Numerical methods for a fixed domain formulation of the glacier profile problem with alternative boundary conditions

In this paper we develop a set of numerical techniques for the simulation of the profile evolution of a valley glacier in the framework of isothermal shallow ice approximation models. The different mathematical formulations are given in terms of a highly nonlinear parabolic equation. A first nonlinearity comes from the free boundary problem associated with the unknown basal extension of the glacier region. This feature is treated using a fixed domain complementarity formulation which is solved numerically by a duality method. The nonlinear diffusive term is explicitly treated in the time marching scheme. A convection dominated problem arises, so a characteristic scheme is proposed for the time discretization, while piecewise linear finite elements are used for the spatial discretization. The presence of infinite slopes in polar regimes motivates an alternative formulation based on a prescribed flux boundary condition at the head of the glacier instead a homogeneous Dirichlet one. Finally, several numerical examples illustrate the performance of the proposed methods.     

高精度高分辨率迎风格式应用于不同速度范围内粘性流动

提出了一种适合于不同速度范围的高精度高分辨率的迎风有限差分格式，并基于此数植模型发展了适应于速度范围极宽的非定常粘性流动通用软件，不仅适用于超音速下捕捉强间断面，跨音速及高亚音速下捕捉弱间断面和滑移面，还可以精确地模拟低速情况下的粘性流动。此软件可分别用于研究内流和外流的流动特性以及预估其粘性损失。     

三维有限差分波束传输法:用于Ti∶ LiNbO_3方向耦合器的模拟

提出求解三维傍轴近似波动方程的交替方向隐式差分格式，并用它模拟Ｔｉ∶ＬｉＮｂＯ３方向耦合器     

Convergence analysis for second‐order accurate schemes for the periodic nonlocal Allen‐Cahn and Cahn‐Hilliard equations

In this paper, we provide a detailed convergence analysis for fully discrete second‐order (in both time and space) numerical schemes for nonlocal Allen‐Cahn and nonlocal Cahn‐Hilliard equations. The unconditional unique solvability and energy stability ensures ? ⁴ stability. The convergence analysis for the nonlocal Allen‐Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn‐Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an H ^?1 inner‐product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O (s ³+h ⁴) convergence in norm, in which s and h denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint s ≤C h . Here, we also prove convergence of the scheme in the maximum norm under the same constraint.     

Decoupled schemes for unsteady MHD equations. I. time discretization

In this article, a time discretization decoupled scheme for two‐dimensional magnetohydrodynamics equations is proposed. The almost unconditional stability and convergence of this scheme are provided. The optimal error estimates for velocity and magnet are provided, and the optimal error estimate for pressure are deduced as well. Finite element spatial discretization and numerical implementation are considered in our article (Zhang and He, Comput Math Appl 69 (2015), 1390–1406). © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 956–973, 2017     

A Hybrid Finite Volume Method for Advection Equations and Its Applications in Population Dynamics

We present in this article a very adapted finite volume numerical scheme for transport type‐equation. The scheme is an hybrid one combining an anti‐dissipative method with down‐winding approach for the flux (Després and Lagoutière, C R Acad Sci Paris Sér I Math 328(10) (1999), 939–944; Goudon, Lagoutière, and Tine, Math Method Appl Sci 23(7) (2013), 1177–1215) and an high accurate method as the WENO5 one (Jiang and Shu, J Comput Phys 126 (1996), 202–228). The main goal is to construct a scheme able to capture in exact way the numerical solution of transport type‐equation without artifact like numerical diffusion or without “stairs” like oscillations and this for any regular or discontinuous initial distribution. This kind of numerical hybrid scheme is very suitable when properties on the long term asymptotic behavior of the solution are of central importance in the modeling what is often the case in context of population dynamics where the final distribution of the considered population and its mass preservation relation are required for prediction. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1114–1142, 2017     

High‐order compact schemes for fractional differential equations with mixed derivatives

In this article, we consider two‐dimensional fractional subdiffusion equations with mixed derivatives. A high‐order compact scheme is proposed to solve the problem. We establish a sufficient condition and show that the scheme converges with fourth order in space and second order in time under this condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2141–2158, 2017     

Strong convergence of composite iterative methods for equilibrium problems and fixed point problems

We introduce a new composite iterative scheme by viscosity approximation method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping. Our results substantially improve the corresponding results of Takahashi and Takahashi [A. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506-515]. Essentially a new approach for finding solutions of equilibrium problems and the fixed points of nonexpansive mappings is provided.     

Initial-boundary problem for the 1-D Euler-Boltzmann equations in radiation hydrodynamics

We study the initial-boundary value problem for the one dimensional EulerBoltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions(U_(△t,d), I_(△t,d)) and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.     

A meshless method for Burgers’ equation using MQ-RBF and high-order temporal approximation

In this paper, a new numerical method is proposed to solve one-dimensional Burgers’ equation using multiquadric (MQ) radial basis function (RBF) for spatial approximation and a second-order compact finite difference scheme for temporal approximation. The numerical results obtained by this way for different Reynolds number have been compared with the existing numerical schemes to show the accuracy and efficiency of the approach. To show the superiority of this meshless method, numerical experiments with non-uniform MQ interpolation node distribution are also performed.