Localized point sources (monopoles) in an acoustical domain are implemented to a three dimensional non-singular Helmholtz boundary element method in the frequency domain. It allows for the straightforward use of higher order surface elements on the boundaries of the problem. It will been shown that the effect of the monopole sources ends up on the right hand side of the resulting matrix system. Some carefully selected examples are studied, such as point sources near and within a concentric spherical core-shell scatterer (with theoretical verification), near a curved focusing surface and near a multi-scale and multi-domain acoustic lens.

In this paper, we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations (VFSDEs). We further constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the method. In particular, when the fractional order is no longer varying, the conclusions obtained are consistent with the relevant conclusions in the existing literature. Finally, the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.

Accurate and efficient analysis of the coupled electroelastic behavior of piezoelectric structures is a challenging task in the community of computational mechanics. During the past few decades, the method of fundamental solutions (MFS) has emerged as a popular and well-established meshless boundary collocation method for the numerical solution of many engineering applications. The classical MFS formulation, however, leads to dense and non-symmetric coefficient matrices which will be computationally expensive for large-scale engineering simulations. In this paper, a localized version of the MFS (LMFS) is devised for electroelastic analysis of two-dimensional (2D) piezoelectric structures. In the LMFS, the entire computational domain is divided into a set of overlapping small sub-domains where the MFS-based approximation and the moving least square (MLS) technique are employed. Different to the classical MFS, the LMFS will produce banded and sparse coefficient matrices which makes the method very attractive for large-scale simulations. Preliminary numerical experiments illustrate that the present LMFM is very promising for coupled electroelastic analysis of piezoelectric materials.

In this work, we study the coercivity of a family of quadratic finite volume element (FVE) schemes over triangular meshes for solving elliptic boundary value problems. The analysis is based on the standard mapping from the trial function space to the test function space so that the coercivity result can be naturally incorporated with most existing theoretical results such as $H^1$ and $L^2$ error estimates. The novelty of this paper is that, each element stiffness matrix of the quadratic FVE schemes can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. As a result, we reach a sufficient condition to guarantee the existence, uniqueness and coercivity result of the FVE solution on general triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic and computable expressions are obtained. By comparison, the existing minimum angle conditions were obtained numerically from a computer program. Theoretical findings are conformed with the numerical results.

A partial Runge-Kutta Discontinuous Galerkin (RKDG) method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations written in semi-Lagrangian formulation on moving quadrilateral meshes. In this method, the fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation is discretized by the RKDG method as our previous paper [47]. The numerical magnetic field in the remaining two directions (i.e., $x$ and $y$) are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD. Since the divergence of the magnetic field in 2D is independent of its $z$-direction component, an exactly divergence-free numerical magnetic field can be obtained by this treatment. We propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes. A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD equations. Some numerical examples are presented to demonstrate the accuracy, non-oscillatory property and preservation of the exactly divergence-free property of our method.

The Coiflet wavelet-homotopy Galerkin method is extended to solve unsteady nonlinear wave equations for the first time. The Korteweg-de Vries (KdV) equation, the Burgers equation and the Korteweg-de Vries-Burgers (KdVB) equation are examined as illustrative examples. Validity and accuracy of the proposed method are assessed in terms of relative variance and the maximum error norm. Our results are found in good agreement with exact solutions and numerical solutions reported in previous studies. Furthermore, it is found that the solution accuracy is closely related to the resolution level and the convergence-control parameter. It is also found that our proposed method is superior to the traditional homotopy analysis method when dealing with unsteady nonlinear problems. It is expected that this approach can be further used to solve complicated unsteady problems in the fields of science and engineering.

