In this paper, we construct high order energy dissipative and conservative local discontinuous Galerkin methods for the Fornberg-Whitham type equations. We give the proofs for the dissipation and conservation for related conservative quantities. The corresponding error estimates are proved for the proposed schemes. The capability of our schemes for different types of solutions is shown via several numerical experiments. The dissipative schemes have good behavior for shock solutions, while for a long time approximation, the conservative schemes can reduce the shape error and the decay of amplitude significantly.

The multiple scattering method T-matrix (MSTMM) can be used to solve the electromagnetic response of systems consisting of many compact scatterers, retaining a good level of accuracy while using relatively few degrees of freedom, largely surpassing other methods in the number of scatterers it can deal with. Here we extend the method to infinite periodic structures using Ewald-type lattice summation, and we exploit the possible symmetries of the structure to further improve its efficiency, so that systems containing tens of thousands of particles can be studied with relative ease. We release a modern implementation of the method, including the theoretical improvements presented here, under GNU General Public Licence.

We consider reduced order modelling of elastodynamics with proper orthogonal decomposition and isogeometric analysis, a recent novel and promising discretization method for partial differential equations. The generalized-$α$ method for transient problems is used for additional flexibility in controlling high frequency dissipation. We propose a fully discrete scheme for the elastic wave equation with isogeometric analysis for spatial discretization, generalized-$α$ method for time discretization, and proper orthogonal decomposition for model order reduction. Numerical convergence and dispersion are shown in detail to show the feasibility of the method. A variety of numerical examples in both 2D and 3D are provided to show the effectiveness of our method.

In this paper, we investigate the ideal magnetohydrodynamic (MHD) equations with random inputs based on generalized polynomial chaos (gPC) stochastic Galerkin approximation. A special treatment with symmetrization is carried out for the gPC stochastic Galerkin method so that the resulting deterministic gPC Galerkin system is provably symmetric hyperbolic in the spatially one-dimensional case. We discretize the hyperbolic gPC Galerkin system with a high-order path-conservative finite volume weighted essentially non-oscillatory scheme in space and a third-order total variation diminishing Runge-Kutta method in time. The method is also extended to two spatial dimensions via the operator splitting technique. Several numerical examples are provided to illustrate the accuracy and effectiveness of the numerical scheme.

This paper develops an efficient positivity-preserving finite volume scheme for the two-dimensional nonequilibrium three-temperature radiation diffusion equations on general polygonal meshes. The scheme is formed as a predictor-corrector algorithm. The corrector phase obtains the cell-centered solutions on the primary mesh, while the predictor phase determines the cell-vertex solutions on the dual mesh independently. Moreover, the flux on the primary edge is approximated with a fixed stencil and the nonnegative cell-vertex solutions are not reconstructed. Theoretically, our scheme does not require any nonlinear iteration for the linear problems, and can call the fast nonlinear solver (e.g. Newton method) for the nonlinear problems. The positivity, existence and uniqueness of the cell-centered solutions obtained on the corrector phase are analyzed, and the scheme on quasi-uniform meshes is proved to be $L^2$- and $H^1$-stable under some assumptions. Numerical experiments demonstrate the accuracy, efficiency and positivity of the scheme on various distorted meshes.

Ionic flow carries electrical signals for cells to communicate with each other. The permanent charge of an ion channel is a crucial protein structure for flow properties while boundary conditions play a role of the driving force. Their effects on flow properties have been analyzed via a quasi-one-dimensional Poisson-Nernst-Planck model for small and relatively large permanent charges. The analytical studies have led to the introduction of flux ratios that reflect permanent charge effects and have a universal property. The studies also show that the flux ratios have different behaviors for small and large permanent charges. However, the existing analytical techniques can reveal neither behaviors of flux ratios nor transitions between small and large permanent charges. In this work we present a numerical investigation on flux ratios to bridge between small and large permanent charges. Numerical results verify the analytical predictions for the two extremal regions. More significantly, emergence of non-trivial behaviors is detected as the permanent charge varies from small to large. In particular, saddle-node bifurcations of flux ratios are revealed, showing rich phenomena of permanent charge effects by the power of combining analytical and numerical techniques. An adaptive moving mesh finite element method is used in the numerical studies.

