Search

In this paper, we consider one- and two-dimensional singularly perturbed parabolic reaction-diffusion problems. We propose a parameter-uniform numerical scheme to solve these problems. The continuous problem is first discretized in the space variable using a fitted operator finite difference method. The partial differential equation is thus transformed into a system of initial value problems which are then integrated in time with the Crank–Nicolson finite difference method. A convergence analysis shows that the scheme is second-order ε-uniform convergent in space and time. Richardson extrapolation of the space variable results in a fourth order ε-uniform convergence. Numerical experiments on two test examples confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

We consider a method of lines (MOL) approach to determine prices of European and American exchange options when underlying asset prices are modeled with stochastic volatility and jump-diffusion dynamics. As with any other numerical scheme for partial differential equations (PDEs), the MOL becomes increasingly complex when higher dimensions are involved, so we first simplify the problem by transforming the exchange option into a call option written on the ratio of the yield processes of the two assets. This is achieved by taking the second asset yield process as the numéraire. Under the equivalent martingale measure induced by this change of numéraire, we derive the exchange option pricing integro-partial differential equations (IPDEs) and investigate the early exercise boundary of the American exchange option. We then discuss a numerical solution of the IPDEs using the MOL, its implementation using computing software and possible alternative boundary conditions at the far limits of the computational domain. Our analytical and numerical investigation shows that the near-maturity behavior of the early exercise boundary of the American exchange option is significantly influenced by the dividend yields and the presence of jumps in the underlying asset prices. Furthermore, with the numerical results generated by the MOL, we are able to show that key jump and stochastic volatility parameters significantly affect the early exercise boundary and exchange option prices. Our numerical analysis also verifies that the MOL performs more efficiently, than other finite difference methods or simulation approaches for American options, since the MOL integrates the computation of option prices, greeks and the early exercise boundary, and does so with the least error. [ABSTRACT FROM AUTHOR]

A new combined technique is developed for studying free convection of magnetohydrodynamic unsteady incompressible flow in a square cavity that partially heated from lower wall and regular cooling from lateral walls. Convection has been studied for several lengths of heat source, Hartman and the Rayleigh numbers. The local and average Nusselt numbers are calculated and presented graphically along the heat source portion. The proposed technique is a combination of the method of lines (MOL) and a developed penalty-artificial compressibility technique (PACT). Unsteady penalty compressibility governing equations are used instead of solving steady-state balances. The penalty and artificial compressibility factors are estimated using a deduced stability restriction. The validation of the MOL-PACT is verified by comparing the current data with other numerical and experimental ones existing in the literature. The proposed technique provides a new choice through the investigation of MHD mass and heat transfer. [ABSTRACT FROM AUTHOR]

CO2 emissions from flue gases can be restricted via a promising post-combustion capture technique, namely, adsorption. In the present work, behavior of adsorbents, prepared from PET (polyethylene terephthalate) waste, for CO2 capture studies was investigated to understand the fixed bed adsorption dynamics. The effect of activation on textural properties, surface functional groups and elemental composition was studied through nitrogen adsorption/desorption, XPS, FTIR, and EDX analysis. Thermal stability of adsorbent samples was also evaluated. The kinetics of the adsorption process was studied using Lagergren's pseudo-first-order model, pseudo-second, fractional order, Elovich, and Avrami models. Freundlich isotherm fit confirmed the heterogeneous nature of adsorbent's surface. Adsorption column breakthrough profiles were modeled using linear driving force (LDF) approximation for mass transfer. Model equations were solved using the Method of lines in MATLAB environment. Results show that the model closely approximated the column breakthrough curves for different CO2 concentrations (5–12.5%) and different adsorption temperatures (30–100 °C). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a new numerical method for solving a class of stochastic partial differential equations with additive multi-noise. Based on the domain decomposition method, we combine the deterministic method of lines and the stochastic Itô-Taylor method to construct high-order stochastic numerical method. For numerical approximation of the interface solutions, we introduce the Barycentric interpolation method. The solution is then carried out by collecting the interior solutions on the subdomains and the updated interface solutions. Finally, we computationally analyse on meaningful subdomains with linear and nonlinear interfaces, the case of a stochastic advection–diffusion with additive multi-noise and Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

