Second-order Space Derivative Research Articles

Probability distribution functions of derivatives and increments for decaying burgers turbulence

A Lagrangian method is used to show that the power law with a -7/2 exponent in the negative tail of the probability distribution function (PDF) of the velocity gradient and of velocity increments, predicted by E et al. [Phys. Rev. Lett. 78, 1904 (1997)] for forced Burgers turbulence, is also present in the unforced case. The theory is extended to the second-order space derivative whose PDF has power-law tails with exponent -2 at both large positive and negative values and to the time derivatives. PDF's of space and time derivatives have the same (asymptotic) functional forms. This is interpreted in terms of a random Taylor hypothesis.

On the viscous and resistive dissipation of magnetohydrodynamic waves

The dissipative magnetohydrodynamic (MHD) equations, in the presence of Ohmic electrical resistance and bulk and shear viscosities, are linearized relative to a mean state of rest, and eliminated for the velocity perturbation. The result is a vector dissipative MHD equation consisting of (i) nondissipative terms, involving second-order space and time derivatives; (ii) dissipative terms, linear or quadratic in the diffusivities, involving fourth- and fifth-order space–time derivatives. Considering plane wave solutions, the dispersion matrix shows that Alfvén waves decouple, and are damped by Ohmic electrical resistance and shear viscosity. The latter two, and the bulk viscosity, damp the coupled slow and fast modes, which satisfy a dispersion relation of degree five in the frequency; it reduces to degree three in the case of magnetosonic waves, i.e., wave vector transverse to the direction of propagation. The roots of these dispersion relations of degree three to five, are obtained by a novel method, which applies first to the case of weak damping, and then can be extended to strong damping, e.g., taking into account products of diffusivities. For most applications, e.g., MHD waves in the solar atmosphere, the second-order effects of products of diffusivities are small.

Kinetically induced multiplicity of steady states in integral catalytic reactors

The multiplicity features of integral catalytic reactors are studied numerically for the noble metal catalyzed oxidation of carbon monoxide with oxygen. The kinetics of the reaction are described with an elementary step model, displaying so-called kinetically induced rate multiplicity for certain conditions, i.e. multiplicity purely caused by the kinetics of the elementary steps. Implementation of the kinetic model requires separate continuity equations for the adsorbed species. An infinite number of steady states is predicted for an isothermal ideal plug flow reactor, because ignition of the catalytic surface can occur at any axial position of the reactor. Superposition of axial back-mixing upon the plug flow as a feedback mechanism does not change this feature. If the reactor is operated in the region of kinetic multiplicity, an infinite number of steady states is also shown if heat conduction through the solid phase in axial direction is incorporated in the model. The latter is in contrast with previous studies of nonisothermal reactors. The only model extension leading to a finite number of steady states, i.e. five or three, in case of kinetic multiplicity is the introduction of a second-order space derivative in the continuity equation of the adsorbed species. The physical interpretation of the latter model extension is not clear however. For a nonisothermal reactor, the multiplicity source can change from purely kinetic to thermo-kinetic. The latter can result in a transition from a region with an infinite number of steady states to a region with a maximum of either three or five steady states. It is shown that the identification of the actual multiplicity source is important for a correct analysis of observed phenomena and that it is crucial for selecting an optimal operation strategy of industrial reactors.

Maximum principle for the optimal control of a hyperbolic equation in one space dimension, part 1: Theory

A maximum principle is developed for a class of problems involving the optimal control of a damped-parameter system governed by a linear hyperbolic equation in one space dimension that is not necessarily separable. A convex index of performance is formulated, which consists of functionals of the state variable, its first- and second-order space derivatives, its first-order time derivative, and a penalty functional involving the open-loop control force. The solution of the optimal control problem is shown to be unique. The adjoint operator is determined, and a maximum principle relating the control function to the adjoint variable is stated. The proof of the maximum principle is given with the help of convexity arguments. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of structural elements for vibration suppression.

Novel explicit finite-difference schemes for time-dependent Schrödinger equations

We present several new finite-difference schemes that can be used to numerically integrate the time-dependent Schrödinger equation. These schemes are explicit and use an Euler-type expression for the discrete time derivative. However, the second-order space derivative is modeled by a novel form not before seen in the research literature.

Implementation of a variable stepsize variable formula method in the time-integration part of a code for treatment of long-range transport of air pollutants

A mathematical model consisting of two partial differential equations is used to study the long-range transport of sulphur di-oxide and sulphate over Europe. The discretization of the first-order space derivatives ( the advection terms) is carried out by a pseudospectral (Fourier) algorithm. A special technique is applied in the discretization of the second-order space derivatives ( the diffusion terms). Two large systems of ordinary differential equations are solved at each time-step. It is shown that these systems can efficiently be treated by a variable stepsize variable formula method (VSVFM) based on the use of predictor-corrector schemes. The stepsize selection strategy and the formula selection strategy are discussed in detail. An attempt to carry out both an accuracy control and a stability control is made at each time-step. The great efficiency of the VSVFM implemented in our software as well as the reliability of the results are illustrated by numerical experiments, in which real meteorological data (for 1979) at the grid-points of a space domain covering the whole of Europe were used. The main ideas, implemented in the time-integration part, might be applied in many other situations, where the systems of ordinary differential equations arising after the space discretization are only moderately stiff (so that the stability requirements are dominant over the accuracy requirements on a large part of the time-interval but the use of implicit time-integration algorithms that require solving systems of algebraic equations at each time-step is not justified). As an illustration only it should be mentioned that such an application has been carried out in connection with models describing long-range transport of nitrogen pollutants over Europe.

On two integro-differential equations arising in particle transport theory

Two integro-differential equations arising in particle transport theory are solved explicitly using a technique involving difference equations. The physical problems to which these equations apply concern the energy-time and energy-space distributions of fast particles (neutrons, atoms, gamma -rays, etc.) as they slow down in a host medium. One of the equations involves the first-order derivative with respect to time or space and describes particles which scatter essentially in the forward direction. The other equation assumes a diffusive motion with almost isotropic scattering and hence involves a second-order space derivative. Solutions are obtained in heterogeneous media where the number density of scatterers varies continuously in space and also for a series of contiguous slabs in which the material properties remain constant but change discontinuously from slab to slab. The slowing-down density and energy deposition functions are discussed and evaluated explicitly in some special cases.