We develop a numerical method to simulate a two-phase compressible flow with sharp phase interface on Eulerian grids. The scheme makes use of a levelset to depict the phase interface numerically. The overall scheme is basically a finite volume scheme. By approximately solving a two-phase Riemann problem on the phase interface, the normal phase interface velocity and the pressure are obtained, which is used to update the phase interface and calculate the numerical flux between the flows of two different phases. We adopt an aggregation algorithm to build cell patches around the phase interface to remove the numerical instability due to the breakdown of the CFL constraint by the cell fragments given by the phase interface depicted using the levelset function. The proposed scheme can handle problems with tangential sliping on the phase interface, topological change of the phase interface and extreme contrast in material parameters in a natural way. Though the perfect conservation of the mass, momentum and energy in global is not achieved, it can be quantitatively identified in what extent the global conservation is spoiled. Some numerical examples are presented to validate the numerical method developed.

In this paper, a hybrid lattice Boltzmann flux solver (LBFS) is proposed for simulation of viscous compressible flows. In the solver, the finite volume method is applied to solve the Navier-Stokes equations. Different from conventional Navier-Stokes solvers, in this work, the inviscid flux across the cell interface is evaluated by local reconstruction of solution using one-dimensional lattice Boltzmann model, while the viscous flux is still approximated by conventional smooth function approximation. The present work overcomes the two major drawbacks of existing LBFS [28–31], which is used for simulation of inviscid flows. The first one is its ability to simulate viscous flows by including evaluation of viscous flux. The second one is its ability to effectively capture both strong shock waves and thin boundary layers through introduction of a switch function for evaluation of inviscid flux, which takes a value close to zero in the boundary layer and one around the strong shock wave. Numerical experiments demonstrate that the present solver can accurately and effectively simulate hypersonic viscous flows.

This paper is devoted to a unified a priori and a posteriori error analysis of CIP-FEM (continuous interior penalty finite element method) for second-order elliptic problems. Compared with the classic a priori error analysis in literature, our technique can easily apply for any type regularity assumption on the exact solution, especially for the case of lower H1+s weak regularity under consideration, where 0 ≤ s ≤ 1/2. Because of the penalty term used in the CIP-FEM, Galerkin orthogonality is lost and Céa Lemma for conforming finite element methods can not be applied immediately when 0≤s≤1/2. To overcome this difficulty, our main idea is introducing an auxiliary C1 finite element space in the analysis of the penalty term. The same tool is also utilized in the explicit a posteriori error analysis of CIP-FEM.

In this paper, a switch function-based gas-kinetic scheme (SF-GKS) is presented for the simulation of inviscid and viscous compressible flows. With the finite volume discretization, Euler and Navier-Stokes equations are solved and the SF-GKS is applied to evaluate the inviscid flux at cell interface. The viscous flux is obtained by the conventional smooth function approximation. Unlike the traditional gas-kinetic scheme in the calculation of inviscid flux such as Kinetic Flux Vector Splitting (KFVS), the numerical dissipation is controlled with a switch function in the present scheme. That is, the numerical dissipation is only introduced in the region around strong shock waves. As a consequence, the present SF-GKS can well capture strong shock waves and thin boundary layers simultaneously. The present SF-GKS is firstly validated by its application to the inviscid flow problems, including 1-D Euler shock tube, regular shock reflection and double Mach reflection. Then, SF-GKS is extended to solve viscous transonic and hypersonic flow problems. Good agreement between the present results and those in the literature verifies the accuracy and robustness of SF-GKS.

In this paper, an inverse source problem for the time-fractional diffusion equation is investigated. The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure. Here the sparsity is understood with respect to the pixel basis, i.e., the source has a small support. By an elastic-net regularization method, this inverse source problem is formulated into an optimization problem and a semismooth Newton (SSN) algorithm is developed to solve it. A discretization strategy is applied in the numerical realization. Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

This paper is devoted to time domain numerical solutions of two-dimensional (2D) material interface problems governed by the transverse magnetic (TM) and transverse electric (TE) Maxwell's equations with discontinuous electromagnetic solutions. Due to the discontinuity in wave solutions across the interface, the usual numerical methods will converge slowly or even fail to converge. This calls for the development of advanced interface treatments for popular Maxwell solvers. We will investigate such interface treatments by considering two typical Maxwell solvers – one based on collocation formulation and another based on Galerkin formulation. To restore the accuracy reduction of the collocation finite-difference time-domain (FDTD) algorithm near an interface, the physical jump conditions relating discontinuous wave solutions on both sides of the interface must be rigorously enforced. For this purpose, a novel matched interface and boundary (MIB) scheme is proposed in this work, in which new jump conditions are derived so that the discontinuous and staggered features of electric and magnetic field components can be accommodated. The resulting MIB time-domain (MIBTD) scheme satisfies the jump conditions locally and suppresses the staircase approximation errors completely over the Yee lattices. In the discontinuous Galerkin time-domain (DGTD) algorithm – a popular GalerkinMaxwell solver, a proper numerical flux can be designed to accurately capture the jumps in the electromagnetic waves across the interface and automatically preserves the discontinuity in the explicit time integration. The DGTD solution to Maxwell interface problems is explored in this work, by considering a nodal based high order discontinuous Galerkin method. In benchmark TM and TE tests with analytical solutions, both MIBTD and DGTD schemes achieve the second order of accuracy in solving circular interfaces. In comparison, the numerical convergence of the MIBTD method is slightly more uniform, while the DGTD method is more flexible and robust.