An adaptive Cartesian method combined with the ghost cell method is proposed to solve problems of unsteady aircraft icing simulation in this paper. In the grid generation module, Cartesian method is used to generate the background grids around the clean and iced geometries. Boundary self-adaptive method is developed to update the grids as the ice accumulates and the geometry is changing over time. Besides, local encryption is carried out around the boundary grids in order to improve the prediction accuracy. An improved ray method is used to classify the grids into four types, named fluid grids, solid grids, boundary grids and ghost grids. The flow field is obtained by solving Euler equations, and the ghost cell method is introduced to provide the boundary conditions due to the non-body-fitted feature of Cartesian grids. Droplet trajectories are calculated using Lagrangian method. And a new and efficient droplet location judgment method is proposed to determine whether the droplet impinges on the surface. Besides, the ice accretion behaviors are predicted using Messinger model. With the method proposed in this paper, extensive numerical tests in various icing temperatures are simulated. And, the computational results are compared with test results. It can be seen that the proposed new method can predict the unsteady process of aircraft ice accretion and the compared results also show better agreements for both glaze and rime ice.

In this work, two fully discrete grad-div stabilized finite element schemes for the fluid-fluid interaction model are considered. The first scheme is standard grad-div stabilized scheme, and the other one is modular grad-div stabilized scheme which adds to Euler backward scheme an update step and does not increase computational time for increasing stabilized parameters. Moreover, stability and error estimates of these schemes are given. Finally, computational tests are provided to verify both the numerical theory and efficiency of the presented schemes.

It is reasonable to assume that a discrete convolution structure dominates the local truncation error of any numerical Caputo formula because the fractional time derivative and its discrete approximation have the same convolutional form. We suggest an error convolution structure (ECS) analysis for a class of interpolation-type approximations to the Caputo fractional derivative. Our assumptions permit the use of adaptive time steps, such as is appropriate for accurately resolving the initial singularity of the solution and also certain complex behavior away from the initial time. The ECS analysis of numerical approximations has two advantages: (i) to localize (and simplify) the analysis of the approximation error of a discrete convolution formula on general nonuniform time grids; and (ii) to reveal the error distribution information in the long-time integration via the global consistency error. The core result in this paper is an ECS bound and a global consistency analysis of the nonuniform Alikhanov approximation, which is constructed at an offset point by using linear and quadratic polynomial interpolation. Using this result, we derive a sharp $L^2$-norm error estimate of a second-order Crank-Nicolson-like scheme for linear reaction-subdiffusion problems. An example is presented to show the sharpness of our analysis.

In many of the plasmas generated in many fields of the high energy density physics, the radiation can significantly alter the material response. A proper microscopic description of these plasmas entails integrated computer codes that self-consistently combines large-scale atomic kinetics and radiation transport. Due to the inherent complexity of this type of codes and its interest in the area of high energy density physics, new developments in this field are welcomed. In this work, we present MIXKIP/RAPCAL, an integrated computational package to perform 1D and 2D large-scale non-local thermodynamic equilibrium atomic kinetics and radiation transfer coupled simulations for high energy density plasmas. This package includes different modules that allow simulations of non-equilibrium plasmas under different degrees of detail and accuracy, depending on the requirements of the situations to analyze. Comparisons with experimental results of homogeneous optically thick plasmas of gold and xenon are presented in order to check its accuracy. The influence of different approaches of the spatial discretization of the plasma in the radiation dependent atomic kinetics simulations is also analyzed. Finally, this study is also made for a non-homogeneous optically thick aluminum plasma.