A variational difference scheme for the one-dimensional diffusion equation

In this paper we construct a homogeneous variational difference scheme for the diffusion equation assuming its coefficients to be bounded and measurable; the order of convergence of the scheme is O(h²). We consider the boundary value problem (1) $$\frac{d}{{dx}}\left( {K(x)\frac{{du}}{{dx}}} \right) - g(x)u = - \frac{{dF}}{{dx}},0< x< X$$ subject to the boundary conditions (2) $$u(0) = a,u(X) = b$$ .     

High order asymptotic-preserving schemes for the Boltzmann equation

In this Note we discuss the construction of high order asymptotic preserving numerical schemes for the Boltzmann equation. The methods are based on the use of Implicit–Explicit (IMEX) Runge–Kutta methods combined with a penalization technique recently introduced in Filbet and Jin (2010) [6].     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

Locally one-dimensional difference scheme for the convective diffusion equation

We consider the mixed boundary-value problem for the nonstationary convective diffusion equation in a rectangular region. The summation approximation method is applied to construct a locally homogeneous difference scheme with O(t ^1/2 + h^3/2) rate of convergence in the L₂ grid metric.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 64–67, 1992;     

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.      

An asymptotic-preserving well-balanced scheme for the hyperbolic heat equationsUn schema « équilibre » et « asymptotic-preserving » pour les equations de la chaleur hyperboliques

We propose here a well-balanced numerical scheme for the one-dimensional Goldstein–Taylor system which is endowed with all the stability properties inherent to the continuous problem and works in both rarefied and diffusive regimes. To cite this article: L. Gosse, G. Toscani, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 337–342.     

Locally one-dimensional difference scheme for a fractional tracer transport equation

A locally one-dimensional scheme for a fractional tracer transport equation of order is considered. An a priori estimate is obtained for the solution of the scheme, and its convergence is proved in the uniform metric.     

A linear backward Euler scheme for the saturation equation: Regularity results and consistency

We consider a linearization of a numerical scheme for the saturation equation (or porous medium equation) , through first order expansions of the fractional function f and the inverse of the function , after a regularization of the porous medium equation. We establish a regularity result for the Continuous Galerkin Method and a regularity result for the linearized scheme analogous to the corresponding nonlinear scheme. We then show that the linearized scheme is consistent with the nonlinear scheme analyzed in a previous work.     

A new finite difference scheme adapted to the one-dimensional Schrödinger equation

We present a new discretisation scheme for the Schrödinger equation based on analytic solutions to local linearisations. The scheme generates the normalised eigenfunctions and eigenvalues simultaneously and is exact for piecewise constant potentials and effective masses. Highly accurate results can be obtained with a small number of mesh points and a robust and flexible algorithm using continuation techniques is derived. An application to the Hartree approximation for SiGe heterojunctions is discussed in which we solve the coupled Schrödinger-Poisson model problem selfconsistently.To the memory of my good friend Erik van Loon, taken from us so soon     

Locally one-dimensional difference scheme for a pseudoparabolic equation with nonlocal conditions

We study a two-dimensional linear pseudoparabolic equation with nonlocal integral boundary conditions in one coordinate direction and use a locally one-dimensional method for solving this problem. We prove the stability of a finite-difference scheme based on the structure of spectrum of the difference operator with nonlocal conditions.     

Fundamental solution of the stationary Dirac equation with a linear potential

Theoretical and Mathematical Physics - We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier...     

A precise pseudodifferential Foldy-Wouthuysen transform for the Dirac equation

We will discuss existence of a unitary pseudodifferential operator U in our algebra of strictly classical pseudodifferential operators on such that U precisely decouples the electronic and positronic part of the Dirac equation, for rather general potentials, and without supersymmetry. Interestingly, an obstruction appears: On may have to remove a finite dimensional space of electronic states, and declare them as positronic, or, vice versa, depending on a certain deficiency index. Possibly, this index is nonzero if electronic bound states penetrate into the positronic continuous spectrum, or vice versa.     

Average implicit linear difference scheme for generalized Rosenau-Burgers equation

In this paper, a linear three-level average implicit finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-Burgers equation is presented. Existence and uniqueness of numerical solutions are discussed. It is proved that the finite difference scheme is convergent in the order of O(τ² + h²) and stable. Numerical simulations show that the method is efficient.     

A class of potentials for the nonstationary Dirac equation

Journal of Mathematical Sciences - A class of nonstationary potentials for the massive Dirac equation in two-dimensional space -time is constructed and studied by the inverse scattering problem...     

Locally one-dimensional scheme for a loaded heat equation with Robin boundary conditions

The third boundary value problem for a loaded heat equation in a p-dimensional parallelepiped is considered. An a priori estimate for the solution to a locally one-dimensional scheme is derived, and the convergence of the scheme is proved.     

A well-balanced van Leer-type numerical scheme for shallow water equations with variable topography

A well-balanced van Leer-type numerical scheme for the shallow water equations with variable topography is presented. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The construction of our scheme is based on exact solutions in computational form of local Riemann problems. Numerical tests are conducted, where comparisons between this van Leer-type scheme and a Godunov-type scheme are provided. Data for the tests are taken in both the subcritical region as well as supercritical region. Especially, tests for resonant cases where the exact solutions contain coinciding waves are also investigated. All numerical tests show that each of these two methods can give a good accuracy, while the van Leer -type scheme gives a better accuracy than the Godunov-type scheme. Furthermore, it is shown that the van Leer-type scheme is also well-balanced in the sense that it can capture exactly stationary contact discontinuity waves.