Constant-coefficient Equation Research Articles

Painlevé Property and Allowed Transformations of the Variable-Coefficient Zakharov-Kuznetsov Equation

In this letter, we discuss the Painlevé property and allowed transformation for the variable coefficient Zakharov–Kuznetsov equation which governs nonlinear ion-acoustic waves in a magnetized plasma. The general solution of the singular manifold equation and stability solution of the VCZK equation are obtained. To prove some information of the integrability of the ZK equation, we prove the constraints that the variable-coefficient functions for the equation to possess the Painlevé property are not equivalent in order that the equations may be transformed into the constant coefficient equation. So we confirm that the ZK equation is not integrable.

Analytical Solutions for Advection and Advection-Diffusion Equations with Spatially Variable Coefficients

sidered here are consistent with the problem of the transport of pollutant in an open channel where the flow in the channel is augmented by steady, unpolluted lateral inflow distributed along the whole length of the channel, such as a steady inflow of ground water. Therefore the analytical solutions are solu­ tions to a practical problem. The simple expressions considered here for the spatial var­ iation of the coefficients facilitate the process of obtaining an­ alytical solutions to these equations. The spatially variable co­ efficient equations reduce to constant coefficient equations through a simple transformation. Consequently, many of the ~nown analytical solutions to the constant coefficient equa­ tions can be used to obtain analytical solutions to the spatially variable coefficient equations. Analytical solutions are provided for the advection of a sud­ den release of pollutant into the channel and for the solution of advection-diffusion equation. The advection of an initial quasi-Gaussian concentration profile in the channel is also considered. The analytical solutions are simple to evaluate and are useful for validating numerical schemes for solving the advection and advection-diffusion equation with spatially var­ iable coefficients written in either conservative or nonconser­ vative form (Zoppou and Knight 1994). The conservative and nonconservative forms of the equa­ tions are valid equations describing different physical prob­ lems. The analytical solutions to the conservative and noncon­ servative forms of the governing equations will be used to illustrate the importance of selecting the equation relevant to the physical problem, when spatially variable coefficients are involved.

Osmotic coefficients and enthalpies of dilution of aqueous sodium n-alkane-1-sulphonates at the temperatures (298.15, 313.15, 328.15, and 343.15) K

Osmotic coefficients of aqueous solutions of sodium n-octane-1-sulphonate, sodium n-decane-1-sulphonate, sodium n-dodecane-1-sulphonate, and sodium n-tetradecane-1-sulphonate at the temperature T= 328.15 K and at molalities ranging from (0.01 to 0.30) mol·kg -1were estimated from vapour pressure osmometer measurements. Integral enthalpies of dilution for aqueous solutions of the same surfactants from T= 298.15 K to T= 343.15 K and at final molalities ranging from about (0.1 to 3.0)· m( cmc) were also measured, where m( cmc) denotes the critical micelle molality. Data for each surfactant were treated using a mass-action model that includes cation-anion interaction terms in the activity coefficient expressions for one anionic micellar species and the monomeric surfactant species. Ion-ion interaction parameters were estimated from the osmotic coefficients, and the temperature coefficients of these parameters were estimated from the enthalpies of dilution. Treatment of the enthalpies of dilution with the model also led to estimated values of Δ r H m ofor the formation reaction of micelles and to the corresponding values m= m( cmc). A simultaneous fit of all the enthalpies for each surfactant yielded numerical values of a small number of parameters for simple equations that describe the rather complex behaviour of aqueous surfactant solutions in terms of an equilibrium constant and activity coefficient equations and their temperature derivatives.

Modeling of non-isothermal gas flow through a heterogeneous medium

A porous media model as a system of two interacting imaginary media is proposed to solve the problem of studying the process of heating of solar thermal station reactor, which is a large long tube filled with rock. The heating is the result of a motioned heat gas flow with the initial constant rate and temperature. The process is described by two one-dimensional non-stationary differential equations of heat transfer convecion and complemented by the equation of mass conservation and temperature dependence for gas density. For the difference realization of obtained problems with the solution, which has a large gradient in a restricted area, a differential approximation method with artificial viscosity coefficient introduction is used. The behavior of a difference solution in the vicinity of area with abrupt temperature change, depending on introduced pseudo viscosity, is studied. The specific applied problem simplified model with the constant coefficient equation, which allows essentially reduced computing time, has been used.

