An asymptotically optimal finite element scheme for the arch problem

Received November 22, 1993     

To the theory of asymptotically stable second-order accurate two-stage scheme for an inhomogeneous parabolic initial-boundary value problem

An asymptotically stable two-stage difference scheme applied previously to a homogeneous parabolic equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial condition is extended to the case of an inhomogeneous parabolic equation with an inhomogeneous Dirichlet boundary condition. It is shown that, in the class of schemes with two stages (at every time step), this difference scheme is uniquely determined by ensuring that high-frequency spatial perturbations are fast damped with time and the scheme is second-order accurate and has a minimal error. Comparisons reveal that the two-stage scheme provides certain advantages over some widely used difference schemes. In the case of an inhomogeneous equation and a homogeneous boundary condition, it is shown that the extended scheme is second-order accurate in time (for individual harmonics). The possibility of achieving second-order accuracy in the case of an inhomogeneous Dirichlet condition is explored, specifically, by varying the boundary values at time grid nodes by O(τ ²), where τ is the time step. A somewhat worse error estimate is obtained for the one-dimensional heat equation with arbitrary sufficiently smooth boundary data, namely, $O\left( {\tau ^2 \ln \frac{T} {\tau }} \right) $ , where T is the length of the time interval.     

An asymptotically stable discretization for the Euler–Poisson system in the quasi-neutral limit

We are interested in the modeling of a plasma in the quasi-neutral limit using the Euler–Poisson system. When this system is discretized with a standard numerical scheme, it is subject to a severe numerical constraint related to the quasi-neutrality of the plasma. We propose an asymptotically stable discretization of this system in the quasi-neutral limit. We present numerical simulations for two different one-dimensional test cases that confirm the expected stability of the scheme in the quasi-neutral limit. To cite this article: P. Crispel et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation

In this paper, we propose a linearized implicit finite difference scheme for solving the fractional Ginzburg-Landau equation. The scheme, which involves three time levels, is unconditionally stable and second-order accurate in both time and space variables. Moreover, the unique solvability, the unconditional stability, and the convergence of the method in the \(L^{\infty }\)-norm are proved by the energy method and mathematical induction. Compared with the implicit midpoint difference scheme (Wang and Huang J. Comput. Phys. 312, 31–49, 2016), current linearized method generally reduces the computational cost. Finally, numerical results are presented to confirm the theoretical results.     

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation, is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.     

A scheme for stable numerical differentiation

A method for stable numerical differentiation of noisy data is proposed. The method requires solving a Volterra integral equation of the second kind. This equation is solved analytically. In the examples considered its solution is computed analytically. Some numerical results of its application are presented. These examples show that the proposed method for stable numerical differentiation is numerically more efficient than some other methods, in particular, than variational regularization.     

A non-isentropic Euler–Poisson model for a collisionless plasma

We analyse transonic solutions of the one-dimensional Euler–Poisson model for a collisionless gas of charged particles in the non-isentropic steady-state case. The model consists of the conservation of mass, momentum and energy equations. The electric field is modelled self-consistently (Coulomb field). Boundary conditions on the particle density and particle temperature are imposed. The analysis is based on representing solutions piecewise as orbits in the particle-density-electric-field phase plane and connecting the orbit segments by the jump and entropy conditions. We characterize the set of all solutions of the Euler–Poisson problem. In particular, we show that, depending upon the length of the interval on which the boundary value problem is posed, fully subsonic, one-shock and (in certain cases) two-shock transonic and smooth transonic solutions exist. Also, numerical computations illustrating the structure of the solutions are reported.     

An algorithm for evaluating stable densities in Zolotarev's (M) parameterization

One of the major obstacles to the use of stable distributions in applications is the lack of formulas for their densities. A program is described to calculate a general α-stable density for α > 0.75 and any skewness value β.     

An asymptotically exact polynomial algorithm for equipartition problems

We describe an approximate algorithm for a special ‘quadratic semi-assignment problem’ arising from ‘equipartition’ applications, where one wants to cluster n objects with given weights w_i into p classes, so as to minimize the variance of the class-weights. The algorithm can be viewed both as a list scheduling method and as a special case of a heuristic procedure, due to Nemhauser and Carlson, for quadratic semi-assignment problems. Our main result is that the relative approximation error is O(1/n) when p and r = (maxw_i)/(min w_i) are bounded.     

An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory

Summary. We propose a finite difference scheme to approximate the Fokker-Planck collision operator in 3 velocity dimensions. The principal feature of this scheme is to provide a decay of the numerical entropy. As a consequence, it preserves the collisional invariants and its stationary solutions are the discrete Maxwellians. We consider both the whole velocity-space problem and the bounded velocity problem. In the latter case, we provide artificial boundary conditions which preserve the decay of the entropy. Received October 18, 1993     

An unconditionally stable spline difference scheme of O(k2+h4) for solving the second-order 1D linear hyperbolic equation

In this paper, the second-order linear hyperbolic equation is solved by using a new three-level difference scheme based on quartic spline interpolation in space direction and finite difference discretization in time direction. Stability analysis of the scheme is carried out. The proposed scheme is second-order accurate in time direction and fourth-order accurate in space direction. Finally, numerical examples are tested and results are compared with other published numerical solutions.     

An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation

We report a new unconditionally stable implicit alternating direction implicit (ADI) scheme of O(k² + h²) for the difference solution of linear hyperbolic equation u_tt + 2αu_t + β²u = u_xx + u_yy + f(x, y, t), αβ ≥ 0, 0 < x, y < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where α > 0 and β ≥ 0 are real numbers. The resulting system of algebraic equations is solved by split method. Numerical results are provided to demonstrate the efficiency and accuracy of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 684–688, 2001     

Generalized viscosity implicit scheme with Meir-Keeler contraction for asymptotically nonexpansive mapping in Banach spaces

Numerical Algorithms - Our focus in this paper is on introducing an iterative scheme based on the generalized implicit method and viscosity approximation method with Meir-Keeler contraction for...     

Every composition operator is (mean) asymptotically Toeplitz

Nazarov and Shapiro recently showed that, while composition operators on the Hardy space H² can only trivially be Toeplitz, or even “Toeplitz plus compact,” it is an interesting problem to determine which of them can be “asymptotically Toeplitz.” I show here that if “asymptotically” is interpreted in, for example, the Cesàro (C,α) sense (α>0), then every composition operator on H² becomes asymptotically Toeplitz.     

An asymptotically exact algorithm for the high-multiplicity bin packing problem

The bin packing problem consists of finding the minimum number of bins, of given capacity D, required to pack a set of objects, each having a certain weight. We consider the high-multiplicity version of the problem, in which there are only C different weight values. We show that when C=2 the problem can be solved in time O( log D). For the general case, we give an algorithm which provides a solution requiring at most C−2 bins more than the optimal solution, i.e., an algorithm that is asymptotically exact. For fixed C, the complexity of the algorithm is O(poly( log D)), where poly(·) is a polynomial function not depending on C.