Explicit Hybrid Numerical Method for the Allen-Cahn Type Equations on Curved Surfaces

We present a simple and fast explicit hybrid numerical scheme for the motion by mean curvature on curved surfaces in three-dimensional (3D) space. We numerically solve the Allen-Cahn (AC) and conservative Allen-Cahn (CAC) equations on a triangular surface mesh. We use the operator splitting method and an explicit hybrid numerical method. For the AC equation, we solve the diffusion term using a discrete Laplace-Beltrami operator on the triangular surface mesh and solve the reaction term using the closed-form solution, which is obtained using the separation of variables. Next, for the CAC equation, we additionally solve the time-space dependent Lagrange multiplier using an explicit scheme. Our numerical scheme is computationally fast and efficient because we use an explicit hybrid numerical scheme. We perform various numerical experiments to demonstrate the robustness and efficiency of the proposed scheme.     

A numerical method for the Landau–Lifshitz equation with magnetostriction

We present a numerical scheme for Landau–Lifshitz–Gilbert equation coupled with the equation of elastodynamics. The considered physical model describes the behaviour of ferromagnetic materials when magnetomechanical coupling is taken into account. The time‐discretization is based on the backward Euler method with projection. In the numerical approximation, the two equations are decoupled by a suitable linearization in order to solve the magnetic and mechanic part separately. The resulting semi‐implicit scheme is linear and allows larger time‐steps than explicit methods. We prove stability and error estimates for the presented time discretization in 2D. Finally, we test the accuracy of the scheme on an academic numerical example with known exact solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonstandard discretizations of the generalized Nagumo reaction‐diffusion equation

A competitive nonstandard semi‐explicit finite‐difference method is constructed and used to obtain numerical solutions of the diffusion‐free generalized Nagumo equation. Qualitative stability analysis and numerical simulations show that this scheme is more robust in comparison to some standard explicit methods such as forward Euler and the fourth‐order Runge‐Kutta method (RK4). The nonstandard scheme is extended to construct a semi‐explicit and an implicit scheme to solve the full Nagumo reaction‐diffusion equation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 363–379, 2003.     

On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev′e Equation: an Operator-Splitting Approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevé transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painlevé ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.     

Evolution of solitary waves for a perturbed nonlinear Schrödinger equation

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of solitary wave tails occur, the location of zeros and the location and amplitude of peaks, in the solitary wave tail. Two examples, the near-continuum limit of a discrete NLS equation and an explicit numerical scheme for the NLS equation, are considered in detail. For the discrete NLS equation it is found that three qualitatively different types of solitary wave tail can occur, while for the explicit finite-difference scheme, only one type of solitary wave tail occurs. An excellent comparison between the perturbation solution and numerical simulations, for the solitary wave tail, is found for both examples.     

RLW方程的一个新的显式多辛格式

以多辛Euler-box格式为基础对正则长波(RLW)方程的初边值问题进行了讨论,推导了一个新的显式10点格式.模拟孤立波的数值实验表明,这个新的多辛格式是行之有效的,能很好的反映出RLW方程的非弹性性质.     

An asymptotically stable Particle-in-Cell (PIC) scheme for collisionless plasma simulations near quasineutrality

We propose a new Particle-in-Cell scheme for the Vlasov–Poisson equation. This scheme remains stable when the Debye length and plasma period tend to zero without any restriction on the size of the time and length step. It relies on a semi-implicit integration of the particle trajectories. The numerical integration cost is that of the standard explicit method thanks to the use of a reformulation of the Poisson equation. To cite this article: P. Degond et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On uniform attractors of explicit approximations

We prove a theorem stating that the uniform attractors of a family of semiprocesses that do not necessarily have a common time semigroup depend on the parameter uppersemicontinuously. We consider an explicit finite-difference scheme for a nonautonomous system of ordinary differential equations and an explicit spectral-difference scheme for the vorticity equation with time-dependent bounded right-hand side on a sphere. We obtain theorems on the existence of uniform attractors of numerical schemes and their closeness to true attractors of the original differential problems.     

Numerical study using explicit multistep Galerkin finite element method for the MRLW equation

In this article, an explicit multistep Galerkin finite element method for the modified regularized long wave equation is studied. The discretization of this equation in space is by linear finite elements, and the time discretization is based on explicit multistep schemes. Stability analysis and error estimates of our numerical scheme are derived. Numerical experiments indicate the validation of the scheme by L₂– and L_∞– error norms and three invariants of motion.4 © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1875–1889, 2015     

An explicit spectral collocation method using nonpolynomial basis functions for the time‐dependent Schrödinger equation

We propose a spectral collocation method for the numerical solution of the time‐dependent Schrödinger equation, where the newly developed nonpolynomial functions in a previous study are used as basis functions. Equipped with the new basis functions, various boundary conditions can be imposed exactly. The preferable semi‐implicit time marching schemes are employed for temporal discretization. Moreover, the new basis functions build in a free parameter λ intrinsically, which can be chosen properly so that the semi‐implicit scheme collapses to an explicit scheme. The method is further applied to linear Schrödinger equation set in unbounded domain. The transparent boundary conditions are constructed for time semidiscrete scheme of the linear Schrödinger equation. We employ spectral collocation method using the new basis functions for the spatial discretization, which allows for the exact imposition of the transparent boundary conditions. Comprehensive numerical tests both in bounded and unbounded domain are performed to demonstrate the attractive features of the proposed method.     

