One-space Dimension Research Articles

Lyapunov spectral analysis of a nonequilibrium Ising-like transition.

By simulating a nonequilibrium coupled map lattice that undergoes an Ising-like phase transition, we show that the Lyapunov spectrum and related dynamical quantities such as the dimension correlation length~$\xi_\delta$ are insensitive to the onset of long-range ferromagnetic order. As a function of lattice coupling constant~$g$ and for certain lattice maps, the Lyapunov dimension density and other dynamical order parameters go through a minimum. The occurrence of this minimum as a function of~$g$ depends on the number of nearest neighbors of a lattice point but not on the lattice symmetry, on the lattice dimensionality or on the position of the Ising-like transition. In one-space dimension, the spatial correlation length associated with magnitude fluctuations and the length~$\xi_\delta$ are approximately equal, with both varying linearly with the radius of the lattice coupling.

Global existence and boundedness of large solutions to nonlinear equations of viscoelasticity with hardening

For the solutions of an initial-boundary value problem for the equations of viscoelasticity with isotropic hardening we derive a uniform bound under a growth condition for the nonlinearities in the case of one-space dimension. Global-in-time existence of solutions to large initial data is a consequence of the existence of this bound. In the most simple form, the equations we consider are Open image in new window with suitable functionsg1,g2,h satisfyingg2≥0 and Open image in new window X

Nonlinearly Stable Compact Schemes for Shock Calculations

In this paper the authors discuss the applications of high-order compact finite difference methods for shock calculations. The main idea is the definition of a local mean that serves as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while keeping the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total variation stable in one-space dimension and maximum norm stable in multispace dimensions. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. The idea in this paper can also be applied to other implicit schemes such as the continuous Galerkin finite element methods.

Convergence and error-bound analysis for mixed problems in compressible flow

We prove convergence and derive an error bound for a finite difference approximation to the discontinuous solution of the Na-vier-Stokes equations for nonisentropic, compressible flow in one-space dimension. The scheme can be implemented under appropriate mesh conditions, and it is shown that the approximations converge at a rate O(δx¼). The error is measured in a norm which dominates the sup-norm of the error in the discontinuous variable.

A flame dictionary method for an unsteady, premixed laminar flame

A numerical method is presented that models a premixed, laminar flame in a single or several space dimensions. The method models the reaction zone of the flame as an interface followed by a preheat zone. The downstream edge of the preheat zone characterizes an entry in a flame dictionary. The entries in the flame dictionary constitute a compendium of one-dimensional, steady-state flame profiles under chemical and flow field conditions encountered by the flame as it progresses through the flow field. The dictionary is independent of the number of space dimensions, geometry and boundary conditions. This method overcomes the problems associated with multiple time and length scales associated with chemically reactive flows. Numerical results in one-space dimension demonstrate the accuracy and efficiency of the flame dictionary method. Results are also presented in two space dimensions for the interaction of a premixed, laminar flame with a cold wall.

BOSONIZATION OF NON-RELATIVISTIC FERMIONS AND W-INFINITY ALGEBRA

We discuss the bosonization of non-relativistic fermions in one-space dimension in terms of bilocal operators which are naturally related to the generators of W-infinity algebra. The resulting system is analogous to the problem of a spin in a magnetic field for the group W-infinity. The new dynamical variables turn out to be W-infinity group elements valued in the coset W-infinity/H where H is a Cartan subalgebra. A classical action with an H gauge invariance is presented. This action is three-dimensional. It turns out to be similar to the action that describes the color degrees of freedom of a Yang–Mills particle in a fixed external field. We also discuss the relation of this action with the one recently arrived at in the Euclidean continuation of the theory using different coordinates.

A Simple proof of the generalized Lieb-Thirring inequalities in one-space dimension

The Lieb-Thirring inequalities give a sharp upper bound for the L p -norm of a function which is the pointwise sum of the squares of a finite orthonormal sequence of functions that are elements of a suitable Sobolev space [LT]. Originally proven for the functions defined on the whole n-dimensional Euclidean space, they were later extended to bounded domains and to suborthogonal sequences of functions [GMT]. Here, we present a simple proof of these inequalities for bounded intervals in one space dimension utilizing simple Sobolev inequalities and standard results from Hilbert space theory.

