Decay rate for perturbations of stationary discrete shocks for convex scalar conservation laws

This paper is to study the decay rate for perturbations of stationary discrete shocks for the Lax-Friedrichs scheme approximating the scalar conservation laws by means of an elementary weighted energy method. If the summation of the initial perturbation over is small and decays at the algebraic rate as , then the solution approaches the stationary discrete shock profiles at the corresponding rate as . This rate seems to be almost optimal compared with the rate in the continuous counterpart. Proofs are given by applying the weighted energy integration method to the integrated scheme of the original one. The selection of the time-dependent discrete weight function plays a crucial role in this procedure.

     

Global smooth solutions for a one-dimensional nonisentropic hydrodynamic model with non-constant lattice temperature

In this paper, a one-dimensional nonisentropic hydrodynamic model for semiconductors with non-constant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. The existence and uniqueness of the corresponding stationary solutions are investigated carefully under proper conditions. Then, global existence of the smooth solutions for the Cauchy problem with initial data, which are perturbations of stationary solutions, is established. It is shown that these smooth solutions tend to the stationary solutions exponentially fast as t → ∞.      

A fully discrete scheme for diffusive-dispersive conservation laws

Summary.   We introduce a fully discrete (in both space and time) scheme for the numerical approximation of diffusive-dispersive hyperbolic conservation laws in one-space dimension. This scheme extends an approach by LeFloch and Rohde [4]: it satisfies a cell entropy inequality and, as a consequence, the space integral of the entropy is a decreasing function of time. This is an important stability property, shared by the continuous model as well. Following Hayes and LeFloch [2], we show that the limiting solutions generated by the scheme need not coincide with the classical Oleinik-Kruzkov entropy solutions, but contain nonclassical undercompressive shock waves. Investigating the properties of the scheme, we stress various similarities and differences between the continuous model and the discrete scheme (dynamics of nonclassical shocks, nucleation, etc). Received November 15, 1999 / Revised version received May 27, 2000 / Published online March 20, 2001     

On the convergence of finite difference schemes for the heat equation with concentrated capacity

Summary. We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is represents the magnitude of the heat capacity concentrated at the point . An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolev's norms, compatible with the smoothness of the solution are obtained. Received November 2, 1999 / Revised version received July 24, 2000 / Published online May 4, 2001     

Convergence rates to the discrete travelling wave for relaxation schemes

This paper is concerned with the asymptotic convergence of numerical solutions toward discrete travelling waves for a class of relaxation numerical schemes, approximating the scalar conservation law. It is shown that if the initial perturbations possess some algebraic decay in space, then the numerical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time, provided the sums of the initial perturbations for the -component equal zero. A polynomially weighted norm on the perturbation of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate.

     

A non-parameterized entropy correction for Roe's approximate Riemann solver

Summary. The main drawback with Roe's approximate Riemann solver is that non-physical expansion shocks can occur in the vicinity of sonic points. Previous work aimed at enforcing the entropy condition is based on the representation of sonic rarefaction waves. We propose a new non-parameterized approach which is based on a nonlinear Hermite interpolation of an approximate flux function and the exact resolution of non convex scalar Riemann problems. Convergence and consistency with the entropy condition are proved for scalar convex conservation laws with arbitrarily large initial data. When considering strictly hyperbolic systems of conservation laws, consistency of the resulting scheme with the entropy condition is also proved for initial data sufficiently close to a constant. Numerical results on a one-dimensional shock-tube and a two-dimensional supersonic forward facing step confirm our theoretical results. Received March 1, 1993 / Revised version received August 26, 1994     

Nonlinear Stability of Stationary Discrete Shocks for Nonconvex Scalar Conservation Laws

This paper is to study the asymptotic stability of stationary discrete shocks for the Lax-Friedrichs scheme approximating nonconvex scalar conservation laws, provided that the summations of the initial perturbations equal to zero. The result is proved by using a weighted energy method based on the nonconvexity. Moreover, the stability is also obtained. The key points of our proofs are to choose a suitable weight function.

     

Staircasing in semidiscrete stabilised inverse linear diffusion algorithms

We consider a semidiscrete model problem for the approximation of stabilised inverse linear diffusion processes. The work is motivated by an important observation on fully discrete schemes concerning the so-called staircasing phenomenon: when sharpening monotone data profiles, fully discrete methods generally introduce stepfunction-type solutions reminiscent of staircases. In this work, we show by an analysis of dynamical systems in corresponding semidiscrete formulations that already the semidiscrete numerical model contains the relevant information on the occurrence of staircasing. Numerical experiments confirm and complement the theoretical results.     

Vibration analysis of Kirchhoff plates by the Morley element method

Vibration analysis of Kirchhoff plates is of great importance in many engineering fields. The semi-discrete and the fully discrete Morley element methods are proposed to solve such a problem, which are effective even when the region of interest is irregular. The rigorous error estimates in the energy norm for both methods are established. Some reasonable approaches to choosing the initial functions are given to keep the good convergence rate of the fully discrete method. A number of numerical results are provided to illustrate the computational performance of the method in this paper.     