A multisymplectic explicit scheme for the modified regularized long-wave equation

In this paper, we derive a new 10-point multisymplectic scheme for the modified regularized long-wave equation. The new scheme is an explicit scheme in the sense that the third time level does not include nonlinear terms. Numerical results indicate that the new scheme not only provides satisfied numerical solutions, but also preserves three invariants of motion very well.     

A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

In this article, we study an explicit scheme for the solution of sine‐Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo‐spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth‐order Runge–Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo‐spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix–vector multiplications. Therefore, we propose a multidomain pseudo‐spectral method for the numerical simulation of the sine‐Gordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long‐time numerical behavior for the sine‐Gordon equation in one and two dimensions. Copyright © 2013 John Wiley & Sons, Ltd.     

Error estimate for a fully discrete spectral scheme for Korteweg-de Vries-Kawahara equation

We are concerned with convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (Kawahara equation, in short), which is a transport equation perturbed by dispersive terms of the 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the standard Fourier-Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L ²-error bound of spectral accuracy in space and of second-order accuracy in time.     

Numerical solution to the Kolmogorov-Feller equation

A finite-difference method is proposed for solving the Kolmogorov-Feller integro-differential equation. The numerical scheme constructed is an unconditionally stable marching scheme, and the boundary conditions are determined on the basis of an explicit solution to the original equation at boundary points.     

A new explicit three-level difference scheme for the solution of the heat flow equation

A new explicit three-level difference scheme for the numerical solution of the heat flow equation is proposed. The main features of the new scheme are: i) it is unconditionally stable, ii) it is very highly accurate from the point of view of the truncation error, and iii) its solution converges to the solution of the heat equation even if the time and the distance increments tend to zero independently.     

An inverse problem of finding a source parameter in a semilinear parabolic equation

An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.     

A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation

A three-time level finite-difference scheme based on a fourth order in time and second order in space approximation has been proposed for the numerical solution of the nonlinear two-dimensional sine-Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor–corrector scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector has been modified by considering as known the already evaluated corrected values instead of the predictor ones. This modified scheme has been tested on the line and circular ring soliton and the numerical experiments have proved that there is an improvement in the accuracy over the standard predictor–corrector implementation. This research was co-funded by E.U. (75%) and by the Greek Government (25%).     

A regular perturbation analytical‐numerical method for the evolution of precancerous lesions caused by the human papillomavirus

A numerical method is developed to analyze the behavior of the evolution of the lesions at the cervical cells caused by the human papillomavirus. The model to be solved consists in a one‐dimensional nonlinear advection–diffusion‐reaction equation. Such equation is approximated by a consistent explicit difference scheme which is based on regular perturbation theory. A constructive procedure for the numerical scheme is given and finally an illustrative example of the evolution of a mild dysplasia is included. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 847–855, 2015     

A fast mass-conserving explicit splitting method for the stochastic space-fractional nonlinear Schrödinger equation with multiplicative noise

A fast mass-conserving explicit splitting method is proposed to solve the stochastic space-fractional nonlinear Schrödinger equation which includes a nonlinear term, a fractional Laplacian term and a multiplicative noise term resulting from the nonlocal property, the random variation of the media and the exterior random disturbance. The method splits the original equation into two sub-equations so that appropriate numerical methods can be applied to each sub-equation. A variety of numerical examples in both one- and multi-dimensional spaces show that the scheme has good mass conservative property, first-order strong convergence in time and high efficiency.     

A Large Time Step explicit scheme (CFL>1) on unstructured grids for 2D conservation laws: Application to the homogeneous shallow water equations

In this work, a Large Time Step (LTS) explicit finite volume scheme designed to allow CFL > 1 is applied to the numerical resolution of 2D scalar and systems of conservation laws on triangular grids. Based on the flux difference splitting formulation, a special concern is put on finding the way of packing the information to compute the numerical solution when working on unstructured grids. Not only the cell areas but also the length of the interfaces and their orientation are questions of interest to send the information from each edge or interface. The information to update the cell variables is computed according to the local average discrete velocity and the orientation of the edges of the cells involved. The performance of these ideas is tested and compared with the conventional explicit first order and second order schemes in academic configurations for the 2D linear scalar equation and for 2D systems of conservation laws (in particular the shallow water equations) without source terms. The LTS scheme is demonstrated to preserve or even gain accuracy and save computational time with respect to the first order scheme.