Two-Dimensional Riemann Problem for a Single Conservation Law

The entropy solutions to the partial differential equation \[ (\partial /\partial t)u(t,x,y) + (\partial /\partial x)f(u(t,x,y)) + (\partial /\partial y)g(u(t,x,y)) = 0,\] with initial data constant in each quadrant of the $(x,y)$ plane, have been constructed and are piecewise smooth under the condition $f''(u) \ne 0, g''(u) \ne 0, (f''(u)/g''(u))\prime \ne 0$. This problem generalizes to several space dimensions the important Riemann problem for equations in one-space dimension. Although existence and uniqueness of solutions are well known, little is known about the qualitative behavior of solutions. It is this with which we are concerned here.

On the integrability of multidimensional nonlinear evolution equations

The integrability-test scheme of Chen, Lee, and Liu [H. H. Chen, Y. C. Lee, and C. S. Liu, Phys. Scr. 20, 490 (1979)] from one-space dimension to multispace dimensions is generalized. The temporal equation of the Lax pair is still the linearized perturbed equation that defines the symmetries. But the spectral operator in the Lax pair is no longer the linear recursion operator for symmetries. The absence of the linear recursion operator for symmetries in higher spatial dimensions therefore presents no direct obstacle to the Chen–Lee–Liu test scheme. The Kadomtsev–Petviashvili equation is shown as an example.

Bound state perturbation theory for the one-space and one-time dimension Klein–Gordon equation

We present a perturbation theory for an arbitrary bound state in the one-space and one-time dimension Klein–Gordon equation in the presence of a scalar potential and a vector (fourth component only) potential by reducing it to a Ricatti equation with the method of logarithmic perturbation expansions. All corrections to the energies and wavefunctions, including corrections to the positions of the nodes in excited states, are expressed in quadratures in a hierarchical scheme, without the use of either the Green’s function or the sum over intermediate states.

The method of lines applied to a simple hyperbolic equation

This paper is concerned with the semidiscrete approximation of hyperbolic equations in one-space dimension. A finite-difference approximation is used for the spatial derivative, but the time derivative is retained to yield a system of ordinary-differential equations. We compare fixed time step methods such as the leapfrog and fourth-order Runge-Kutta with variable time step ODE solvers such as Runge-Kutta-Fehlberg and the Adams method of Shampine. We are especially interested in the determination of the optimal time step for the fixed time step methods. A graphical method for this determination is described. We also test two predictor-corrector schemes, one based on the Milne corrector, the other based on an implicit Runge-Kutta formula.

Equation formulation considerations for efficient numerical modeling of semiconductor phenomena

This paper highlights the importance of equation formulation and associated programming efficiency with respect to modeling of semiconductor phenomena via numerical methods. Two numerical modeling efforts are developed in this paper for one-space dimension device modeling. It is shown on a per iteration basis that the ratio of the computational effort between the two methods is a factor of sixteen. The reduction in the computational effort between the two methods was realized by reformulating the mathematical equations and by reconsidering the effect of programming efficiency. A by-product of the reformulation was a factor of two to three improvement in the convergence rate of the nonlinear iteration. With all considerations, the overall improvement in the solution times for one-space dimension device numerical modeling was determined to be a factor of 30–50. When considering equation formulation alone, the per iteration improvement is a factor of 1·43. Coupled with the two to three convergence rate improvement the overall improvement due to equation formulation was approximately 3·67.

Boundary-Layer Formation in the Pinch

A study is made of various processes that occur prior to the pinch effect when an electric field is applied to a deuterium gas. The variables of the problem are the percentage of ionization, the electron and ion temperatures, the resistivity of the gas, and the current density. A one-dimensional problem is considered in which the above variables are determined as functions of one-space dimension and the time. The equations determining these variables are: the heat developed as the current flows through the gas equals the rate of increase of the internal energy of the plasma, the equation describing the rate of ionization, the equation describing the rate of transfer of energy from electrons to ions and the electromagnetic field equations. These equations are a generalization of the plasma equations solved by Wyld and Watson in that spatial dependence is included, i.e., a current layer is calculated instead of assuming a constant current. The equations are solved numerically using an IBM 704 computer.