A mass and energyconservativeexponential time differencing scheme using the method of lines is proposed for the numerical solution of a certain family of first-order time-dependent PDEs. The resulting nonlinear system is solved with an unconditionally stable modified predictor–corrector method using a second-order explicit scheme. The efficiency of the method introduced is analyzed and discussed by applying it to the nonlinear cubic Schrödinger equation. The results arising from the experiments for the single, the double soliton waves and the system of two Schrödinger equations are compared with relevant known ones. [ABSTRACT FROM AUTHOR]

A 2D radiation model based on method of lines (MOL) solution of discrete ordinates method (DOM) coupled with spectral line-based weighted sum of gray gases model (SLW) is developed to predict radiative heat fluxes along the dilute zone of the lignite-fired 150 kWtMiddle East Technical University (METU) circulating fluidized bed combustor (CFBC) under both air-fired and oxy-fired conditions. The dilute zone is treated as an axisymmetric cylindrical enclosure containing a non-gray, absorbing-emitting-isotropically scattering medium. Radiative properties of particles are evaluated by using geometric optics approximation. Input data for the radiation model under air-fired conditions is obtained from predictions of a comprehensive model previously developed and benchmarked against measurements on the same CFBC burning low calorific value indigenous lignite with high volatile matter/fixed carbon (VM/FC) ratio in its own ash. Input data for the radiation model under oxy-fired conditions, on the other hand, is provided by the predictions of a model of oxy-fired retrofit of the same test rig. Predictions of the radiation model reveal similar axial radiative heat flux distributions for air- and retrofitted oxy-fired combustion. In an attempt to investigate the effect of particle load, radiative fluxes were calculated with and without the participating effect of particles for the same input data in each case. Predictions are found to be higher with particle load and much less sensitive to higher carbon dioxide concentration under oxy-fired conditions. [ABSTRACT FROM PUBLISHER]

Predictive accuracy and computationally efficiency of method of lines (MOL) solution of the discrete ordinate method (DOM) coupled with different radiative property estimation techniques (GG, SLW, SNBCK) are assessed by applying them to the prediction of incident radiative fluxes along the freeboard walls of a 0.3 MWt atmospheric bubbling fluidized bed combustor (ABFBC) and comparing their predictions with measurements generated previously from two runs one without and the other with recycle. Freeboard is treated as a three-dimensional rectangular enclosure containing a gray/non-gray, absorbing-emitting-isotropically scattering medium. Radiative properties of particles are evaluated by using geometric optics approximation. A comparative study is also provided between the source term distributions predicted by MOL solution of DOM with gray gas (GG), spectral line-based weighted sum of gray gases (SLW), and statistical narrow-band correlated-k (SNBCK) along the centerline of the freeboard for both runs. Comparisons reveal that the MOL solution of DOM with SLW model and geometric optics approximation provides accurate and computationally efficient solutions for wall fluxes and source term distributions in the freeboard of fluidized bed combustors containing particle laden combustion gases. [ABSTRACT FROM PUBLISHER]

The inverse problems for a parabolic equation with an unknown source (space or time dependent) on the right-hand side are considered. The numerical method is based on the method of lines. The results of numerical experiments on test problems are given. [ABSTRACT FROM PUBLISHER]

In this article, we propose an algorithm for identifying a moving boundary in a 2D heat conduction equation without using initial data. The algorithm combines the method of lines and a quasi-reversibility method. The latter replaces the heat equation with a fourth-order partial differential equation, thereby stabilizing the problem. With the regularization parameters chosen suitably, the method can give an accurate approximation to the sought-for moving boundary. Numerical examples show that the proposed method is effective and stable. [ABSTRACT FROM AUTHOR]

