A numerical method for non-linear water waves

A numerical method based on a complex potential formulation is developed for the computation on non-linear waves. It is applied to solve the problem of waves generated by a submerged doublet. Results for the dominant steady wave trains are presented. Good agreement with non-linear theory is obtained.     

A hybrid numerical method and its application to inviscid compressible flow problems

An improved version of the artificially upstream flux vector scheme, is developed to efficiently compute inviscid compressible flow problems. This numerical scheme, named AUFSR (Tchuen et al. 2011), is obtained by hybridizing the AUFS scheme with Roe’s solver. This approach handles difficulties encountered by the AUFS scheme, in the case where the flux vector does not check the homogeneous property. The present scheme for multi-dimensional flows introduces a certain amount of numerical dissipation to shear waves, as Roe’s splitting. The AUFSR scheme is not only robust for shock-capturing, but also accurate for resolving shear layers. Numerical results for 1D Riemann problems and several 2D problems are investigated to show the capability of the method to accurately compute inviscid compressible flow when compared to AUFS, and Roe solvers.     

A multiscale numerical method for the heterogeneous cable equation

Several interesting problems in neuroscience are of multiscale type, i.e. possess different temporal and spatial scales that cannot be disregarded. Such characteristics impose severe burden to numerical simulations since the need to resolve small scale features pushes the computational costs to unreasonable levels. Classical numerical methods that do not resolve the small scales suffer from spurious oscillations and lack of precision.This paper presents an innovative numerical method of multiscale type that ameliorates these maladies. As an example we consider the case of a cable equation modeling heterogeneous dendrites. Our method is not only easy to parallelize, but it is also nodally exact, i.e., it matches the values of the exact solution at every node of the discretization mesh, for a class of problems.To show the validity of our scheme under different physiological regimes, we describe how the model behaves whenever the dendrites are thin or long, or the longitudinal conductance is small. We also consider the case of a large number of synapses and of large or low membrane conductance.     

A numerical method for transmission problems for the Helmholtz equation

This method consists in decoupling the transmission problem into two boundary value problems, which can be solved separately by well known procedures. A convergence proof is given with the help of the integral equation method and convergence results on projection methods.     

A numerical method for solving a coupled channel Schrödinger equation

An approximation method involving spherical delta functions is presented for the solving of coupled channel differential equations, in particular the Schrödinger equation. A specific example is worked out in detail.     

A new numerical method for kinetic equations in several dimensions

We present first results of a numerical method solving inhomogeneous partial differential equation of first order with a conservation property. The method is based on the Finite Particle Schemes for homogeneous PDE's of the first order as the Vlasov-Poisson system in kinetic theory. The inhomogeneity is redefined as a flux. For the associated ‘velocity-field’ given by the Radon-Nikodym derivative of the flux, we give a numerical approximation. Together with the ‘velocity-field’ given, by the derivative terms of first order this gives the right hand side of the equations of motion of the particles. The computation can be done in a very efficient way and the results are in good agreement with the exact solution.     

A method for fuzzy rules extraction directly from numerical dataand its application to pattern classification

In this paper, we discuss a new method for extracting fuzzy rules directly from numerical input-output data for pattern classification. Fuzzy rules with variable fuzzy regions are defined by activation hyperboxes which show the existence region of data for a class and inhibition hyperboxes which inhibit the existence of data for that class. These rules are extracted from numerical data by recursively resolving overlaps between two classes. Then, optimal input variables for the rules are determined using the number of extracted rules as a criterion. The method is compared with neural networks using the Fisher iris data and a license plate recognition system for various examples     

A numerical method for diffusion-convection equation using high-order difference schemes

In this paper, we propose a simple general form of high-order approximation of O(c²+ch²+h⁴) to solve the two-dimensional parabolic equation αuxx+βuyy=F(x,y,t,u,ux,uy,ut), where α and β are positive constants. We apply the compact form for solving diffusion-convection equation. The results of numerical experiments are presented and compared with analytical solutions to confirm the higher accuracy of the presented scheme.     

The multisymplectic numerical method for Gross-Pitaevskii equation

For a Bose-Einstein Condensate placed in a rotating trap and confined in the z-axis, a multisymplectic difference scheme was constructed to investigate the evolution of vortices in this paper. First, we look for a steady state solution of the imaginary time G-P equation. Then, we numerically study the vortices's development in real time, starting with the solution in imaginary time as initial value.     

A numerical method for the Hartree equation of the Helium atom

We describe in this paper a numerical method for computing the normalized pointwise positive solution of the Hartree equation for the Helium atom. The method consists of minimizing the Hartree energy by a decomposition coordination method via an augmented Lagrangian. Some numerical results are presented.     

