Three-Dimensional Lattice Boltzmann Model for the Complex Ginzburg–Landau Equation

In this paper, a lattice Boltzmann model for the three-dimensional complex Ginzburg–Landau equation is proposed. The multi-scale technique and the Chapman–Enskog expansion are used to describe higher-order moments of the complex equilibrium distribution function and a series of complex partial differential equations. The modified partial differential equation of the three-dimensional complex Ginzburg–Landau equation with the third order truncation error is obtained. Based on the complex lattice Boltzmann model, some motions of the stable scroll, such as the scroll wave with a straight filament, scroll ring, and helical scroll are simulated. The comparisons between results of the lattice Boltzmann model with those obtained by the alternative direction implicit scheme are given. The numerical results show that this model can be used to simulate the three-dimensional complex Ginzburg–Landau equation.     

An Efficient Lattice Boltzmann Model for Steady Convection–Diffusion Equation

In this paper, an efficient lattice Boltzmann model for n-dimensional steady convection–diffusion equation with variable coefficients is proposed through modifying the equilibrium distribution function properly, and the Chapman–Enskog analysis shows that the steady convection–diffusion equation with variable coefficients can be recovered exactly. Detailed simulations are performed to test the model, and the results show that the accuracy and efficiency of the present model are better than previous models.     

A Coupled Lattice Boltzmann Method to Solve Nernst–Planck Model for Simulating Electro-osmotic Flows

In this paper, we focus on the nonlinear coupling mechanism of the Nernst–Planck model and propose a coupled lattice Boltzmann method (LBM) to solve it. In this method, a new LBM for the Nernst–Planck equation is developed, a multi-relaxation-time (MRT)-LBM for flow field and an LBM for the Poisson equation are used. And then, we discuss the choice of the model and found that the MRT-LBM is much more stable and accurate than the LBGK model. A reasonable iterative sequence and evolution number for each LBM are proposed by considering the properties of the coupled LBM. The accuracy and stability of the presented coupled LBM are also discussed through simulating electro-osmotic flows (EOF) in micro-channels. Furthermore, to test the applicability of it, the EOF with non-uniform surface potential in micro-channels based on the Nernst–Planck model is simulated. And we investigate the effects of non-uniform surface potential on the pattern of the EOF at different external applied electric fields. Finally, a comparison of the difference between the Nernst–Planck model and the Poisson–Boltzmann model is presented.     

Development and Application of the Cauchy–Poisson Method to Layer Elastodynamics and the Timoshenko Equation

We consider a generalization of the Cauchy–Poisson method to an n-dimensional Euclidean space and its application to the construction of hyperbolic approximations. In Euclidean space, constraints on derivatives are introduced. The principle of hyperbolic degeneracy in terms of parameters is formulated and its implementation in the form of necessary and sufficient conditions is given. As the particular case of a four-dimensional space with preserving operators up to the sixth order a generalized hyperbolic equation is obtained for bending vibrations of plates with coefficients dependent only on the Poisson number. As special cases, this equation includes all the well-known Bernoulli–Euler, Kirchhoff, Rayleigh, and Timoshenko equations. As a development of Maxwell’s and Einstein’s research on the propagation of perturbations with finite velocity in a continuous medium, Tymoshenko’s non-trivial construction of the equation for bending vibrations of a beam is noted.     

A Prolate-Element Method for Nonlinear PDEs on the Sphere

A p-type spectral-element method using prolate spheroidal wave functions (PSWFs) as basis functions, termed as the prolate-element method, is developed for solving partial differential equations (PDEs) on the sphere. The gridding on the sphere is based on a projection of the prolate-Gauss-Lobatto points by using the cube-sphere transform, which is free of singularity and leads to quasi-uniform grids. Various numerical results demonstrate that the proposed prolate-element method enjoys some remarkable advantages over the polynomial-based element method: (i) it can significantly relax the time step size constraint of an explicit time-marching scheme, and (ii) it can increase the accuracy and enhance the resolution.     