Time-stepping algorithms and their implementations are a critical component within the solution of time-dependent partial differential equations (PDEs). In this article, we present a generic framework - both in terms of algorithms and implementations - that allows an almost seamless switch between various explicit, implicit and implicit-explicit (IMEX) time-stepping methods. We put particular emphasis on how to incorporate time-dependent boundary conditions, an issue that goes beyond classical ODE theory but which plays an important role in the time-stepping of the PDEs arising in computational fluid dynamics. Our algorithm is based upon J.C. Butcher's unifying concept of general linear methods that we have extended to accommodate the family of IMEX schemes that are often used in engineering practice. In the article, we discuss design considerations and present an object-oriented implementation. Finally, we illustrate the use of the framework by applications to a model problem as well as to more complex fluid problems. [ABSTRACT FROM AUTHOR]

This article is concerned with the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonstationary convection-diffusion problem with nonlinear convection and nonlinear diffusion. Optimal estimates in the L∞(L2)-norm are derived for the symmetric interior penalty (SIPG) scheme in two dimensions. The error analysis is carried out for nonconforming triangular meshes under the assumption that the exact solution of the problem and the solution of a linearized elliptic dual problem are sufficiently regular. [ABSTRACT FROM AUTHOR]

Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the numerical method of lines are given. An explicit Euler method is proposed for the numerical solution of systems thus obtained. This leads to difference scheme for the original problem. A complete convergence analysis for the method is given. [ABSTRACT FROM AUTHOR]

In this paper, two methods are developed for linear parabolic partial differential equation with variable coefficients, which are based on rational approximation to the matrix exponential functions. These methods are L-stable, third-order accurate in space and time. In the development of these methods, second-order spatial derivatives are approximated by third-order finite-difference approximations, which give a system of ordinary differential equations whose solution satisfies a recurrence relation that leads to the development of algorithms. These algorithms are tested on heat equation with variable coefficients, subject to homogeneous and/or time-dependent boundary conditions, and no oscillations are observed in the experiments. The method is also modified for a nonlinear problem. All these methods do not require complex arithmetic, and based on partial fraction technique, which is very useful for parallel processing. [ABSTRACT FROM AUTHOR]

New second- and fourth-order accurate semianalytical methods are introduced for the numerical solution of the one-dimensional nonlinear cubic Schrödinger equation. These methods are based on the combination of the Method of Lines and the Lanczos's Tau Method. The methods are self-starting and proved to be stable, accurate, and energy conservative for long time-integration periods. Approximate solutions are sought, on segmented sets of parallel lines, as finite expansions in terms of a given orthogonal polynomial basis. We have carried out numerical application concerning several cases for the propagation, collision and the bound states of N solitons, 2 ≤ N ≤ 5. Accurate results have been obtained using both shifted Chebyshev and Legendre polynomials. These results compare competitively with some other published results obtained using different methods. [ABSTRACT FROM AUTHOR]

A novel parallel computational fluid dynamic (CFD) code based on method of lines (MOL) approach was developed for the numerical simulation of multi-component reacting flows using detailed transport and thermodynamic models. The code was applied to the prediction of a confined axi-symmetric laminar co-flowing methane-air diffusion flame for which experimental data were available in the literature. 1-, 5- and 10-step reduced finite-rate reaction mechanisms were employed for methane-air combustion sub-model. Steady-state velocity, temperature and species profiles obtained by all the mechanisms were validated against experimental data and were found to be in reasonably good agreement with measurements. The code was found to be a useful tool for the prediction and understanding of transient combustion systems. [ABSTRACT FROM AUTHOR]

A mathematical model of multicomponent vacuum desorption, which occurs in vacuum freeze-drying process, was developed. In freeze-drying porous biomaterials and pharmaceutical are considered and the vacuum freeze-drying process, especially the moisture desorption in its final stage, is investigated. In this article, the drying with conductive healing and constant contact surface temperature was considered. Pressure drop is taken into account in the model formulation but was neglected in process simulation because of thin material layers undergoing freeze-drying. Model equations were solved by numerical method of lines. Moisture content and temperature distributions within the drying material were predicted from the model as a function of drying time. [ABSTRACT FROM AUTHOR]