Dead-beat tracking of complex signals

The dead-beat response problem of discrete-time systems was thoroughly studied in the past within the framework of the control systems theory, but has been traditionally restricted to prototype time-domain inputs only. This article presents an extension to the tracking of complex signals that can be generated by systems described by homogeneous linear constant-coefficient difference equations. The proposed approach takes advantage of a general theory for the computation of the response of a linear system to time-domain inputs and the moment-and-cumulant theory for a discrete-time signal. >

Stability and error analysis for time integrators applied to strain-softening materials

The behavior of explicit time integrators is examined for the field equations of dynamic viscoplasticity in one dimension. The analysis proceeds by linearizing the field equations about the current state and freezing coefficients so that the solution to the system of ordinary differential equations can be computed explicitly. Amplification matrices for the central difference method (with rate tangent constitutive update) and fourth-order Runge-Kutta schemes are obtained by applying these schemes to the resulting linear, constant coefficient equations. These amplification matrices lead to local truncation error estimates for the temporal integrators as well as to estimates of the resulting growth rate. Analysis of the cental difference method with a forward Euler constitutive update indicates that the local truncation error involves as the product of the third power of the strain-rate and the square of the time step. In viscoplastic models, the viscoplastic strain-rate is an increasing (decreasing) function of the viscoplastic strain in softening (hardening). Also, the eigenvalues of the rate tangent method increase with strain-rate in softening. Thus, it is recommended that the time step be reduced in localization problems to keep the strain-increment within a prescribed tolerance. As a byproduct of the analysis, conditions for numerical stability for the central difference method with a rate tangent constitutive update are obtained for strain-hardening dynamic viscoplasticity. In the softening regime, the resulting linearized differential equations admit exponentially growing solution. Thus, conventional definitions of stability such as Neumann stability and absolute stability are inappropriate. In order to characterize stability of such systems, we introduce the concept of g-rel stability which combines the concept of relative and absolute stability. We show conditions under which the resulting system is g-rel stable and thus characterize the nature of the numerical stability that can be expected in such problems.

New lod and adi methods for the two-dimensional diffusion equation

This paper extends earlier work on the solution of the constant-coefficient two-dimensional diffusion equation by considering two classes of time-split finite-difference methods, namely locally one-dimensional (LOD) schemes and alternating direction implicit (ADI) methods. Two new fourth-order techniques are described and tested. Firstly, a LOD method based on the fourth-order explicit Noye-Hayman procedure for the one-dimensional diffusion equation is described. Proper treatment of values at, and adjacent to, the boundary at intermediate time levels is necessary, otherwise the method degenerates to second-order. Secondly, an unconditionally stable ADI method based on a (3,9) two-dimensional computational molecule is developed and tested.

Variable-coefficient wave equations with exact spreading solutions

We discuss a family of variable-coefficient linear wave equations that are exactly solved by a generalization of progressing waves that spread out as they propagate. This can be viewed as a generalization to some variable-coefficient equations of the dispersion theory method that is standard for constant coefficient equations. The method is illustrated by solving some physically interesting equations.

The calculation of upstream concentrations from downstream data for a two-layer flow

Is it possible to infer the time-dependent cross-sectionally uniform concentration at the entry to a sampling tube from concentration flux measurements at the exit? Alas, if the shear dispersion within the sampling tube is modelled by a constant-coefficient diffusion equation, the answer is no: the inversion problem is ill-posed. However, the closely related and more accurate telegraph equation can be inverted. As the separation between entry and exit increases, the deconvolution formula grows exponentially and oscillates more rapidly. This has severe repercussions upon the frequency and accuracy with which the exit measurements need to be made

An infinitesimal Liouville-Arnold theorem as a criterion of reducibility for variational Hamiltonian equations

We prove a criterion for the variational equation about a quasiperiodic solution of a Hamiltonian equation being reducible to a constant coefficient equation. We discuss applications of this criterion to the stability problem for lower dimensional invariant tori.

A high-resolution euler solver based on multigrid, semi-coarsening, and defect correction

In an earlier paper, an O(N) method for the computation of stationary solutions to the Euler equations of inviscid compressible gas dynamics has been described. The method is a variant of the multigrid technique and employs semi-coarsening in all co-ordinate directions simultaneously. It provides good convergence rates for first-order upwind discretisations even in the case of alignment, the flow being aligned with the grid. Here we discuss the application of this scheme to higher-order discretisations. Two-grid analysis for the linear constant-coefficient case shows that it is difficult to obtain uniformly good convergence rates for a higher-order scheme, because of waves perpendicular to stream lines. The defect correction technique suffers from the same problem. However, convergence to a point where the residual of the total error (the sum of the iteration error and the discretisation error) is of the order of the truncation error can be obtained in about seven defect correction cycles, according to estimates for the linear constant-coefficient equations. This result is explored for the nonlinear case by some illustrative numerical experiments.

Explictt two-level finite-difference methods for the two-dimensional diffusion equation

Two new two-level implicit finite-difference methods for solving the constant-coefficient two-dimensional diffusion equation are developed by including weights in the discretisation of the equation. Firstly, a finite-difference equation is found which uses the same computational molecule as the Crank-Nicolson equation, is unconditionally stable and second-order accurate, but is found to be more accurate for the numerical tests that were run. Then a higher-order method is developed by elimination of the largest terms from the truncation error obtained from the modified equivalent equation. The resulting unconditionally stable fourth-order method is found to be much more accurate than both of the second-order methods. This new method is more stable than the fourth-order explicit method developed by Noye and Hayman in [1], but it requires more CPU time to run when the same spatial grid is used.

A model for stability of the semi-implicit backward differentiation formulas

A model is presented for stability for an extension of linear multistep methods for stiff ordinary differential equations. The method is based on a prediction followed by a fixed number of corrections obtained by a Newton scheme with inexact Jacobian matrix. The impact on stability of error in the matrix over a broad range of linear, constant coefficient equations is modeled. The model provides practical guidance for implementation of software for stiff equations.

A model for stability of the semi-implicit backward differentiation formulas

A model is presented for stability for an extension of linear multistep methods for stiff ordinary differential equations. The method is based on a prediction followed by a fixed number of corrections obtained by a Newton scheme with inexact Jacobian matrix. The impact on stability of error in the matrix over a broad range of linear, constant coefficient equations is modeled. The model provides practical guidance for implementation of software for stiff equations.

An Extension of a Theorem of Perron

We give sufficient conditions for a linear differential equation of order n to have a solution whose logarithmic derivative is asymptotically in a small neighbourhood of a given constant. This asymptotic behavior is specified by means of a general comparison function $\varphi $. An appropriate formulation of the growth properties of $\varphi $ is introduced. The differential equation is regarded as a perturbation of some constant coefficient equation. Our smallness conditions permit conditional convergence of some corresponding improper integrals.

Evaluation of the Filtered Leapfrogtrapezoidal Time Integration Method

An analysis and evaluation are presented for a new method of time integration for fluid dynamic proposed by Dietrich. The method, called the filtered leapfrog-trapezoidal (FLT) scheme, is analyzed for the one-dimensional constant-coefficient advection equation and is shown to have some advantages for quasi-steady flows. A modification (FLTW) using a weighted combination of FLT and leapfrog is developed which retains the advantages for steady flows, increases accuracy for time-dependent flows, and involves little coding effort. Merits and applicability are discussed.

Numerical solution of the generalized Laplace equation by the boundary element method

The boundary element method (BEM) has, in general, some advantages with respect to domain methods inasmuch as no internal discretization of the domain is required. This article shows that the generalized Laplace equation (GLE) can also be dealt with advantageously by BEM. The basic technique to achieve this consists of transforming the starting equation GLE into a constant-coefficient equation to which the standard BEM can be applied. The procedure is applied to solve numerically three test problems with known analytical solutions.

The Cauchy problem for time-dependent abstract parabolic equations of higher order

After the pioneering works by Chazarain [2] and Dubinskii [3] in the early seventies, several papers were devoted to higher order differential equations in a Banach space, especially for either constant coefficient or second order equations. Detailed references can be find in [4, Sect. 25(c)]. Very recent results in the constant coefficient case are due to Sandefur [S] for second order semilinear equations and to Neubrander [S] for linear arbitrary order equations. In a recent paper [7], we considered an equation such as

Time-accurate solution of advection-diffusion problems by finite elements

The Taylor-Galerkin method, recently proposed for the spatial and temporal discretization of hyperbolic equations, is employed to derive accurate and efficient numerical schemes for the solution of time-dependent advection-diffusion problems. Two distinct numerical strategies are discussed: the first is suitable for evolutionary problems, while the second is appropriate for situations in which a steady state is eventually reached. For purely evolutionary problems the Taylor-Galerkin method is applied to the complete advection-diffusion equation and, through modifications of the standard ‘mass’ matrix, is shown to generate an incremental form of the Crank-Nicolson time-stepping method. Such an incremental form also lends itself to the derivation of computationally simple, but accurate, explicit algorithms. To deal with transient situations which evolve towards a highly convective steady state, the ‘global’ Taylor-Galerkin method is found to be ineffective since it reduces to the standard Galerkin formulation as the temporal term vanishes. For such cases we suggest the use of a splitting-up method in which advection and diffusion are treated separately by appropriate Taylor-Galerkin methods. The proposed methods are illustrated first for linear, constant-coefficient equations. They are successively extended to deal with nonlinear and multi-dimensional problems. Numerical applications indicate their effectiveness.

On Mapping Linear Partial Differential Equations to Constant Coefficient Equations

A constructive algorithm is developed to determine whether or not a given linear p.d.e. can be mapped into a linear p.d.e. with constant coefficients. The algorithm is based on analyzing the infinitesimals of the Lie group of transformations leaving invariant the given p.d.e. As consequences, in two dimensions, necessary and sufficient conditions are given for mapping: (1) a parabolic p.d.e. into the heat equation; (2) a hyperbolic p.d.e. into the wave equation; (3) an elliptic p.d.e. into Laplace’s equation or the Helmholtz equation.The corresponding mappings are given explicitly.