Based on Broad Learning System with preprocessing, the impenetrable obstacles were reconstructed. Firstly, the far-field data were preprocessed by Random Forest, and the shapes of the obstacles were classified by dividing the far-field data into different categories. Secondly, the broad learning system was employed for reconstructing the unknown scatterer. The far-field data of the scatterer were regarded as the input nodes of mapped features in the network, and all the mapped features were connected with the enhancement nodes of random weights to the output layer. Subsequently, the coefficient of the output can be obtained by the pseudoinverse. This method for the recovery of the scattering obstacles is named RF-BLS. Finally, numerical experiments revealed that the proposed method is effective, and that the training speed was significantly improved, compared with the deep learning method.

The main theme of the current article is to investigate the heat transfer in the pulsatile flow of an electrically conducting viscous fluid in a constricted channel under the effect of the magnetic field and thermal radiation. The unsteady governing equations simplified for low conducting fluids are solved numerically by finite difference method using stream-vorticity function formulation. The influence of the flow parameters such as the Hartmann number (magnetic parameter), Strouhal number (flow pulsation parameter), Prandtl number, and radiation parameter is studied on the relevant flow profiles. The influence of different emerging parameters on the skin friction coefficient and Nusselt number are examined, as well. In general, the profiles are observed to exhibit a relatively more regular pattern upstream of the construction than that downstream of the constriction.

To solve conservation laws, efficient schemes such as essentially non-oscillatory (ENO) and weighted ENO (WENO) have been introduced to control the Gibbs oscillations. Based on radial basis functions (RBFs) with the classical WENO-JS weights, a new type of WENO schemes has been proposed to solve conservation laws [J. Guo et al., J. Sci. Comput., 70 (2017), pp. 551–575]. The purpose of this paper is to introduce a new formulation of conservative finite difference RBFWENO schemes to solve conservation laws. Unlike the usual method for reconstructing the flux functions, the flux function is generated directly with the conservative variables. Comparing with Guo and Jung (2017), the main advantage of this framework is that arbitrary monotone fluxes can be employed, while in Guo and Jung (2017) only smooth flux splitting can be used to reconstruct flux functions. Several 1D and 2D benchmark problems are prepared to demonstrate the good performance of the new scheme.

The critical transition Reynolds number is the lowest value at which the turbulent flow can hold in real flows. The determination of the critical transition Reynolds number not only is a scientific problem, but also is important for some engineering problems. However, there is no available theoretical method to search the critical value. For the hypersonic boundary layer with significant importance for engineering problems, there is no available experimental method to search the critical value so far. Consequently, it is imperative to take numerical method to search it. In this paper, direct numerical simulations (DNS) method is employed to determine the critical transition Reynolds number for the incompressible flat-plate boundary layer. Firstly, under the assumption of parallel flow, the temporal mode DNS is performed to determine the critical value as $Re_{xpcr} =43767,$ which is quite close to the numerical results of other people. Secondly, under the condition of nonparallel flow, the spatial mode DNS is performed to determine the critical transition Reynolds number as $Re_{xcr}=3×10^5$, which is well consistent with the experimental results. In principle, the proposed method in this paper can be extended to the supersonic/hypersonic boundary layer, and that problem will be discussed in the subsequent papers.

In this paper, we propose an approach for constructing conservative and maximum-principle-preserving finite volume schemes by using the method of undetermined coefficients, which depend nonlinearly on the linear non-conservative one-sided fluxes. In order to facilitate the derivation of expressions of these undetermined coefficients, we explicitly provide a simple constriction condition with a scaling parameter. Such constriction conditions can ensure the final schemes are exact for linear solution problems and may induce various schemes by choosing different values for the parameter. In particular, when this parameter is taken to be 0, the nonlinear terms in our scheme degenerate to a harmonic average combination of the discrete linear fluxes, which has often been used in a variety of maximum-principle-preserving finite volume schemes. Thus our method of determining the coefficients of the nonlinear terms is more general. In addition, we prove the convergence of the proposed schemes by using a compactness technique. Numerical results demonstrate that our schemes can preserve the conservation property, satisfy the discrete maximum principle, possess a second-order accuracy, be exact for linear solution problems, and be available for anisotropic problems on distorted meshes.