In this paper, our main purpose is to establish the existence of positive solution of the following system where are constants. F(x,u,υ) = λp(x)[g(x)a(u)+f(υ)], H(x,u,υ)=θq(x)[g1(x)b(υ)+h(u)], λ, θ>0 are parameters, p(x), q(x) are radial symmetric functions, is called p(x)-Laplacian. We give the existence results and consider the asymptotic behavior of the solutions. In particular, we do not assume any symmetric condition, and we do not assume any sign condition on F(x,0,0) and H(x,0,0) either.

Aerosol modeling is very important to study the behavior of aerosol dynamics in atmospheric environment. In this paper we consider numerical methods for the nonlinear aerosol dynamic equations on time and particle size. The finite volume element methods based on the linear interpolation and Hermite interpolation are provided to approximate the aerosol dynamic equation where the condensation and removal processes are considered. Numerical examples are provided to show the efficiency of these numerical methods.

The differential quadrature method (DQM) has been successfully used in a variety of fields. Similar to the conventional point discrete methods such as the collocation method and finite difference method, however, the DQM has some difficulty in dealing with singular functions like the Dirac-delta function. In this paper, two modifications are introduced to overcome the difficulty encountered in solving differential equations with Dirac-delta functions by using the DQM. The moving point load is work-equivalent to loads applied at all grid points and the governing equation is numerically integrated before it is discretized in terms of the differential quadrature. With these modifications, static behavior and forced vibration of beams under a stationary or a moving point load are successfully analyzed by directly using the DQM. It is demonstrated that the modified DQM can yield very accurate solutions. The compactness and computational efficiency of the DQM are retained in solving the partial differential equations with a time dependent Dirac-delta function.

This paper studies a system of semi-linear fractional diffusion equations which arise in competitive predator-prey models by replacing the second-order derivatives in the spatial variables with fractional derivatives of order less than two. Moving finite element methods are proposed to solve the system of fractional diffusion equations and the convergence rates of the methods are proved. Numerical examples are carried out to confirm the theoretical findings. Some applications in anomalous diffusive Lotka-Volterra and Michaelis-Menten-Holling predator-prey models are studied.

A counterexample is constructed. It confirms that the error of conforming finite element solution is proportional to the coefficient jump, when solving interface elliptic equations. The Scott-Zhang operator is applied to a nonconforming finite element. It is shown that the nonconforming finite element provides the optimal order approximation in interpolation, in L2-projection, and in solving elliptic differential equation, independent of the coefficient jump in the elliptic differential equation. Numerical tests confirm the theoretical finding.

In this paper, we study a new stabilized method based on the local pressure projection to solve the semi-linear elliptic equation. The proposed scheme combines nonconforming finite element pairs NCP1–P1 triangle element and two-level method, which has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, but have more favorable stability and less support sets. Stability analysis and error estimates have been done. Finally, numerical experiments to check estimates are presented.

This paper derives a higher order hybrid stress finite element method on quadrilateral meshes for linear plane elasticity problems. The method employs continuous piecewise bi-quadratic functions in local coordinates to approximate the displacement vector and a piecewise-independent 15-parameter mode to approximate the stress tensor. Error estimation shows that the method is free from Poisson-locking and has second-order accuracy in the energy norm. Numerical experiments confirm the theoretical results.

A mathematical multi-zone ice accretion model used in the numerical simulation of icing on airfoil surface based on three water states, namely, continuous film, rivulets and beads is studied in this paper. An improved multi-zone roughness model is proposed. According to the flow state of liquid water and film flow, rivulets flow governing equations are established to calculate film mass distribution, film velocity, rivulet wetness factor and rivulet mass distribution. Force equilibrium equations of droplet are used to establish the critical conditions of water film broken into rivulets and rivulets broken into beads. The temperature conduction inside the water layer and ice layer is considered. Using the proposed model ice accretion on a NACA0012 airfoil profile with a 4° angle of attack under different icing conditions is simulated. Different ice shapes like glaze ice, mixed ice and rime ice are obtained, and the results agree well with icing wind tunnel experiment data. It can be seen that, water films are formed on the surface, and heights of the films vary with icing time and locations. This results in spatially-temporally varying surface roughness and heat transfer process, ultimately affects the ice prediction. Model simulations indicate that the process of water film formation and evolution cannot be ignored, especially under glaze ice condition.