Analysis of the “Toolkit” Method for the Time-Dependent Schrödinger Equation

The goal of this paper is to provide an analysis of the “toolkit” method used in the numerical approximation of the time-dependent Schrödinger equation. The “toolkit” method is based on precomputation of elementary propagators and was seen to be very efficient in the optimal control framework. Our analysis shows that this method provides better results than the second order Strang operator splitting. In addition, we present two improvements of the method in the limit of low and large intensity control fields.     

Guidelines for Poisson Solvers on Irregular Domains with Dirichlet Boundary Conditions Using the Ghost Fluid Method

We consider the variable coefficient Poisson equation with Dirichlet boundary conditions on irregular domains. We present numerical evidence for the accuracy of the solution and its gradients for different treatments at the interface using the Ghost Fluid Method for Poisson problems of Gibou et al. (J. Comput. Phys. 176:205–227, 2002; 202:577–601, 2005). This paper is therefore intended as a guide for those interested in using the GFM for Poisson-type problems (and by consequence diffusion-like problems and Stefan-type problems) by providing the pros and cons of the different choices for defining the ghost values and locating the interface. We found that in order to obtain second-order-accurate gradients, both a quadratic (or higher order) extrapolation for defining the ghost values and a quadratic (or higher order) interpolation for finding the interface location are required. In the case where the ghost values are defined by a linear extrapolation, the gradients of the solution converge slowly (at most first order in average) and the convergence rate oscillates, even when the interface location is defined by a quadratic interpolation. The same conclusions hold true for the combination of a quadratic extrapolation for the ghost cells and a linear interpolation. The solution is second-order accurate in all cases. Defining the ghost values with quadratic extrapolations leads to a non-symmetric linear system with a worse conditioning than that of the linear extrapolation case, for which the linear system is symmetric and better conditioned. We conclude that for problems where only the solution matters, the method described by Gibou, F., Fedkiw, R., Cheng, L.-T. and Kang, M. in (J. Comput. Phys. 176:205–227, 2002) is advantageous since the linear system that needs to be inverted is symmetric. In problems where the solution gradient is needed, such as in Stefan-type problems, higher order extrapolation schemes as described by Gibou, F. and Fedkiw, R. in (J. Comput. Phys. 202:577–601, 2005) are desirable.     

A New Family of Mixed Methods for the Reissner-Mindlin Plate Model Based on a System of First-Order Equations

The mixed method for the biharmonic problem introduced in (Behrens and Guzmán, SIAM J. Numer. Anal., 2010) is extended to the Reissner-Mindlin plate model. The Reissner-Mindlin problem is written as a system of first order equations and all the resulting variables are approximated. However, the hybrid form of the method allows one to eliminate all the variables and have a final system only involving the Lagrange multipliers that approximate the transverse displacement and rotation at the edges of the triangulation. Mixed finite element spaces for elasticity with weakly imposed symmetry are used to approximate the bending moment matrix. Optimal estimates independent of the plate thickness are proved for the transverse displacement, rotations and bending moments. A post-processing technique is provided for the displacement and rotations variables and we show numerically that they converge faster than the original approximations.     

Stable Boundary Treatment for the Wave Equation on Second-Order Form

A stable and accurate boundary treatment is derived for the second-order wave equation. The domain is discretized using narrow-diagonal summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a stable and efficient scheme. The analysis is verified by numerical simulations in one-dimension using high-order finite difference discretizations, and in three-dimensions using an unstructured finite volume discretization.     

Analysis of the Finite Element Method for the Laplace–Beltrami Equation on Surfaces with Regions of High Curvature Using Graded Meshes

We derive error estimates for the piecewise linear finite element approximation of the Laplace–Beltrami operator on a bounded, orientable, \(C^3\), surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error estimates that do not depend on the curvature on the high curvature region. Numerical experiments are provided.     

A Rectangular Finite Volume Element Method for a Semilinear Elliptic Equation

In this paper we extend the idea of interpolated coefficients for semilinear problems to the finite volume element method based on rectangular partition. At first we introduce bilinear finite volume element method with interpolated coefficients for a boundary value problem of semilinear elliptic equation. Next we derive convergence estimate in H ¹-norm and superconvergence of derivative. Finally an example is given to illustrate the effectiveness of the proposed method. This work is supported by Program for New Century Excellent Talents in University of China State Education Ministry, National Science Foundation of China, the National Basic Research Program under the Grant (2005CB321703), the key project of China State Education Ministry (204098), Scientific Research Fund of Hunan Provincial Education Department, China Postdoctoral Science Foundation (No. 20060390894) and China Postdoctoral Science Foundation (No. 20060390894).     

