of heat transfer enhancement are investigated. It is found that, adding particles destabilizes the flows in a square cavity and enhances heat transfer.

We consider the time dependent Darcy problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of these equations in the case of general solution. This discretization relies on a backward Euler's scheme for the time variable and finite elements for the space variables. We prove a priori error estimates both for the time steps and the meshes.

In the present study, a numerical procedure for analyzing the seismic response of the linear elastic half-plane including buried cavities subjected to incident P and SV waves is proposed by means of the novel singular boundary method (SBM). The SBM is a recently developed boundary-type meshless collocation method, which applies the singular fundamental solutions as basis functions. In order to avoid the singularities of the fundamental solutions, the SBM introduces the concept of origin intensity factors at the origin. With the aid of the origin intensity factors of the Laplace and the plane-strain elastostatic problems, this study first derives the origin intensity factors for the traction boundary condition as well as the origin intensity factors for the flat boundary, in order to form the SBM formulation for wave scattering problems in the linear elastic half-plane including buried cavities. Results obtained with the SBM model are compared with the results obtained with the finite element method, which shows that the method is quite promising for studying seismic wave scattering problems.

Second-order finite-difference schemes are developed to solve the corresponding Fokker-Planck equation of Brownian motion with dry friction, which is one of the simplest models of stochastic piecewise-smooth systems. For the Fokker-Planck equation with a discontinuous drift, both explicit and implicit second order schemes are derived by finite volume method. The proposed schemes are proved to be stable both for the one-variable (related to the velocity only) and two-variable (related to the velocity and displacement) cases. Numerical experiments are implemented for both the two cases. Some known analytical results of the considered model are used to confirm the effectiveness and desired accuracy of the schemes.

The initial-boundary value problems for parabolic equations with nonlocal conditions have been widely applied in various fields. In this work, we firstly build an implicit Euler scheme for an initial-boundary value problem of one dimensional parabolic equations with an integral two-space-variables condition. Then we prove that this scheme can reach the asymptotic optimal error estimate in the maximum norm. Next, the formulas used to approximate the solution derivatives with respect to time and spatial variables are presented, and it is proved for the first time that they have superconvergence. In the end, numerical experiments demonstrate the theoretical results.

Numerical investigation of multiple shock/turbulent flow interaction is carried out using large eddy simulation for a supersonic channel flow with an inlet free stream Mach number 1.61. Various fundamental mechanisms dictating the flow phenomena including shock train, shear layer and turbulence behavior are investigated. It is found that the existence of the shock train and separated shear layer has an important influence on turbulence features. The turbulence intensities and turbulent kinetic energy (TKE) are strengthened in the region of the multiple shock because of the unsteadiness of the shocks. The investigation on the transport equations of the TKE and the pressure fluctuation reveals that the multiple shock and the roll-up vortices of shear layer can promote the generation of the TKE and the pressure fluctuation. The unsteady behavior of flow field is further analyzed by means of the proper orthogonal decomposition method. It is found that the multiple shock and the separated shear layer dominate the unsteady feature.

In this work, we investigate a novel two-level discretization method for semilinear reaction-diffusion equations. Motivated by the two-grid method for nonlinear partial differential equations (PDEs) introduced by Xu [18] on physical space, our discretization method uses a two-grid finite element discretization method for semilinear partial differential equations on physical space and a two-level finite difference method for the corresponding time space. Specifically, we solve a semilinear equations on a coarse mesh $\mathcal{T}_H(\Omega)$ (partition of domain $\Omega$ with mesh size $H$) with a large time step size $\Theta$ and a linearized equations on a fine mesh $\mathcal{T}_h(\Omega)$ (partition of domain $\Omega$ with mesh size $h$) using smaller time step size $\theta$. Both theoretical and numerical results show that when $h=H^2, \theta=\Theta^2$, the novel two-grid numerical solution achieves the same approximate accuracy as that for the original semilinear problem directly by finite element method with $\mathcal{T}_h(\Omega)$ and $\theta$.

We consider the approximation of eigenvalue problems for elasticity equations with interface. This kind of problems can be efficiently discretized by using immersed finite element method (IFEM) based on Crouzeix-Raviart P1-nonconforming element. The stability and the optimal convergence of IFEM for solving eigenvalue problems with interface are proved by adopting spectral analysis methods for the classical eigenvalue problem. Numerical experiments demonstrate our theoretical results.

Techniques for the efficient approximate solution of systems of convection-diffusion partial differential equations modelling the sedimentation of droplets of different sizes in a viscous fluid are introduced. These techniques comprise the use of Polynomial Viscosity Matrix (PVM) methods for the convective numerical fluxes and implicit treatment of the nonlinear diffusion terms. Numerical examples based on [A. Abeynaike et al., Chem. Eng. Sci., 79 (2012), pp. 125–137] are presented.

In this paper, we investigate the superconvergence property of a time-dependent nonlinear Schrödinger equation with the bilinear finite element method on the rectangular mesh. We prove the superclose error estimate in $H^1$-norm with order $\mathcal{O}(h^2)$ between the approximated solution and the elliptic projection of the exact solution. Moreover, we obtain the global superconvergence result in $H^1$-norm with order $\mathcal{O}(h^2)$ by the interpolation post-processing operator.

Buckling and post-buckling have been expensively considered as critical limit states of slender/thin members over the centuries. Recent research interests have been shifted to exploit the potential "smart applications" of buckling-induced elastic instability at multiscale. For example, buckling elements have been implemented to develop monostable, bistable and multistable mechanisms for different functional utilities. To design buckling-induced techniques and optimize outputs, it is of necessity to accurately control the mechanical response of buckling elements. This review paper aims at showcasing the theoretical studies of multiscale buckling and post-buckling analysis mainly conducted during the last two decades. The studies of micro/nanoscale buckling and post-buckling analysis are discussed with respect to the nonlocal elasticity, non-classical couple stress and strain gradient elasticity theories, respectively, while the macroscale buckling analysis is categorized into the small and large deformation studies. A comparison is presented between the small and large deformation models at macroscale to identify the applicability of the theories. Recent research trends are reviewed on the optimal design of buckling-induced mechanisms and techniques. In the end, we discuss the potential research avenues for future innovations.