Computational fluid dynamics (CFD) has been used by numerous researchers for the simulation of flows around wind turbines. Since the 2000s, the experiments of NREL phase VI blades for blind comparison have been a de-facto standard for numerical software on the prediction of full scale horizontal axis wind turbines (HAWT) performance. However, the characteristics of vortex structures in the wake, whether for modeling the wake or for understanding the aerodynamic mechanisms inside, are still not thoroughly investigated. In the present study, the flow around N-REL phase VI blades was numerically simulated, and the results of the wake field were compared with the experimental ones of a one-to-eight scaled model in a low-speed wind tunnel. A good agreement between simulation and experimental results was achieved for the evaluation of overall performances. The simulation captured the complete formation procedure of tip vortex structure from the blade. Quantitative analysis showed the streamwise translation movement of vortex cores. Both the initial formation and the damping of vorticity in near wake field were predicted. These numerical results showed good agreements with the measurements. Moreover, wind tunnel wall effects were also investigated on these vortex structures, and it revealed further radial expansion of the helical vortical structures in comparison with the free-stream case.

In this paper, we investigate the performance of the seventh-order hybrid cell-edge and cell-node dissipative compact scheme (HDCS-E8T7) on curvilinear mesh for noise prediction in subsonic flow. In order to eliminate the errors due to surface conservation law (SCL) is dissatisfied on curvilinear meshes, the symmetrical conservative metric method (SCMM) is adopted to calculate the grid metric derivatives for the HDCS-E8T7. For the simulation of turbulence flow which may have main responsibility for the noise radiation, the new high-order implicit large eddy simulation (HILES) based on the HDCS-E8T7 is employed. Three typical cases, i.e., scattering of acoustic waves by multiple cylinder, sound radiated from a rod-airfoil and subsonic jet noise from nozzle, are chosen to investigate the performance of the new scheme for predicting aeroacoustic problem. The results of scattering of acoustic waves by multiple cylinder indicate that the HDCS-E8T7 satisfying the SCL has high resolution for the aeroacoustic prediction. The potential of the HDCS-E8T7 for aeroacoustic problems on complex geometry is shown by the predicting sound radiated from a rod-airfoil configuration. Moreover, the subsonic jet noise from nozzle has been successfully predicted by the HDCS-E8T7.

In this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the H1 norm and the pressure in the L2 norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.

Numerical instability may occur when simulating high Reynolds number flows by the lattice Boltzmann method (LBM). The multiple-relaxation-time (MRT) model of the LBM can improve the accuracy and stability, but is still subject to numerical instability when simulating flows with large single-grid Reynolds number (Reynolds number/grid number). The viscosity counteracting approach proposed recently is a method of enhancing the stability of the LBM. However, its effectiveness was only verified in the single-relaxation-time model of the LBM (SRT-LBM). This paper aims to propose the viscosity counteracting approach for the multiple-relaxation-time model (MRT-LBM) and analyze its numerical characteristics. The verification is conducted by simulating some benchmark cases: the two-dimensional (2D) lid-driven cavity flow, Poiseuille flow, Taylor-Green vortex flow and Couette flow, and three-dimensional (3D) rectangular jet. Qualitative and Quantitative comparisons show that the viscosity counteracting approach for the MRT-LBM has better accuracy and stability than that for the SRT-LBM.

A problem of two equal, semi-permeable, collinear cracks, situated normal to the edges of an infinitely long piezoelectric strip is considered. Piezoelectric strip being prescribed out-of-plane shear stress and in-plane electric-displacement. The Fourier series and integral equation methods are adopted to obtain analytical solution of the problem. Closed-form analytic expressions are derived for various fracture parameters viz. crack-sliding displacement, crack opening potential drop, field intensity factors and energy release rate. An numerical case study is considered for poled PZT–5H, BaTiO3 and PZT–6B piezoelectric ceramics to study the effect of applied electro-mechanical loadings, crack-face boundary conditions as well as inter-crack distance on fracture parameters. The obtained results are presented graphically, discussed and concluded.

In this article, Sinc collocation method is considered to obtain the numerical solution of integral algebraic equation of index-1 by reducing it to an explicit system of algebraic equation. It is shown that Sinc collocation solution can produce an error of order . Moreover, Sinc method is applied to several examples to illustrate the accuracy and implementation of the method.