Lattice Boltzmann Model Based on the Rebuilding-Divergency Method for the Laplace Equation and the Poisson Equation

In this paper, a new lattice Boltzmann model based on the rebuilding-divergency method for the Poisson equation is proposed. In order to translate the Poisson equation into a conservation law equation, the source term and diffusion term are changed into divergence forms. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales and several higher-order moments of equilibrium distribution functions are obtained. Thus, by rebuilding the divergence of the source and diffusion terms, the Laplace equation and the Poisson equation with the second accuracy of the truncation errors are recovered. In the numerical examples, we compare the numerical results of this scheme with those obtained by other classical method for the Green-Taylor vortex flow, numerical results agree well with the classical ones.     

A linear discrete scheme for the Ginzburg–Landau equation

This paper considers a 2D Ginzburg–Landau equation with a periodic initial-value condition. A fully discrete Galerkin–Fourier spectral approximation scheme, which is a linear scheme, is constructed and the dynamical behaviour of the discrete system is then analysed. First, the existence and convergence of global attractors of the discrete system are obtained by a priori estimates and the error estimates of the discrete solution without any restriction on the time step, and the convergence of the discrete scheme is then obtained. The numerical stability of the discrete scheme is proved.     

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.     

An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations

In this paper, we propose a split-step quasi-compact finite difference method to solve the nonlinear fractional Ginzburg–Landau equations both in one and two dimensions. The original equations are split into linear and nonlinear subproblems. The Riesz space fractional derivative is approximated by a fourth-order fractional quasi-compact method. Furthermore, an alternating direction implicit scheme is constructed for the two dimensional linear subproblem. The unconditional stability and convergence of the schemes are proved rigorously in the linear case. Numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method.     

Chebyshev polynomials for the numerical solution of fractal–fractional model of nonlinear Ginzburg–Landau equation

Engineering with Computers - This paper introduces a new version for the nonlinear Ginzburg–Landau equation derived from fractal–fractional derivatives and proposes a computational...     

Uniform regularity for a 3D time-dependent Ginzburg–Landau model in superconductivity

In this paper, we prove the uniform-in-$\sigma$ regularity for 3D time-dependent Ginzburg–Landau model in superconductivity in the case of Coulomb gauge. Here $\sigma$ is the normal conductivity of the superconducting material.     

Convergence of linearized backward Euler–Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with temporal gauge

We present error analysis of fully discrete Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with the temporal gauge, where a linearized backward Euler scheme is used for the time discretization. We prove that the convergence rate is O(τ+h^r) if the finite element space of piecewise polynomials of degree r is used. Due to the degeneracy of the problem, the convergence rate is one order lower than the optimal convergence rate of finite element methods for parabolic equations. Numerical examples are provided to support our theoretical analysis.     

Analysis of running waves stability in the Ginzburg–Landau equation with small diffusion

The local dynamics of the Ginzburg–Landau equation with small diffusion and periodic boundary conditions in the neighborhood of running waves is analyzed. Sufficient conditions for the instability of running waves and sufficient conditions for their stability are found.     

A Least-Squares/Relaxation Method for the Numerical Solution of the Three-Dimensional Elliptic Monge–Ampère Equation

In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.     

Lattice Boltzmann models for two-dimensional coupled Burgers’ equations

In this paper, two lattice Boltzmann models for two-dimensional coupled Burgers’ equations are proposed through treating the part or all of convection items as the source term, where the spatial gradient can be calculated by the distribution function. The models can exactly recover the Burgers’ equations without any assumptions. Some numerical tests are also performed to validate the present models. It is found that the proposed models are more accurate and efficient in solving two-dimensional coupled Burgers’ equations.     

Convergence of a decoupled mixed FEM for the dynamic Ginzburg–Landau equations in nonsmooth domains with incompatible initial data

In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg–Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform \(L^{3+\delta }\) regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a \(\ell ^2(W^{1,3+\delta })\) estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge.     

