In this paper, we consider the optimization problem of identifying the pollution sources of convection-diffusion-reaction equations in a groundwater process. The optimization model is subject to a convection-diffusion-reaction equation with pumping point and pollution point sources. We develop a linked optimization and simulation approach combining with the Differential Evolution (DE) optimization algorithm to identify the pumping and injection rates from the data at the observation points. Numerical experiments are taken with injections of constant rates and time-dependent variable rates at source points. The problem with one pumping point and two pollution source points is also studied. Numerical results show that the proposed method is efficient. The developed optimized identification approach can be extended to high-dimensional and more complex problems.

The forced-vibration response of a simply-supported isotropic thick-walled hollow elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surfaces is studied within the framework of linear elastodynamics. Exact semi-analytical solutions for the steady-state displacement field of the cylinder are constructed using recently-published parametric solutions to the Navier-Lame equation. Formal application of the standing-wave boundary conditions generates three parameter-dependent $6\times6$ linear systems, each of which can be numerically solved in order to determine the parametric response of the cylinder's displacement field under various conditions. The method of solution is direct and demonstrates a general approach that can be applied to solve many other elastodynamic forced-response problems involving isotropic elastic cylinders. As an application, and considering several examples, the obtained solution is used to compute the steady-state frequency response in a few specific low-order excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a unique subset of the natural frequencies of the simply-supported cylinder. The considered problem is of general theoretical interest in structural mechanics and acoustics and more practically serves as a benchmark forced-vibration problem involving a thick-walled hollow elastic cylinder.

In this paper, a fifth-order weighted essentially nonoscillatory scheme is presented for simulating dam-break flows in a finite difference framework. The new scheme is a convex combination of two quadratic polynomials with a fourth-degree polynomial in a classical WENO fashion. The distinguishing feature of the present method is that the same five-point information is used but smaller absolute truncation errors and the same accuracy order in the smooth region are obtained. The new nonlinear weights are presented by Taylor expansion of the smoothness indicators of the small stencils to sustain the optimal fifth-order accuracy. The linear advection equation, nonlinear scalar Burgers equation, and one- and two-dimensional Euler equations are used to validate the high-order accuracy and excellent resolution of the presented method. Finally, one- and two-dimensional Saint-Venant equations are tested by using the new fifth-order scheme to simulate a dam-break flow.

In this paper, a novel immersed boundary method is presented for simulating incompressible flows governed by Navier-Stokes equations. A virtual boundary is formed by the cell edges (for two-dimensional cases) in the vicinity of the immersed boundary. In the domain with the virtual boundary, the governing equations can be solved in the conventional way. Reconstructed velocity is imposed on the virtual boundary, which is determined via the interpolation along the direction normal to the wall and in conjunction with the no-slip condition for the actual boundary. For "freshly cleared nodes" on the virtual boundary encountered in moving-boundary problems, pressure at the previous time step is reconstructed by solving the local simplified momentum equation. In the test case for an analytical solution, the local accuracy of pressure is verified to be of the second order. In order to further validate the present method, various flows over the stationary and/or moving circular cylinder and NACA0012 airfoil have been simulated. The obtained results agree well with the available numerical or experimental data in the published literatures.

In this paper, we propose a novel Legendre neural network combined with the extreme learning machine algorithm to solve variable coefficients linear delay differential-algebraic equations with weak discontinuities. First, the solution interval is divided into multiple subintervals by weak discontinuity points. Then, Legendre neural network is used to eliminate the  hidden layer by expanding the input pattern using Legendre polynomials on each subinterval. Finally, the parameters of the neural network are obtained by training with the extreme learning machine. The numerical examples show that the proposed method can effectively deal with the difficulty of numerical simulation caused by the discontinuities.

In framework of the fictitious domain methods with immersed interfaces for the elasticity problem, the present contribution is to study and numerically validate the jump-integrated boundary conditions method with sharp interface for the vector elasticity system discretized by a proposed finite volume method. The main idea of the fictitious domain approach consists in embedding the original domain of study into a geometrically larger and simpler one called the fictitious domain. Here, we present a cell-centered finite volume method to discretize the fictitious domain problem. The proposed method is numerically validated for different test cases. This work can be considered as a first step before more challenging problems such as fluid-structure interactions or moving interface problems.

A convolutional neural network is developed for rapidly predicting multiphase flow in heterogeneous porous media. Some direct numerical methods can acquire accurate results of multiphase flow in porous media. However, once the geometry of the porous media changes, it takes much computational time to perform a new simulation. Here, a deep neural network model in the field of semantic segmentation is developed. It takes the two-dimensional microstructure of heterogeneous porous media as inputs and is able to predict corresponding multiphase flow fields (pressure and saturation fields). Compared to the direct lattice Boltzmann simulations, the inference time on new geometry of porous media can be reduced by several orders of magnitude. Our results show that the machine learning method is a good prediction tool in a wide range of porosity and heterogeneity. Besides, to better understand the inherent process, a visible explanation is presented on what our neural networks have learned.

In this paper, the recently-developed singular boundary method is applied to address free boundary problems. This mesh-less numerical method is based on the use of the origin intensity factors with fundamental solutions. Three numerical examples and their results are compared with the results obtained using traditional methods. The comparisons indicate that the proposed scheme yields good results in determining the position of the free boundary.

In this paper, some two-grid finite-volume methods are constructed for solving the steady-state Schrödinger equation. The method projects the original coupled problem onto a coarser grid, on which it is less expensive to solve, and then prolongates the approximated coarse solution back to the fine grid, on which it is not much more difficult to solve the decoupled problem. We have shown, both theoretically and numerically,  that our schemes are more efficient and achieve asymptotically optimal accuracy as long as the mesh sizes satisfy $h=\mathcal{O}(H^2)$.

In this paper, we propose a finite element time-domain (FETD) method for the Maxwell's equations in chiral metamaterials (CMMs). The time-domain model equations are constructed by the auxiliary differential equations (ADEs) method. The source excitation method entitled total-field and scattered-field (TF/SF) decomposition technique is applied to FETD method for the first time in simulating the propagation of electromagnetic wave in CMMs, based on which a unified ADE-FETD-UPML-TF/SF scheme is proposed to simulate the wave in CMMs. The following properties of CMMs can be observed successfully from the numerical experiments based on our method, i.e., the ability of the polarization rotation, and the negative phase velocity. The amplitude of reflected wave can effectively be controlled by the physical parameters of CMMs.

An energy-preserving scheme is proposed for the coupled Gross-Pitaevskii equations.  The scheme is constructed by high order compact method in the spatial direction and average vector field method in the temporal direction, respectively. The scheme is energy-preserving, stable, and of sixth order in space and of second order in time. Numerical experiments verify the theoretical results. The dynamic behavior modeled by the coupled Gross-Pitaevskii equations is also numerically investigated.

In this paper, we propose a new conservative gradient discretization method (GDM) for one-dimensional parabolic partial differential equations (PDEs). We use the implicit Euler method for the temporal discretization and conservative gradient discretization method  for spatial discretization. The method is based on a new cell-centered meshes, and it is locally conservative. It has smaller truncation error than the classical finite volume method on uniform meshes. We use the framework of the gradient discretization method to analyze the stability and convergence. The numerical experiments show that the new method has second-order convergence. Moreover, it is more accurate than the classical finite volume method in flux error, $L^2$ error and $L^{\infty}$ error.