Even for state-of-the-art implicit LES (ILES) methods, where the truncation error acts as physically-motivated subgrid-scale model, simultaneously resolving turbulent and genuine non-turbulent subgrid scales is an open challenge. For the purpose of dealing with non-turbulent subgrid scales, such as shocks, extra sensors, which often are case-dependent, are generally employed. The problem originates in the lack of scale-separation between low-wavenumber resolved-scale regions, high-wavenumber resolved or non-resolved fluctuations, and discontinuities. The targeted ENO (TENO) approach allows for separately designing the dispersive and dissipative truncation error components. Thus it provides a suitable environment to develop an implicit LES model. In this paper, we extend previous work and propose a variant of TENO family scheme [Fu et al., JCP 305 (2016): 333-359], which can separate resolved and nonresolved scales effectively. The novel idea is to propose a nonlinear dissipation-control strategy by adapting the cut-off parameter C_T dynamically while measuring the nonsmoothness based on the first-order undivided difference. Low-wavenumber smooth scales are handled by an optimized linear scheme while high-wavenumber components, that involve nonresolved fluctuations and discontinuities, are subjected to adaptive nonlinear dissipation. A set of benchmark simulations with a wide range of length-scales and with discontinuities has been conducted without specific parameter adaptation. Numerical experiments demonstrate that the proposed TENO8-A scheme exhibits robust shock-capturing and high wave-resolution properties, and that it is suitable for simulating flow fields that contain isotropic turbulence and shocks. It is a promising alternative to other viable approaches.

Compositional gas flow in a heterogeneous porous medium and in the coupled atmospheric boundary layer above the porous medium surface is of interest in many applications, which requires reliable numerical tools for modeling of very complex physical processes. But there are still many important effects which are very often ignored in contemporary models of this flow. One of them is compressibility. So far, no models of non-isothermal compositional compressible gas flow in a porous medium and in the coupled atmospheric boundary layer above its surface has been reported in the literature. Therefore, we propose mathematical and numerical models for the description of the above scenario. In order to assess the reliability of our numerical model, we analyze its convergence by quantitative computational studies. We also present several qualitative computational studies which present the dynamics of the non-isothermal compositional compressible gas flow in free flow–porous medium flow interaction.

In this paper, we present a finite volume method for solving Poisson-Nernst-Planck (PNP) equations in one spatial dimension. To reduce computational cost, an adaptive moving mesh strategy is employed in order to resolve thin Debye layers near the boundary. In addition to the standard monitor functions, we propose two new ones for the moving mesh partial differential equations to improve the accuracy of the numerical solution. The method guarantees the strict mass conservation. We have proved that the scheme maintains positivity on the adaptive moving mesh which has not been done for PNP.

A new approach for numerically solving 3-T diffusion equations on 2-D scattered point distributions is developed by the finite point method. In this paper, a new method for selecting neighboring points is designed, which is robust and well reflects variations of gradients of physical quantities. Based on this, a new discretization method is proposed for the diffusion operator, which results in a new scheme with the stencil of minimal size for numerically solving nonlinear diffusion equations. Distinguished from most of meshless methods often involving dozens of neighboring points, this method needs only five neighbors of the point under consideration. Numerical simulations show the good performance of the proposed methodology.

We propose and study a double source transfer domain decomposition method (Double STDDM) for solving the truncated perfectly matched layer approximation in the bounded domain of Helmholtz problems. Based on the decomposition of the domain into non-overlapping layers and instead of transferring the source along one direction in STDDM [Z. Chen and X. Xiang, 2013], Double STDDM transfers the source in each layer along two directions, which can capture of the reflection information for heterogenous media. Double STDDM is an iterative scheme, and in each iteration, it first transfers the source from down to up and produces the Up wave (the wave propagating from down to up), and then transfers the source from up to down and produces the Down wave (the wave propagating from up to down). The output of Double STDDM is the summation of the Up and Down waves that are produced during the iteration. By using the fundamental solution of the PML equation, the convergence of Double STDDM is proved for the case of a constant wavenumber.
Numerical examples are included to show the efficient performance of using Double STDDM as a preconditioner both for the problems with constant and heterogenous wavenumbers. For problems with a low velocity contrast, the number of iterations is independent of the wavenumber and mesh size, whereas for problems with a high velocity contrast, double STDDM performs much better than STDDM.

In this paper, we reformulate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Gaussian wave packet transform. With this approach, the highly oscillatory Schrödinger equation is equivalently transformed into another Schrödinger type wave equation, the w equation, which is essentially not oscillatory and thus requires much less computational effort. We propose two numerical methods to solve the w equation, where the Hamiltonian is either divided into the kinetic, the potential and the convection part, or into the kinetic and the potential-convection part. The convection, or the potential-convection part is treated by a semi-Lagrangian method, while the kinetic part is solved by the Fourier spectral method. The numerical methods are proved to be unconditionally stable, spectrally accurate in space and second order accurate in time, and in principle they can be extended to higher order schemes in time. Various one dimensional and multidimensional numerical tests are provided to justify the properties of the proposed methods.

In this paper, we generalize the point integral method to solve the general elliptic PDEs with variable coefficients and corresponding eigenvalue problems with Neumann, Robin and Dirichlet boundary conditions on point cloud. The main idea is using integral equations to approximate the original PDEs. The integral equations are easy to discretize on the point cloud. The truncation error of the integral approximation is analyzed. Numerical examples are presented to demonstrate that PIM is an effective method to solve the elliptic PDEs with smooth coefficients on point cloud.

In this paper, we consider the semi-implicit spectral deferred correction (SDC) methods for hyperbolic systems of conservation laws with stiff relaxation terms. The relaxation term is treated implicitly, and the convection terms are treated by explicit schemes. The SDC schemes proposed are asymptotic preserving (AP) in the zero relaxation limit and can be constructed easily and systematically for any order of accuracy. Weighted essentially non-oscillatory (WENO) schemes are adopted in spatial discretization to achieve high order accuracy. After a description of the asymptotic preserving property of the SDC schemes, several applications will be presented to demonstrate the stiff accuracy and capability of the schemes.

In this paper, we present a simple a posteriori error estimate for the weak Galerkin (WG) finite element method for a model second order elliptic equation. This residual type estimator can be applied to general meshes such as hybrid, polytopal and those with hanging nodes. We prove the reliability and efficiency of the estimator. Extensive numerical tests demonstrate the effectiveness and flexibility of the mesh refinement guided by this error estimator.

In this paper, we present an efficient numerical scheme to calculate N-periodic wave solutions to the Toda-type equations. The starting point is the algebraic condition for having N-periodic wave solutions proposed by Akira Nakamura. The basic idea is to formulate the condition as a nonlinear least square problem and then use the Gauss-Newton method to solve it. By use of this numerical scheme, we calculate the 3-periodic wave solutions to some discrete integrable equations such as the Toda lattice equation, the Lotka-Volterra equation, the differential-difference KP equation and so on.

We develop a second order well-balanced finite volume scheme for compressible Euler equations with a gravitational source term. The well-balanced property holds for arbitrary hydrostatic solutions of the corresponding Euler equations without any restriction on the equation of state. The hydrostatic solution must be known a priori either as an analytical formula or as a discrete solution at the grid points. The scheme can be applied to curvilinear meshes and in combination with any consistent numerical flux function and time stepping routines. These properties are demonstrated on a range of numerical tests.