In this paper we generalize a new type of compact Hermite weighted essentially non-oscillatory (HWENO) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method, which was recently developed in [38] for structured meshes, to two dimensional unstructured meshes. The main idea of this HWENO limiter is to reconstruct the new polynomial by the usage of the entire polynomials of the DG solution from the target cell and its neighboring cells in a least squares fashion [11] while maintaining the conservative property, then use the classical WENO methodology to form a convex combination of these reconstructed polynomials based on the smoothness indicators and associated nonlinear weights. The main advantage of this new HWENO limiter is the robustness for very strong shocks and simplicity in implementation especially for the unstructured meshes considered in this paper, since only information from the target cell and its immediate neighbors is needed. Numerical results for both scalar and system equations are provided to test and verify the good performance of this new limiter.

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total $\mathcal{O}$($N_S$$N_T$) memory and $\mathcal{O}$(($N_S$$N^2_T$) work, where $N_T$ and $N_S$ represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative $^C_0$$D^α_t$$f(t)$ of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel $t^{−1−α}$ on the interval [∆t,T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials $N_{exp}$ needed is of order $\mathcal{O}$($log$$N_T$) for T≫1 or $\mathcal{O}$($log^2N_T$) for T≈1 for fixed accuracy ε. The resulting algorithm requires only $\mathcal{O}$($N_SN_{exp}$) storage and $\mathcal{O}$($N_SN_TN_{exp}$) work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

In this paper, we present an efficient rescaling scheme for computing the long-time dynamics of expanding interfaces. The idea is to design an adaptive time-space mapping such that in the new time scale, the interfaces evolves logarithmically fast at early growth stage and exponentially fast at later times. The new spatial scale guarantees the conservation of the area/volume enclosed by the interface. Compared with the original rescaling method in [J. Comput. Phys. 225(1) (2007) 554–567], this adaptive scheme dramatically improves the slow evolution at early times when the size of the interface is small. Our results show that the original three-week computation in [J. Comput. Phys. 225(1) (2007) 554–567] can be reproduced in about one day using the adaptive scheme. We then present the largest and most complicated Hele-Shaw simulation up to date.

The lattice Boltzmann method is employed to simulate the steady flow in a two-dimensional lid-driven semi-elliptical cavity. Reynolds number (Re) and vertical-to-horizontal semi-axis ratio (D) are in the range of 500-5000 and 0.1-4, respectively. The effects of Re and D on the vortex structure and pressure field are investigated, and the evolutionary features of the vortex structure with Re and D are analyzed in detail. Simulation results show that the vortex structure and its evolutionary features significantly depend on Re and D. The steady flow is characterized by one vortex in the semi-elliptical cavity when both Re and D are small. As Re increases, the appearance of the vortex structure becomes more complex. When D is less than 1, increasing D makes the large vortexes more round, and the evolution of the vortexes with D becomes more complex with increasing Re. When D is greater than 1, the steady flow consists of a series of large vortexes which superimpose on each other. As Re and D increase, the number of the large vortexes increases. Additionally, a small vortex in the upper-left corner of the semi-elliptical cavity appears at a large Re and its size increases slowly as Re increases. The highest pressures appear in the upper-right corner and the pressure changes drastically in the upper-right region of the cavity. The total pressure differences in the semi-elliptical cavity with a fixed D decrease with increasing Re. In the region of the main vortex, the pressure contours nearly coincide with the streamlines, especially for the cavity flow with a large Re.

The focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is "logically" rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as the activator-activator and short-range inhibition; long-range activation.

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is the superconducting phase for which the Cooper pairs have a non-zero total momentum, depending on the splitting of the Fermi surface sheets for electrons with opposite spin. In infinite systems the momentum is a continuous function of the temperature. In this paper, we have shown how the finite size of the system, through the discretized geometry of the Fermi surface, affects the physical properties of the FFLO state by introducing discontinuities in the Cooper pair momentum. Our calculations in an isotropic system show that the superconducting state with two opposite Cooper pair momenta is more stable than state with one momentum also in nano-size systems, where finite size effects play a crucial role.

The Jiles-Atherton (J-A) model is a commonly used physics-based model in describing the hysteresis characteristics of ferromagnetic materials. However, citations and interpretation of this model in literature have been non-uniform. Solution methods for solving numerically this model has not been studied adequately. In this paper, through analyzing the mathematical properties of equations and the physical mechanism of energy conservation, we point out some unreasonable descriptions of this model and develop a relatively more accurate, modified J-A model together with its numerical solution method. Our method employs a fixed point method to compute anhysteretic magnetization. We obtain the susceptibility value of the anhysteretic magnetization analytically and apply the 4th order Runge-Kutta method to the solution of total magnetization. Computational errors are estimated and then precisions of the solving method in describing various materials are verified. At last, through analyzing the effects of the accelerating method, iterative error and step size on the computational errors, we optimize the numerical method to achieve the effects of high precision and short computing time. From analysis, we determine the range of best values of some key parameters for fast and accurate computation.

We present a numerical method to solve the Vlasov-Poisson-Fokker-Planck (VPFP) system using the $NRxx$ method proposed in [4, 7, 9]. A globally hyperbolic moment system similar to that in [23] is derived. In this system, the Fokker-Planck (FP) operator term is reduced into the linear combination of the moment coefficients, which can be solved analytically under proper truncation. The non-splitting method, which can keep mass conservation and the balance law of the total momentum, is used to solve the whole system. A numerical problem for the VPFP system with an analytic solution is presented to indicate the spectral convergence with the moment number and the linear convergence with the grid size. Two more numerical experiments are tested to demonstrate the stability and accuracy of the $NRxx$ method when applied to the VPFP system.

This is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve the second-order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discretize the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple ($Y_t$,$Z_t$,$A_t$,$Γ_t$) of the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effectiveness of the proposed numerical schemes. Applications of our numerical schemes to stochastic optimal control problems are also presented.

Motivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and $C^r$ parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the $L^2$-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

In this paper, we present some efficient numerical schemes to solve a two-phase hydrodynamics coupled phase field model with moving contact line boundary conditions. The model is a nonlinear coupling system, which consists the Navier-Stokes equations with the general Navier Boundary conditions or degenerated Navier Boundary conditions, and the Allen-Cahn type phase field equations with dynamical contact line boundary condition or static contact line boundary condition. The proposed schemes are linear and unconditionally energy stable, where the energy stabilities are proved rigorously. Various numerical tests are performed to show the accuracy and efficiency thereafter.

Defects arise when nematic liquid crystals are under topological constraints at the boundary. Recently the study of defects has drawn a lot of attention because of the growing theoretical and practical significance. In this paper, we investigate the relationship between two-dimensional defects and three-dimensional defects within nematic liquid crystals confined in a shell. A highly accurate spectral method is used to solve the Landau-de Gennes model to get the detailed static structures of defects. Interestingly, the solution is radial-invariant when the thickness of the shell is sufficiently small. As the shell thickness increases, the solution undergoes symmetry break to reconfigure the disclination lines. We study this three-dimensional reconfiguration of disclination lines in detail under different boundary conditions. In particular, we find that the temperature plays an important role in deciding whether the transition between two-dimensional defects and three-dimensional defects is continuous or discontinuous for the shell with planar anchoring condition on both inner and outer surfaces. We also discuss the characterization of defects in two- and three-dimensional spaces within the tensor model.