Variational Space–Time Methods for the Wave Equation

In this work we present some new variational space–time discretisations for the scalar-valued acoustic wave equation as a prototype model for the vector-valued elastic wave equation. The second-order hyperbolic equation is rewritten as a first-order in time system of equations for the displacement and velocity field. For the discretisation in time we apply continuous Galerkin–Petrov and discontinuous Galerkin methods, and for the discretisation in space we apply the symmetric interior penalty discontinuous Galerkin method. The resulting algebraic system of equations exhibits a block structure. First, it is simplified by some calculations to a linear system for one of the variables and a vector update for the other variable. Using the block diagonal structure of the mass matrix from the discontinuous Galerkin discretisation in space, the reduced system can be condensed further such that the overall linear system can be solved efficiently. The convergence behaviour of the presented schemes is studied carefully by numerical experiments. Moreover, the performance and stability properties of the schemes are illustrated by a more sophisticated problem with complex wave propagation phenomena in heterogeneous media.     

Unconditionally Energy Stable Linear Schemes for the Diffuse Interface Model with Peng–Robinson Equation of State

In this paper, we investigate numerical solution of the diffuse interface model with Peng–Robinson equation of state, that describes real states of hydrocarbon fluids in the petroleum industry. Due to the strong nonlinearity of the source terms in this model, how to design appropriate time discretizations to preserve the energy dissipation law of the system at the discrete level is a major challenge. Based on the “Invariant Energy Quadratization” approach and the penalty formulation, we develop efficient first and second order time stepping schemes for solving the single-component two-phase fluid problem. In both schemes the resulted temporal semi-discretizations lead to linear systems with symmetric positive definite spatial operators at each time step. We rigorously prove their unconditional energy stabilities in the time discrete sense. Various numerical simulations in 2D and 3D spaces are also presented to validate accuracy and stability of the proposed linear schemes and to investigate physical reliability of the target model by comparisons with laboratory data.     

Local Discontinuous Galerkin Methods for the Khokhlov–Zabolotskaya–Kuznetzov Equation

Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation is a model that describes the propagation of the ultrasound beams in the thermoviscous fluid. It contains a nonlocal diffraction term, an absorption term and a nonlinear term. Accurate numerical methods to simulate the KZK equation are important to its broad applications in medical ultrasound simulations. In this paper, we propose a local discontinuous Galerkin method to solve the KZK equation. We prove the \(L^2\) stability of our scheme and conduct a series of numerical experiments including the focused circular short tone burst excitation and the propagation of unfocused sound beams, which show that our scheme leads to accurate solutions and performs better than the benchmark solutions in the literature.     

Optimal Path Programming of the Stewart Platform Manipulator Using the Boltzmann–Hamel–d'Alembert Dynamics Formulation Model

In this paper, the dynamics formulation of a general Stewart platform manipulator (SPM) with arbitrary geometry and inertia distribution is addressed. Based on a structured Boltzmann–Hamel–d'Alembert approach, in which the true coordinates are for translations and quasi-coordinates are for rotations, a systematic methodology using the parallelism inherent in the parallel mechanisms is developed to derive the explicit closed-form dynamic equations which are feasible for both forward and inverse dynamics analyses in the task space. Thus, a singularity-free path programming of the SPM for the minimum actuating forces is presented to demonstrate the applications of the developed dynamics model. Using a parametric path representation, the singularity-free path programming problem can be cast to the determination of undetermined control points, and then a particle swarm optimization algorithm is employed to determine the optimal control points and the associated trajectories. Numerical examples are implemented for the moving platform with constant orientations and varied orientations.     

High Order Algorithm for the Time-Tempered Fractional Feynman–Kac Equation

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in Wu et al. (Phys Rev E 93:032151, 2016), being called the time-tempered fractional Feynman–Kac equation named after Richard Feynman and Mark Kac who first considered the model describing the functional distribution of normal motion. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as

$$\begin{aligned} {^S\!}D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t)\!=\!D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t)\!-\!\lambda ^\gamma G(x,p,t) \end{aligned}$$

with \(\widetilde{\lambda }=\lambda + pU(x),\, p=\rho +J\eta ,\, J=\sqrt{-1}\), where

$$\begin{aligned} D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t) =\frac{1}{\varGamma (1-\gamma )} \left[ \frac{\partial }{\partial t}+\widetilde{\lambda } \right] \int _{0}^t{\left( t-z\right) ^{-\gamma }}e^{-\widetilde{\lambda }\cdot (t-z)}{G(x,p,z)}dz, \end{aligned}$$

and \(\lambda \ge 0\), \(0<\gamma <1\), \(\rho >0\), and \(\eta \) is a real number. The designed schemes are unconditionally stable and have the global truncation error \(\mathscr {O}(\tau ^2+h^2)\), being theoretically proved and numerically verified in complex space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the ‘physical’ equation (without artificial source term).