A numerical method for exact boundary controllability problems for the wave equation

The computational approximation of exact boundary controllability problems for the wave equation in two dimensions is studied. A numerical method is defined that is based on the direct solution of optimization problems that are introduced in order to determine unique solutions of the controllability problem. The uniqueness of the discrete finite-difference solutions obtained in this manner is demonstrated. The convergence properties of the method are illustrated through computational experiments. Efficient implementation strategies for the method are also discussed. It is shown that for smooth, minimum L²-norm Dirichlet controls, the method results in convergent approximations without the need to introduce regularization. Furthermore, for the generic case of nonsmooth Dirichlet controls, convergence with respect to L² norms is also numerically demonstrated. One of the strengths of the method is the flexibility it allows for treating other controls and other minimization criteria; such generalizations are discussed. In particular, the minimum H¹-norm Dirichlet controllability problem is approximated and solved, as are minimum regularized L²-norm Dirichlet controllability problems with small penalty constants. Finally, a discussion is provided about the differences between our method and existing methods; these differences may explain why our methods provide convergent approximations for problems for which existing methods produce divergent approximations unless they are regularized in some manner.     

A simple equation solver and its application to financial modelling

We discuss a simple algorithm for solving sets of simultaneous equations. The algorithm can solve systems of linear and some kinds of non-linear equations, although it has nowhere near the power of a general non-linear equation solver. Its principal advantages over more general algorithms are simplicity and speed. Versions of the algorithm have been used in a graphics language and in a system for interactively modifying the equations that constitute financial models. We discuss the second application in more detail here.     

Multisymplectic numerical method for the regularized long-wave equation

In this paper, we derive a 6-point multisymplectic Preissman scheme for the regularized long-wave equation from its Bridges' multisymplectic form. Backward error analysis is implemented for the new scheme. The performance and the efficiency of the new scheme are illustrated by solving several test examples. The obtained results are presented and compared with previous methods. Numerical results indicate that the new multisymplectic scheme can not only obtain satisfied solutions, but also keep three invariants of motion very well.     

Thirteen-moment model kinetic equation and its parameters

We propose the generalized model kinetic equation that represents a hybrid of the Shakhov equation and the ellipsoidal statistical Holway equation; this is the thirteen-moment equation. The constants of the equation are expressed at first in the terms of the transport coefficients, namely, the viscosity of a gas, its heat conductivity, and the self-diffusion coefficient. Next, the transport coefficients are expressed in terms of integral brackets; for the molecule–rigid sphere model these coefficients are brought to a number in the first and second approximations.     

A lattice Boltzmann model for electromagnetic waves propagating in a one-dimensional dispersive medium

A first-order extended lattice Boltzmann (LB) model with special forcing terms for one-dimensional Maxwell equations exerting on a dispersive medium, described either by the Debye or Drude model, is proposed in this study. The time dependent dispersive effect is obtained by the inverse Fourier transform of the frequency-domain permittivity and is incorporated into the LB evolution equations via equivalent forcing effects. The Chapman–Enskog multi-scale analysis is employed to ensure that proposed scheme is mathematically consistent with the targeted Maxwell’s equations at the macroscopic limit. Numerical validations are executed through simulating four representative cases to obtain their LB solutions and compare those with the analytical solutions and existing numerical solutions by finite difference time domain (FDTD). All comparisons show that the differences in numerical values are very small. The present model can thus accurately predict the dispersive effects, and demonstrate first order convergence. In addition to its accuracy, the proposed LB model is also easy to implement. Consequently, this new LB scheme is an effective approach for numerical modeling of EM waves in dispersive media.     

A numerical method for multipoint boundary value problems with application to a restricted three body problem

A type of parallel shooting method is proposed for the solution of nonlinear multipoint boundary value problems. It extends the usual quasilinearization method and a previous shooting method developed for such problems, and reduces to usual multiple shooting techniques for two point boundary value problems. The effectiveness of the method for stiff problems is illustrated by an application to the problem of finding periodic solutions of a restricted three body problem with given Jacobian constant and unknown period.     

Analysis of a multigrid method for a transport equation by numerical Fourier analysis

In this paper, we perform Fourier analysis for a multigrid method with two-cell μ-line relaxation for solving isotropic transport equations. Our numerical results show that the Fourier analysis prediction for convergence rates is more accurate than that previously found by matrix analysis.     

Application of a numerical increased-accuracy method to the solution of the poisson equation

To solve the Poisson equation in a rectangular domain, a numerical increased-accuracy method is used. This method is based on the relationship be Ween the values of the function and its derivatives at neighboring nodes of a net domain. Results of solution of model problems using this method are presented. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 176–182, November–December, 1999.