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     

Long-time Asymptotic Behavior of Lax-Friedrichs Scheme

In this paper we investigate the asymptotic stability of the discrete shocks of the Lax-Friedrichs scheme for hyperholic systems of conservation laws. For single equations, we show that the discrete shocks of the Lax-Friedrichs scheme are asymptotically stable in the sense of I³ and I¹. For the systems of conservation laws, if the summation of initial perturbations equals to zero, we show the l² stability and l¹ boundedness.     

Inverse parabolic problems and discrete orthogonality

Summary This paper considers a discrete sampling scheme for the approximate recovery of initial data for one dimensional parabolic initial boundary value problems on a bounded interval. To obtain a given approximate, data is sampled at a single time and at a finite number of spatial points. The significance of this inversion scheme is the ability to accurately predict the error in approximation subject to choice of sample time and spatial sensor locations. The method is based on a discrete analogy of the continuous orthogonality for Sturm-Liouville systems. This property, which is of independent mathematical interest, is the notion of discrete orthogonal systems, which loosely speaking provides an exact (or approximate) Gauss-type quadrature for the continuous biorthogonality conditions.Supported in part by NSF Grant #DMS8905-344. Texas Advanced Research Program Grant #0219-44-5195 and AFOSR Grant #88-0309Visiting at Texas Tech University, Fall 1989Supported in part by NSF Grant #DMS8905-344, NSA grant #MDA904-85-H009 and NASA Grant #NAQ2-89     

Long memory in a linear stochastic Volterra differential equation

In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined. Moreover, it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential. Under certain conditions upon the kernel, even arbitrarily slow decay rates of the autocovariance function can be achieved. Analogous results are obtained for the corresponding discrete equation.     

Micropolar fluid system in a space of distributions and large time behavior

We analyze the well-posedness of the initial value problem for the generalized micropolar fluid system in a space of tempered distributions and also prove the existence of the stationary solutions. The asymptotic stability of solutions is showed in this space, and as a consequence, a criterium for vanishing small perturbations of initial data (stationary solution) at large time is obtained. A fast decay of the solutions is obtained when we assume more regularity on the initial data.     

Entropy generation and shock resolution in the particle model of compressible fluids

The examination of the particle model of compressible fluids that has been developed by the author [Numer. Math. (1997) 76: 111–142] and that has recently been extended to particles of variable size [Numer. Math. (1999) 82: 143–159], is continued. It is shown that, in the limit of particle sizes tending to zero, both the mass density and the mass flux density and the entropy density and the entropy flux density converge in the weak sense and satisfy the corresponding conservation laws. To incorporate entropy generation in shocks, a new kind of viscous force is introduced. Received November 22, 1996 / Revised version received March 30, 1998     

Waveform relaxation for reaction-diffusion equations

We report a new waveform relaxation (WR) algorithm for general semi-linear reaction-diffusion equations. The superlinear rate of convergence of the new WR algorithm is proved, and we also show the advantages of the new approach superior to the classical WR algorithms by the estimation on iteration errors. The corresponding discrete WR algorithm for reaction-diffusion equations is presented, and further the parallelism of the discrete WR algorithm is analyzed. Moreover, the new approach is extended to handle the coupled reaction-diffusion equations. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

Mixing times with applications to perturbed Markov chains

A measure of the “mixing time” or “time to stationarity” in a finite irreducible discrete time Markov chain is considered. The statistic , where {π_j} is the stationary distribution and m_ij is the mean first passage time from state i to state j of the Markov chain, is shown to be independent of the initial state i (so that η_i = η for all i), is minimal in the case of a periodic chain, yet can be arbitrarily large in a variety of situations. An application considering the effects perturbations of the transition probabilities have on the stationary distributions of Markov chains leads to a new bound, involving η, for the 1-norm of the difference between the stationary probability vectors of the original and the perturbed chain. When η is large the stationary distribution of the Markov chain is very sensitive to perturbations of the transition probabilities.     

Asymptotic Stability of Traveling Wave Solutions for Perturbations with Algebraic Decay

For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation of the traveling wave profile decays at an algebraic rate, then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate, and optimal intermediate results that combine temporal and spatial decay are obtained. The proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations.     

Existence and asymptotic stability of relaxation discrete shock profiles

In this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws. The existence of discrete shocks is established by suitable manifold construction, and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in , provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the relaxing scheme for the single shock solution of the system of hyperbolic conservation laws. These results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks.

     

Error and extrapolation of a compact LOD method for parabolic differential equations

This paper is concerned with a compact locally one-dimensional (LOD) finite difference method for solving two-dimensional nonhomogeneous parabolic differential equations. An explicit error estimate for the finite difference solution is given in the discrete infinity norm. It is shown that the method has the accuracy of the second-order in time and the fourth-order in space with respect to the discrete infinity norm. A Richardson extrapolation algorithm is developed to make the final computed solution fourth-order accurate in both time and space when the time step equals the spatial mesh size. Numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm.