Stiff Nonlinear Differential Equations Research Articles

RBF-PS method and Fourier Pseudospectral method for solving stiff nonlinear partial differential equations

Radial basis function-Pseudospectral method and Fourier Pseudospectral (FPS) method are extended for stiff nonlinear partial differential equations with a particular emphasis on the comparison of the two methods. Fourth-order Runge-Kutta scheme is applied for temporal discretization. The numerical results indicate that RBF-PS method can be more accurate than standard Fourier pseudospectral method for many nonlinear wave equations.

An efficient technique for the point reactor kinetics equations with Newtonian temperature feedback effects

The point reactor kinetics equations of multi-group of delayed neutrons in the presence Newtonian temperature feedback effects are a system of stiff nonlinear ordinary differential equations which have not any exact analytical solution. The efficient technique for this nonlinear system is based on changing this nonlinear system to a linear system by the predicted value of reactivity and solving this linear system using the fundamental matrix of the homogenous linear differential equations. The nonlinear point reactor kinetics equations are rewritten in the matrix form. The solution of this matrix form is introduced. This solution contains the exponential function of a variable coefficient matrix. This coefficient matrix contains the unknown variable, reactivity. The predicted values of reactivity in the explicit form are determined replacing the exponential function of the coefficient matrix by two kinds, Backward Euler and Crank Nicholson, of the rational approximations. The nonlinear point kinetics equations changed to a linear system of the homogenous differential equations. The fundamental matrix of this linear system is calculated using the eigenvalues and the corresponding eigenvectors of the coefficient matrix. Stability of the efficient technique is defined and discussed. The efficient technique is applied to the point kinetics equations of six-groups of delayed neutrons with step, ramp, sinusoidal and the temperature feedback reactivities. The results of these efficient techniques are compared with the traditional methods.

Taylor’s series method for solving the nonlinear point kinetics equations

Abstract Taylor’s series method for solving the point reactor kinetics equations with multi-group of delayed neutrons in the presence of Newtonian temperature feedback reactivity is applied and programmed by FORTRAN. This system is the couples of the stiff nonlinear ordinary differential equations. This numerical method is based on the different order derivatives of the neutron density, the precursor concentrations of i-group of delayed neutrons and the reactivity. The r th order of derivatives are derived. The stability of Taylor’s series method is discussed. Three sets of applications: step, ramp and temperature feedback reactivities are computed. Taylor’s series method is an accurate computational technique and stable for negative step, negative ramp and temperature feedback reactivities. This method is useful than the traditional methods for solving the nonlinear point kinetics equations.

Numerical Stability Study of the Explicit-Implicit Range-Kuta Synthesis Methods (IMEX-ARK)

In this research we studied the numerical stability for Implicit-Explicit Additive Runge-Kutta (IMEX-ARK) methods which have the A[]-stability .This stability is equivalent A-stability of (B()-ARK) methods where B()=B1+(1-)B2 of second and third order for different intervals of values.This methods are suitable for solving stiff non-linear differential equations which has the form : which contains stiff and non-stiff terms. We have used Matlab as programming tool .

High Order Contractive Runge–Kutta Methods for Volterra Functional Differential Equations

A contractivity theory of Runge–Kutta methods for stiff nonlinear Volterra functional differential equations (VFDEs) is established, by means of which a family of high order contractive Runge–Kutta methods for VFDEs are identified. The application of the general theory for stiff nonlinear VFDEs to the special cases of stiff nonlinear delay differential equations and delay integro-differential equations is also presented.