This paper is concerned with the analysis of the discontinuous Galerkin finite element method applied to the space semidiscretization of nonlinear nonstationary convection-diffusion problems. The attention is paid to the derivation of the method and the investigation of error estimates for schemes with a symmetric as well as nonsymmetric discretization of diffusion terms and with the interior and boundary penalty. The theoretical results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

This article aims to present a new approach based on C 1 -cubic splines introduced by Sallam and Naim Anwar [Sallam, S. and Naim Anwar, M. (2000). Stabilized cubic C 1 -spline collocation method for solving first-order ordinary initial value problems, Int. J. Comput. Math. , 74 , 87-96.], which is A-stable, for the time integration of parabolic equations (diffusion or heat equation). The introduced method is an example of the so-called method of lines (the solution is thought to consist of space discretization and time integration), which is an extension of the 1/3-Simpson's finite-difference scheme. Our main objective is to prove the unconditional stability of the proposed method as well as to show that the method is convergent and is of order O ( h 2 ) + O ( k 4 ) i.e. it is a fourth-order in time and second-order in space. Computational results also show that the method is relevant for long time interval problems. [ABSTRACT FROM AUTHOR]

Parallel implementation of a serial code, namely method of lines (MOL) solution for momentum equations (MOLS4ME), previously developed for the solution of transient Navier'Stokes equations for incompressible separated internal flows in regular and complex geometries, is described. The serial code was parallelized by using a domain decomposition strategy with overlapped boundary at the interfaces for information exchange between the sub-domains. A parallel virtual machine (PVM) and dual-processor personal computer (PC) connected over a switch are employed as the message passing software and the parallel computing platform, respectively. Performance of the parallelization strategy was first tested on a transient 1-D laminar pipe flow problem for MOL, explicit and implicit finite difference methods. Assessment of the performance of the parallel code (PARMOLS4ME) was then tested by applying it to two test cases; (i) laminar pipe flow with sudden expansion and (ii)turbulent flow in a gas turbine combustor simulator (GTCS) and comparing the results of serial and parallel codes. Comparisons show that the flow fields predicted by parallel codes agree well with those of serial codes at considerably lower execution times. Effects of load balancing are also investigated and it is seen that the load balancing has a significant effect on the execution time of the parallel code. [ABSTRACT FROM AUTHOR]

The article deals with the initial boundary value problem for an infinite system of first order quasilinear functional differential equations. A comparison result concerning infinite systems of differential difference inequalities is proved. A function satisfying such inequalities is estimated by a solution of a suitable Cauchy problem for an ordinary functional differential system. The comparison result is used in an existence theorem and in the investigation of the stability of the numerical method of lines for the original problem. A theorem on the error estimate of the method is given. The infinite system of first order functional differential equations contains, as particular cases, equations with a deviated argument and integral differential equations of the Volterra type. [ABSTRACT FROM AUTHOR]

Algorithms for the determination of eigenmodes and for the analysis of longitudinally varying structures are presented. The media may be anisotropic and the devices may consist of an arbitrary number of layers or waveguide sections, respectively. Starting with generalized transmission line equations, suitable expressions are derived which are implemented in the method of lines, a semianalytical algorithm. To ensure the numerical stability, a transformation of the reflection coefficient is performed. Numerical results are presented for a Bragg-grating with a very high number of periods, for an electro-optical switch, and for an anisotropic groove waveguide. The results prove stability of the algorithm, and comparisons with other methods show the accuracy. [ABSTRACT FROM AUTHOR]

A posteriori error estimates for semidiscrete finite element methods for a nonlinear parabolic initial-boundary value problem are considered. The error estimates are obtained by solving local parabolic or elliptic equations for corrections to the solution on each element. The convergence results improve previous results where unnecessary assumptions are imposed on the approximate solution and the elliptic projection of the exact solution. [ABSTRACT FROM AUTHOR]