Interface problems have wide applications in modern scientific research. Obtaining accurate numerical solutions of multi-domain problems involving triple junction conditions remains a significant challenge. In this paper, we develop an efficient finite element method based on non-body-fitting meshes for solving multi-domain elliptic interface problems. We follow the idea of immersed finite element by modifying local basis functions to accommodate interface conditions. We enrich the local finite element space by adding new basis functions for handling non-homogeneous flux jump. The numerical scheme is symmetric and positive definite. Numerical experiments are provided to demonstrate the features of our method.

An entropy conservative (EC) numerical flux for the multiclass Lighthill-Whitham-Richards (MCLWR) kinematic traffic model based on the general framework by Tadmor [E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, I, Math. Comput., 49 (1987), pp. 91-103] is proposed. The approach exploits the existence of an entropy pair for a particular form of this model. The construction of EC fluxes is of interest since in combination with numerical diffusion terms they allow one to design entropy stable schemes for the MCLWR model. In order to obtain a higher-order accurate scheme and control oscillations near discontinuities, a third-order WENO reconstruction recently proposed by Ray [D. Ray, Third-order entropy stable scheme for the compressible Euler equations, in C. Klingenberg and M. Westdickenberg (eds.), Springer Proc. Math. Stat., 237, pp. 503-515] is used. Numerical experiments for different classes of drivers are presented to test the performance of the entropy stable scheme constructed with the entropy conservative flux proposed.

Quality calculation of the demagnetization field plays an important role in the computational micromagnetics. It is a nontrivial challenge to develop a robust and efficient algorithm to handle the requirements from the practical simulations, since the nonlocality of the demagnetization field evaluation and the irregularity of the computational domain. In [C. J. Garcia-Cervera and A. M. Roma, Adaptive mesh refinement for micromagnetics simulations, IEEE Trans. Magn., 42(6) (2006), PP. 1648-1654], the evaluation of the demagnetization field is split into solving two partial differential equations by finite difference scheme and calculating the integrals on the domain boundary. It is this integral who causes the computational complexity of the algorithm $\mathcal{O}(N^{4/3})$. To partially resolve the efficiency issue and to make the solver more flexible on handling the magnet with complicated geometry, we introduce an $h$-adaptive finite element method for the demagnetization field calculations. It can be observed from the numerical results that i). with the finite element discretization, the domain with curved defects can be resolved well, and ii). with the adaptive methods, the total amount of the mesh grids can be reduced significantly to reach the given accuracy, which effectively accelerates the simulations.

In the paper we consider some plane boundary value problems of asymmetric theory of elasticity for perforated domains. The domain is the square with holes arranged in the definite manner. The formulated problems are solved approximately by using the method of fundamental solutions is used.

The physical model is defined by a coupled system of seepage displacement for simulating chemical oil recovery numerically, formulated by three nonlinear partial differential equations concerning the pressure of Darcy-Forchheimer flow, the concentration and the component saturations. The pressure appears within the concentration equation and saturation equations and the Darcy-Forchheimer velocity controls the concentration and saturations. The flow equation is solved by the conservative mixed finite element method. The order of the accuracy is improved by the velocity. The conservative mixed volume element with characteristics is applied to compute the concentration, i.e., the diffusion is discretized by the mixed volume element and convection is computed by the method of characteristics. The method of characteristics has strong computational stability at sharp fronts and confirms high computational accuracy. The mixed volume element simulates diffusion, concentration and the adjoint vector function simultaneously. The nature of conservation is an important physical feature in the numerical simulation. The saturations of different components are computed by the method of characteristic fractional step difference and the computational work is shortened significantly by decomposing a three-dimensional problem into three successive one-dimensional problems and using the method of speedup. By using the theory and technique of a priori estimates of differential equations, convergence of the optimal second order in $l^2$ norm is obtained. Numerical examples are provided to show the effectiveness and viability of this method. This composite method can solve the challenging benchmark problem well.

In this paper, we propose trigonometric polynomial reconstructions based on one five-point stencil and two two-point stencils, instead of algebraic polynomial reconstructions defined on three three-point stencils [20,35], as a building block for designing a fifth-order trigonometric weighted essentially non-oscillatory (TWENO) scheme to solve hyperbolic conservation laws and highly oscillatory problems. The main objective of the paper is to extremely reduce the difficulty in computing the linear weights, could get less absolute truncation errors in smooth regions, and keep sharp shock transitions in nonsmooth regions. Extensive benchmark numerical tests including some highly oscillatory problems are provided to verify the good performance of the new scheme.

Based on the global interpolating generalized moving least square (IGMLS), an element-free Galerkin (EFG) method for rigid-flexible coupling dynamics of axially moving hub-beam systems is implemented. With the consideration of the geometry nonlinear terms and the longitudinal shrinking induced by the transverse deformation, the first-order approximate coupling (FOAC) model which both the dynamic stiffening and softening effects can be reflected is built. The global interpolating and element-free properties make some advantages of the EFG: the convenient imposition for both displacement and derivative boundary conditions and the direct obtaining of actual nodal values compared with the standard EFG method; convenience in dealing with the rigid-flexible coupling terms, constructing highly continuous shape functions and assembling the whole spatial discretized equations by comparing to FEM. Numerical comparisons between the EFG and analytical solutions on the vibration of a cantilever beam show the good accuracy and efficiency of the EFG based on the proper option of related numerical parameters. Numerical results of the axially moving hub-beam system demonstrate the reasonability of FOAC model and the feasibility of the global IGMLS applied for rigid-flexible coupling dynamics.

The effect of a tail with non-uniform flexibility on flapping foil dynamics is studied systematically in this work. Both the power extraction and propulsion behaviors of a flapping foil are checked. Three distinct kinds of non-uniform flexibility distribution are considered, i.e., linear distribution, piecewise distribution and heterocercal distribution. Based on the numerical results obtained, it is found that the tail when the flexibility decreasing along the length is beneficial for power extraction, especially for the piecewise distribution. The reason is that the non-uniform flexible tail can generate bigger bending deformation than the uniform flexible tail, which is good for the leading edge vortex (LEV) capture and then results in the significant rise of the lift. On the other hand, the tail with the decreasing flexibility is harmful to the thrust generation, whilst the flexibility increasing along the length has little benefit on propulsion. The results obtained may have implications for the optimization of the flapping foil propeller and energy harvester.

In this work, we present an immersed boundary-simplified gas kinetic scheme for simulation of two-dimensional (2D) incompressible flows with curved and moving boundaries. Specifically, a fractional step technique with predictor and corrector processes is introduced to solve the governing equations. In the predictor step, the macroscopic governing differential equations are solved on the fixed Eulerian meshes by the recently developed simplified gas kinetic scheme (GKS). Compared to the conventional GKS, the simplified GKS is simpler and more efficient. At the same time, the simplified GKS inherits the advantage of good robustness of conventional GKS. In the corrector step, the velocity correction is carried out on the Lagrangian points by the implicit boundary condition-enforced immersed boundary method (IBM). Since it strictly originates from the no-slip boundary condition, this approach can avoid completely the unphysical streamline penetration phenomenon. Several numerical experiments show that the 2D incompressible flows with curved and moving boundaries can be well simulated by the developed scheme.

In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated resolution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the high-fidelity numerical solutions of the Poisson-Boltzmann equation upon which the fast RBM algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.

In this paper, we study a fourth-order quasi-compact conservative difference scheme for solving the fractional Klein-Gordon-Schrödinger equations. The scheme constructed in this work can preserve exactly the discrete charge and energy conservation laws under Dirichlet boundary conditions. By the energy method, the proposed quasi-compact conservative difference scheme is proved to be unconditionally stable and convergent with order $\mathcal{O}(\tau^{2}+h^{4})$ in maximum norm. Finally, several numerical examples are given to confirm the theoretical results.

We compare the diagonal form fast multipole method (FMM) with the traditional boundary element method (BEM) for a boundary integral equation (BIE) with oscillatory Hankel kernels which arising in using hybrid numerical-asymptotic boundary integral method to the two-dimensional (2D) scattering of a time-harmonic acoustic incident plane wave. The diagonal form FMM is a very efficient and popular algorithm for the rapid solution of boundary value problems. However, we show that the efficiency of the diagonal form FMM is greatly reduced for this kind of BIE. Numerical examples are given to confirm the proposed results.