Longitudinal dispersion in a horizontal subsurface flow constructed wetland: a numerical solution

We present a numerical solution for the dead zone model which describes the solute transport in a subsurface and horizontal flow constructed wetland. This model is a system of two mass balance equations for two conceptual areas: the main channel and the storage zone. We use finite difference schemes to determine the numerical solution of the system and we study its convergence by presenting properties related to the stability and accuracy of the schemes. Concerning the experimental results, the magnitude of the longitudinal dispersion and the extension of dead volumes is estimated for clean conditions and after a certain operating period under organic loading conditions. The results showed a considerable amount of longitudinal dispersion through the bed, which was very strong near the feeding point, indicating the occurrence of mixing and significant presence of dead zones and short-circuiting. This approach is expected to be useful to determine operating conditions, such as, the control of the incoming organic loading, and also to avoid the increase of dead zones as a means to improve treatment performance. References J. M. Bahr and J. Rubin. Direct comparison of kinetic and local equilibrium formulations for solute transport affected by surface reactions. Water Resource Res. 23(3):438--452, 1987. doi:10.1029/WR023i003p00438 k: E. Bencala and R. A. Walters. Simulation of solute transport in a mountain pool-and-riffle stream: a transient storage model. Water Resour Res, 19(3):718--724, 1983. doi:10.1029/WR019i003p00718 K. Dekker and J. G. Verwer. Stability of Runge-Kutta Methods for Nonlinear Stiff Differential Equations. cwi Monograph 2, Elsevier, Amsterdam, 1984. S. H. Keefe, L. B. Barber, R. L. Runkel, J. N. Ryan, D. M. McKnight and R. D. Wass. Conservative and reactive solute transport in constructed wetlands. Water Resource Res., 40: W01201, 2004. doi:10.1029/2003WR002130 C. J. Martinez and W. R. Wise. Analysis of constructed treatment wetland hydraulics with the transient storage model otis. Ecological Engineering, 20(3):211, 2003. doi:10.1016/S0925-8574(03)00029-6 C. F. Nordin and B. M. Troutman. Longitudinal dispersion in rivers: the persistence of skewness in observed data. Water Resour Res, 16(1):123--128, 1980. doi:10.1029/WR016i001p00123 R. L. Runkel and S. C. Chapra. An efficient numerical solution of the transient storage equations for solute transport in small streams. Water Resour Res, 29(1):211--215, 1993. doi:10.1029/92WR02217 J. G. Verwer and J. M. Sanz-Serna. Convergence of Method of Lines Approximations to Partial Differential Equations. Computing, 33(3-4):297--313, 1984. doi:10.1007/BF02242274 B. J. Wagner and J. W Harvey. Experimental design for estimating parameters of rate-limited mass transfer: analysis of stream tracer studies. Water Resource Res., 33(7): 1731--1741, 1997. doi:10.1029/97WR01067

Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series

The paper presents single-term Haar wavelet series (STHWS) approach to the solution of nonlinear stiff differential equations arising in nonlinear dynamics. The properties of STHWS are given. The method of implementation is discussed. Numerical solutions of some model equations are investigated for their stiffness and stability and solutions are obtained to demonstrate the suitability and applicability of the method. The results in the form of block-pulse and discrete solutions are given for typical nonlinear stiff systems. As compared with the TR BDF2 method of Shampine and Gill’s method, the STHWS turns out to be more effective in its ability to solve systems ranging from mildly to highly stiff equations and is free from stability constraints.

Large-scale Bayesian parameter estimation for a three-compartment cardiac metabolism model during ischemia

The estimation of a large number of parameters in metabolic systems from few measurements is a difficult and important problem. A three-compartment model of the cellular cardiac metabolism depends on a large number of parameters, many of which are nonphysical. The model consists of a large system of stiff nonlinear ordinary differential equations that describe the dynamics of the concentrations in the three compartments during ischemia. The data are limited to measured concentrations at few observation times, thus making the inverse problem severely under-determined and ill-posed. The problem is modelled in a Bayesian framework where we relate the unknown parameters to the data through conditional probability density functions. This setting allows us to import necessary a priori knowledge about the model into the estimation process to guide us towards a meaningful solution. The methods for the parameter estimation combine numerical methods for unconstrained optimization and Monte Carlo-based statistical sampling techniques.

Rate-Type Model for Bituminous Mixtures and Its Application to Sand Asphalt

A variety of hot mix asphalt mixtures are used in highway and runway pavement construction. Each mixture caters to specific needs. Mixtures differ from one another in the type and percentage of aggregates and asphalt used, and their response can be markedly different, and thus there is a need to develop constitutive models that can differentiate between the different kinds of mixtures. In this paper, we outline a general procedure for the constitutive modeling of bituminous mixtures. We illustrate the efficacy of this approach by means of an application to sand asphalt. The governing equations for this special problem reduce to a stiff nonlinear ordinary differential equation and this is solved numerically using Gear’s method. We compare the results of the predictions of the model that we have developed with the compressive creep experiments carried out by Wood and Goetz on a typical sand asphalt mixture and find them to be in good agreement.

Kinetic Model for the Degradation of MTBE by Fenton's Oxidation

Environmental Context.Since the early 1990s, methyl tert-butyl ether (MTBE), a possible human carcinogen, has been used as a gasoline oxygenate at concentrations of up to 15% by volume; however, a fraction of the MTBE produced has inevitably been released to the environment. And, spills at gasoline service stations have resulted in local groundwater contamination levels of MTBE over 100 mg L−1, because of its very high water solubility. Advanced oxidation is a common technique for mineralizing organic contaminants, but the reaction chemistry needs to be better understood to facilitate design of remediation systems. Abstract.A kinetic model for the degradation of methyl tert-butyl ether (MTBE) in batch reactors with Fenton’s oxidation (Fe2+/ H2O2) in aqueous solutions was developed. This kinetic model consists of equations accounting for (1) hydrogen peroxide chemistry in aqueous solution, (2) iron speciation, and (3) MTBE oxidation. The mechanisms of MTBE degradation, and the resultant pathways for the formation and degradation of the byproducts, were proposed on the basis of previous studies. A set of stiff nonlinear ordinary differential equations that describe the rate of formation of each species in this batch system was solved using Matlab (R13) software. The kinetic model was validated with published experimental data. The degradation of MTBE by Fenton’s oxidation is predicted well by the model, as are the formation and degradation of byproducts, especially methyl acetate (MA) and tert-butyl alcohol (TBA). Finally, a sensitivity analysis based on calculating the sum of the squares of the residuals (SSR) after making a perturbation of one rate constant at a time was applied to discern the effect of each reaction on MTBE disappearance.

An alternative semi-implicit Euler method for the integration of highly stiff nonlinear differential equations

A relatively simple method has been developed for the integration of highly stiff sets of differential equations describing important, noncatalytic, gas-solid reaction systems. the method is based on the semi-implicit Euler scheme which makes it possible to solve the resulting algebraic equations separately with the aid of always converging procedures such as the interval halving or regula falsi. The developed, semi-implicit Euler method has been compared to some implicit and other semi-implicit techniques. Although the proposed procedure is not so effective as the conventional methods in standard stiff situations, it works reliably also under such circumstances when the conventional techniques fail.

KINETIC PARAMETER ESTIMATION FOR THE CAUSTICIZING REACTION USING A DIFFERENTIAL ALGEBRAIC OPTIMIZATION TECHNIQUE

Abstract Causticizing is an important reaction in the kraft pulping process in which white liquor is regenerated from green liquor. In spite of its major significance to the pulp & paper mill, the kinetics of causticizing reaction is not yet well understood. A kinetic model for the causticizing reaction is developed in this study. The model doesn't take into account the mass transfer considerations as the experiments were done with analytical grade lime. The pseudo homogeneous kinetic model developed here is a stiff non-linear differential equation. The parameters involved in the model are solved using Differential Algebraic Optimization Technique. A systematic approach for solving the parameter estimation problem is outlined in this paper. The model predictions were found to be within 1 % of the observed values.

On the convergence of waveform relaxation methods for stiff nonlinear ordinary differential equations

This paper concerns the numerical solution of stiff initial value problems for systems of ordinary differential equations. We focus on the class of waveform relaxation methods, which was introduced by Lelarasmee et al. (1982). In waveform relaxation methods, a so-called continuous time iteration is set up, which is based on a decoupling of a given initial value problem into a number of subsystems. The continuous time iteration generates a sequence of functions that approximate the solution to the given initial value problem. After discretization of the initial value problems in the continuous time iteration, one obtains a so-called discrete time iteration. In this paper we investigate the convergence of continuous time and discrete time iteration processes. We consider discrete time iteration processes that are obtained from Runge-Kutta methods, and derive convergence results that are relevant in applications to nonlinear, nonautonomous, stiff initial value problems.

Computing the internal transient voltage response of a transformer with a nonlinear core using Gear's method. II. Verification

An EHV transformer's insulation structure must be designed to withstand the internal electrical stresses generated during system transients. Computer models are employed for predetermination of these stresses. Part I of this paper developed a detailed transformer model and solution method representing the nonlinear, saturable characteristic of tire core during transients. The resulting set of stiff nonlinear differential algebraic equations are solved by application of Gear's method. Part II presents the verification of this methodology. This is accomplished by comparing the computed and measured response of a 765/345/34.5 kV 500 MVA autotransformer during energization and transient excitation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Microscopic nuclear structure on a parallel platform

Numerical techniques, used to solve a system of stiff nonlinear partial differential equations in a Hartree-Fock mean field description of the structure of the nucleus, are described in detail. A full three-dimensional discrete representation is obtained using a collocation basis-spline spatial lattice, which provides a superior method as compared to traditional approaches. The set of Hartree-Fock equations reduce, in this case, to various matrix vector multiplications and other elementary operations. Parallelization is achieved by mapping the state vectors onto the nodes, using the Intel Paragon and iPSC/860. Procedures used to perform global operations and broadcasts, and a discussion on increasing the vectorization efficiency are presented. Special attention is given to reduce the communication overhead. An algorithm to pipeline the message passing procedures within the Gram-Schmidt orthogonalization routine is discussed in detail.

Computing the internal transient voltage response of a transformer with a nonlinear core using Gear's method. Part 1: Theory

An EHV transformer's insulation structure must be designed to withstand the internal stresses generated during transients. Computer models are employed for predetermination of these stresses. This paper develops a detailed transformer model and solution method which represent the nonlinear, saturable characteristic of the magnetic core during transients. The resulting set of stiff nonlinear differential algebraic equations are solved by application of Gear's method. This paper describes the transformer model and the necessary set of equations based on linearization of the iron core's saturable magnetic characteristic at each time interval. >

Normal Forms, Averaging and Resonance Control of Flexible Structures

In this paper, we address a novel averaging method for analysis and control of high dimensional, stiff nonlinear differential equations governing the dynamics of a variety of flexible structures. The method yields significant reduction of the initial equations by the averaging of high oscillatory elastic distortions over slow rigid-body motions which can be given in either analytical or numerical form. The averaging allows the stiff property to be overcome. In the averaging equations only exceptional resonance nonlinear terms persist which actually control the plant’s stability. A stabilization problem is addressed for an averaging nonlinear plant and the error involved in such an approximation is estimated. A nonlinear stabilizing control is then evaluated in terms of the averaging normal form and the resonance feedback is shown to be an efficient design procedure.

Domain Decomposition to Solve Transition Layers and Asymptotics

The author considers the numerical computation of stiff nonlinear partial differential equations (PDEs) that can be studied by the methods of singular perturbation. Two domain decomposition methods that solve numerically the layers of the singular perturbation problem are presented. These numerical methods use at different stages the information given by the asymptotic analysis. The simplified model of reacting flow of Majda [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 1059–1080], a singular perturbation problem with a turning point [L. Abrahamsson and S. Osher, SIAM J. Numer. Anal., 19 (1982), pp. 979–992], and a combustion problem [J. Pelaez, SIAM J. Appl. Math., 47 (1987), pp. 781–799] that models a sequence of two chemical reactions will be considered as test problems.

On the convergence of multistep methods for nonlinear stiff differential equations

Convergence estimates are given forA(?)-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturbation arguments with frequency domain techniques.

On the convergence of multistep methods for nonlinear stiff differential equations

Convergence estimates are given for A ( )-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturba...