We develop fractional buffer layers (FBLs) to absorb propagating waves without reflection in bounded domains. Our formulation is based on variable-order spatial fractional derivatives. We select a proper variable-order function so that dissipation is induced to absorb the coming waves in the buffer layers attached to the domain. In particular, we first design proper FBLs for the one-dimensional one-way and two-way wave propagation. Then, we extend our formulation to two-dimensional problems, where we introduce a consistent variable-order fractional wave equation. In each case, we obtain the fully discretized equations by employing a spectral collocation method in space and Crank-Nicolson or Adams-Bashforth method in time. We compare our results with a finely tuned perfectly matched layer (PML) method and show that the proposed FBL is able to suppress reflected waves including corner reflections in a two-dimensional rectangular domain. We also demonstrate that our formulation is more robust and uses less number of equations.

This paper proposes a high order deep neural network (HOrderDNN) for solving high frequency partial differential equations (PDEs), which incorporates the idea of "high order" from finite element methods (FEMs) into commonly-used deep neural networks (DNNs) to obtain greater approximation ability. The main idea of HOrderDNN is introducing a nonlinear transformation layer between the input layer and the first hidden layer to form a high order polynomial space with the degree not exceeding $p$, followed by a normal DNN. The order $p$ can be guided by the regularity of solutions of PDEs. The performance of HOrderDNN is evaluated on high frequency function fitting problems and high frequency Poisson and Helmholtz equations. The results demonstrate that: HOrderDNNs($p > 1$) can efficiently capture the high frequency information in target functions; and when compared to physics-informed neural network (PINN), HOrderDNNs($p > 1$) converge faster and achieve much smaller relative errors with same number of trainable parameters. In particular, when solving the high frequency Helmholtz equation in Section 3.5, the relative error of PINN stays around 1 with its depth and width increase, while the relative error can be reduced to around 0.02 as $p$ increases (see Table 5).

The discrete duality finite volume method has proven to be a practical tool for discretizing partial differential equations coming from a wide variety of areas of physics on nearly arbitrary meshes. The main ingredients of the method are: (1) use of three meshes, (2) use of the Gauss-Green theorem for the approximation of derivatives, (3) discrete integration by parts. In this article we propose to extend this method to the coupled grey thermal-$P_N$ radiative transfer equations in Cartesian and cylindrical coordinates in order to be able to deal with two-dimensional Lagrangian approximations of the interaction of matter with radiation. The stability under a Courant-Friedrichs-Lewy condition and the preservation of the diffusion asymptotic limit are proved while the experimental second-order accuracy is observed with manufactured solutions. Several numerical experiments are reported which show the good behavior of the method.

Graphene nanoribbons are considered as one of the most promising ways to design electron devices where the active area is made of graphene. In fact, graphene nanoribbons present a gap between the valence and the conduction bands as in standard semiconductors such as Si or GaAs, at variance with large area graphene which is gapless, a feature that hampers a good performance of graphene field effect transistors.
To use graphene nanoribbons as a semiconductor, an accurate analysis of their electron properties is needed. Here, electron transport in graphene nanoribbons is investigated by solving the semiclassical Boltzmann equation with a discontinuous Galerkin method. All the electron-phonon scattering mechanisms are included. The adopted energy band structure is that devised in [1] while according to [2] the edge effects are described as an additional scattering stemming from the Berry-Mondragon model which is valid in presence of edge disorder. With this approach a spacial 1D transport problem has been solved, even if it remains two dimensional in the wave-vector space. A degradation of charge velocities, and consequently of the mobilities, is found by reducing the nanoribbon width due mainly to the edge scattering.

Calculations of properties of materials require performing numerical integrals over the Brillouin zone (BZ). Integration points in density functional theory codes are uniformly spread over the BZ (despite integration error being concentrated in small regions of the BZ) and preserve symmetry to improve computational efficiency. Integration points over an irreducible Brillouin zone (IBZ), a rotationally distinct region of the BZ, do not have to preserve crystal symmetry for greater efficiency. This freedom allows the use of adaptive meshes with higher concentrations of points at locations of large error, resulting in improved algorithmic efficiency. We have created an algorithm for constructing an IBZ of any crystal structure in 2D and 3D. The algorithm uses convex hull and half-space representations for the BZ and IBZ to make many aspects of construction and symmetry reduction of the BZ trivial. The algorithm is simple, general, and available as open-source software.