Direct Discretization Method for the Cahn–Hilliard Equation on an Evolving Surface

We propose a simple and efficient direct discretization scheme for solving the Cahn–Hilliard (CH) equation on an evolving surface. By using a conservation law and transport formulae, we derive the CH equation on evolving surfaces. An evolving surface is discretized using an unstructured triangular mesh. The discrete CH equation is defined on the surface mesh and its dual surface polygonal tessellation. The evolving triangular surfaces are then realized by moving the surface nodes according to a given velocity field. The proposed scheme is based on the Crank–Nicolson scheme and a linearly stabilized splitting scheme. The scheme is second-order accurate, with respect to both space and time. The resulting system of discrete equations is easy to implement, and is solved by using an efficient biconjugate gradient stabilized method. Several numerical experiments are presented to demonstrate the performance and effectiveness of the proposed numerical scheme.     

A Dual-Petrov-Galerkin Method for the Kawahara-Type Equations

An efficient and accurate numerical scheme is proposed, analyzed and implemented for the Kawahara and modified Kawahara equations which model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. The scheme consists of dual-Petrov-Galerkin method in space and Crank-Nicholson-leap-frog in time such that at each time step only a sparse banded linear system needs to be solved. Theoretical analysis and numerical results are presented to show that the proposed numerical is extremely accurate and efficient for Kawahara type equations and other fifth-order nonlinear equations. This work is partially supported by the National Science Council of the Republic of China under the grant NSC 94-2115-M-126-004 and 95-2115-M-126-003. This work is partially supported by NSF grant DMS-0610646.     

Characteristic Line Based Schemes for Solving a Quasilinear Hierarchical Size-Structured Model

New schemes, based on the characteristics line method, for solving a hierarchical size-structured model with nonlinear growth, mortality and reproduction rate are developed. The idea of the schemes is not to follow the characteristics from the initial condition, but for each time-step to find the origins of the grid nodes at the previous time level. Numerical tests, including comparison with exact solutions for the new schemes, are elaborated. Numerical results that confirm the theoretical order of convergence of the new schemes are presented.     

A Neural Network Constructing Method Based on Many Kinds of Neurons Model

In this paper, the geometrical meaning of the neurons in BPNN （Back Propagation Neural Network） and RBFNN （Radial Basis Function Neural Network） in the High Dimensional Space （HDS） is analyzed, and a new type of NN is built based on many kinds of neurons. This new NN solves the covering problem of the complex geometry shape in the high dimensional feature space which is raised by the Biomimetic Pattern Recognition （BPR）. This new NN-Constructing method has broken the traditional thinking mode of constructing NN which only uses one kind of neuron.     

Accuracy and Stability of the Petrov–Galerkin Method for Solving the Stationary Convection-Diffusion Equation

The accuracy and stability of numerical solution of the stationary convection-diffusion equation by the finite element Petrov–Galerkin method are analyzed with the use of weight functions with different stabilization parameters as test functions, and estimates are obtained for the accuracy of the method depending on the choice of a collection of stabilization parameters. The convergence of the method is shown.     

An easy and efficient combination of the Mixed Finite Element Method and the Method of Lines for the resolution of Richards’ Equation

In this work, the Mixed Hybrid Finite Element (MHFE) method is combined with the Method Of Lines (MOL) for an accurate resolution of the Richard's Equation (RE). The combination of these methods is often complicated since hybridization requires a discrete approximation of the time derivative whereas with the MOL, it should remain continuous. In this paper, we use the new mass lumping technique developed in Younes et al. [Younes, A., Ackerer, P., Lehmann, F., 2006. A new mass lumping scheme for the mixed hybrid finite element method. International Journal for Numerical Methods in Engineering 67, pp. 89–107.] for the MHFE method. With this formulation, the MOL is easily implemented and sophisticated time integration packages can be used without significant amount of work.Numerical simulations are performed on both homogeneous and heterogeneous porous media to show the efficiency and robustness of the developed scheme.