Recent analytical solutions for peridynamic (PD) models of transient diffusion and elastodynamics allow us to revisit convergence of 1D PD models to their classical counterparts. We find and explain the reasons for some interesting differences between the convergence behavior for transient diffusion and elastodynamics models. Except for very early times, PD models for transient diffusion converge monotonically to the classical one. In contrast, for elastodynamic problems this convergence is more complex, with some periodic/almost-periodic characteristics present. These special features are investigated for sine waves used as initial conditions. The analysis indicates that the convergence behavior of PD solutions to the classical one can be understood in terms of convergence properties for models using the Fourier series expansion terms of a particular initial condition. The results obtained show new connections between PD models and their corresponding classical versions.

The iterative solution of the sequence of linear systems arising from three-temperature (3-T) energy equations is an essential component in the numerical simulation of radiative hydrodynamic (RHD) problem. However, due to the complicated application features of the RHD problems, solving 3-T linear systems with classical preconditioned iterative techniques is challenging. To address this difficulty, a physical-variable based coarsening two-level (PCTL) preconditioner has been proposed by dividing the fully coupled system into four individual easier-to-solve subsystems. Despite its nearly optimal complexity and robustness, the PCTL algorithm suffers from poor efficiency because of the overhead associated with the construction of setup phase and the solution of subsystems. Furthermore, the PCTL algorithm employs a fixed strategy for solving the sequence of 3-T linear systems, which completely ignores the dynamically and slowly changing features of these linear systems. To address these problems and to efficiently solve the sequence of 3-T linear systems, we propose an adaptive two-level preconditioner based on the PCTL algorithm, referred to as $α$Setup-PCTL. The adaptive strategies of the $α$Setup-PCTL algorithm are inspired by those of $α$Setup-AMG algorithm, which is an adaptive-setup-based AMG solver for sequence of sparse linear systems. The proposed $α$Setup-PCTL algorithm could adaptively employ the appropriate strategies for each linear system, and thus increase the overall efficiency. Numerical results demonstrate that, for 36 linear systems, the $α$Setup-PCTL algorithm achieves an average speedup of 2.2, and a maximum speedup of 4.2 when compared to the PCTL algorithm.

We propose an efficient numerical method for the simulation of the two-phase flows with moving contact lines in three dimensions. The mathematical model consists of the incompressible Navier-Stokes equations for the two immiscible fluids with the standard interface conditions, the Navier slip condition along the solid wall, and a contact angle condition (Ren et al. (2010) [28]). In the numerical method, the governing equations for the fluid dynamics are coupled with an advection equation for a level-set function. The latter models the dynamics of the fluid interface. Following the standard practice, the interface conditions are taken into account by introducing a singular force on the interface in the momentum equation. This results in a single set of governing equations in the whole fluid domain. Similarly, the contact angle condition is imposed by introducing a singular force, which acts in the normal direction of the contact line, into the Navier slip condition. The new boundary condition, which unifies the Navier slip condition and the contact angle condition, is imposed along the solid wall. The model is solved using the finite difference method. Numerical results are presented for the spreading of a droplet on both homogeneous and inhomogeneous solid walls, as well as the dynamics of a droplet on an inclined plate under gravity.

Treecode algorithms are widely used in evaluation of $N$-body pairwise interactions in $\mathcal{O}(N)$ or $\mathcal{O}(NlogN)$ operations. While they can provide high accuracy approximations, a criticism leveled at the methods is that they lack global smoothness. In this work, we study the effect of smoothness on the accuracy of treecodes by comparing three tricubic interpolation based treecodes with differing smoothness properties: a global $\mathcal{C}^1$ continuous tricubic, and two new tricubic interpolants, one that is globally $\mathcal{C}^0$ continuous and one that is discontinuous. We present numerical results which show that higher smoothness leads to higher accuracy for properties dependent on the derivatives of the kernel, nevertheless the global $\mathcal{C}^0$ continuous and discontinuous treecodes are competitive with the $\mathcal{C}^1$ continuous treecode. One advantage of the discontinuous treecode over the $\mathcal{C}^1$ continuous is that, in addition to function evaluations, the discontinuous treecode only requires evaluations of the first derivatives of the kernel while the $\mathcal{C}^1$ continuous treecode requires evaluations up to third order derivatives. When the first derivatives are computed using finite differences, the discontinuous version can be viewed as kernel independent and of utility for a wider array of kernels with minimal effort.

We introduce a new function space, dubbed as the Barron spectrum space, which arises from the target function space for the neural network approximation. We give a Bernstein type sufficient condition for functions in this space, and clarify the embedding among the Barron spectrum space, the Bessel potential space, the Besov space and the Sobolev space. Moreover, the unexpected smoothness and the decaying behavior of the radial functions in the Barron spectrum space have been investigated. As an application, we prove a dimension explicit $L^q$ error bound for the two-layer neural network with the Barron spectrum space as the target function space, the rate is dimension independent.

We propose a hybrid scheme combing the diffuse interface method and the material point method to simulate the complex interactions between the multiphase compressible flow and elastoplastic solid. The multiphase flow is modelled by the multi-component model and solved using a generalized Godunov method in the Eulerian grids, while the elastoplastic solid is solved by the classical material point method in a combination of Lagrangian particles and Eulerian background grids. In order to facilitate the simulation of fluid-solid interactions, the solid variables are further interpolated to the cell center and coexist with the fluid in the same cell. An instantaneous relaxation procedure of velocity and pressure is adopted to simulate the momentum and energy transfers between various materials, and to keep the system within a tightly coupled interaction. Several numerical examples, including shock tube problem, gas-bubble problem, air blast, underwater explosion and high speed impact applications are presented to validate the numerical scheme.

In this paper, we correct the finite volume element methods for diffusion equations on general triangular and quadrilateral meshes. First, we decompose the numerical fluxes of original schemes into two parts, i.e., the principal part with a two-point flux structure and the defective part. And then with the help of local extremums, we transform the original numerical fluxes into nonlinear numerical fluxes, which can be expressed as a nonlinear combination of two-point fluxes. It is proved that the corrected schemes satisfy the discrete strong extremum principle without restrictions on the diffusion coefficient and meshes. Numerical results indicate that the corrected schemes not only satisfy the discrete strong extremum principle but also preserve the convergence order of the original finite volume element methods.

In this paper, we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasi-geostrophic equations, which include a spectral fractional Laplacian determining the vorticity for the transport velocity of the potential temperature. The fractional Laplacian is approximated by a Fourier-Galerkin spectral method, while the time evolution of the potential temperature is discretized by a high order conservative finite difference scheme. Weighted essentially non-oscillatory (WENO) reconstructions are also considered for comparison. Due to a low regularity of problems involving such a fractional Laplacian, especially in the critical or supercritical regime, directly applying the Fourier spectral method leads to a very oscillatory transport velocity associated with the gradient of the vorticity, e.g. around smooth extrema. Instead of using an artificial filter, we propose to reconstruct the velocity from the vorticity with central difference discretizations. Numerical results are performed to demonstrate the good performance of our proposed approach.