In this paper, we propose a local discontinuous Galerkin (LDG) method for the multi-dimensional stochastic Cahn-Hilliard type equation in a general form, which involves second-order derivative $∆u$ in the multiplicative noise. The stability of our scheme is proved for arbitrary polygonal domain with triangular meshes. We get the sub-optimal error estimate $\mathbb{O}(h^k)$ if the Cartesian meshes with $Q^k$ elements are used. Numerical examples are given to display the performance of the LDG method.

Existing mapped WENO schemes can hardly prevent spurious oscillations while preserving high resolutions at long output times. We reveal in this paper the essential reason of such phenomena. It is actually caused by that the mapping function in these schemes can not preserve the order of the nonlinear weights of the stencils. The nonlinear weights may be increased for non-smooth stencils and be decreased for smooth stencils. It is then indicated to require the set of mapping functions to be order-preserving in mapped WENO schemes. Therefore, we propose a new mapped WENO scheme with a set of mapping functions to be order-preserving which exhibits a remarkable advantage over the mapped WENO schemes in references. For long output time simulations of the one-dimensional linear advection equation, the new scheme has the capacity to attain high resolutions and avoid spurious oscillations near discontinuities meanwhile. In addition, for the two-dimensional Euler problems with strong shock waves, the new scheme can significantly reduce the numerical oscillations.

In this paper, a novel mixed wavelet-learning method is developed for predicting macroscopic effective heat transfer conductivities of braided composite materials with heterogeneous thermal conductivity. This innovative methodology integrates respective superiorities of multi-scale modeling, wavelet transform and neural networks together. By the aid of asymptotic homogenization method (AHM), off-line multi-scale modeling is accomplished for establishing the material database with high-dimensional and highly-complex mappings. The multi-scale material database and the wavelet-learning strategy ease the on-line training of neural networks, and enable us to efficiently build relatively simple networks that have an essentially increasing capacity and resisting noise for approximating the high-complexity mappings. Moreover, it should be emphasized that the wavelet-learning strategy can not only extract essential data characteristics from the material database, but also achieve a tremendous reduction in input data of neural networks. The numerical experiments performed using multiple 3D braided composite models verify the excellent performance of the presented mixed approach. The numerical results demonstrate that the mixed wavelet-learning methodology is a robust method for computing the macroscopic effective heat transfer conductivities with distinct heterogeneity patterns. The presented method can enormously decrease the computational time, and can be further expanded into estimating macroscopic effective mechanical properties of braided composites.

In present paper, mesh adaptation is applied for curing the pathological behaviors of the enhanced time-accurate upwind scheme (Loh & Jorgenson, AIAAJ 2016). In the original ETAU (enhanced time-accurate upwind) scheme, a multi-dimensional dissipation model is required to cure the pathological behaviors. The multi-dimensional dissipation model will increase the global dissipation level reducing numerical resolution. In present work, the metric-based mesh adaptation strategy provides an alternative way to cure the pathological behaviors of the shock capturing. The Hessian matrix of flow variables is applied to construct the metric, which represents the curvature of the physical solution. The adapting operation can well refine the anisotropic meshes at the location with large gradients. The numerical results show that the adaptation of mesh provides a possible way to cure the pathological behaviors of upwind schemes.

The near-wall domain decomposition method (NDD) has proved to be very efficient for modeling near-wall fully turbulent flows. In this paper the NDD is extended to non-equilibrium regimes with laminar-turbulent transition (LTT) for the first time. The LTT is identified with the use of the $e^N$-method which is applied to both incompressible and compressible flows. The NDD is modified to take into account LTT in an efficient way. In addition, implementation of the intermittency expands the capabilities of NDD to model non-equilibrium turbulent flows with transition. Performance of the modified NDD approach is demonstrated on various test problems of subsonic and supersonic flows past a flat plate, a supersonic flow over a compression corner and a planar shock wave impinging on a turbulent boundary layer. The results of modeling with and without decomposition are compared in terms of wall friction and show good agreement with each other while NDD significantly reducing computational resources needed. It turns out that the NDD can reduce the computational time as much as three times while retaining practically the same accuracy of prediction.