Backward Finite Difference Research Articles

Abstract Approach to Evolution Equations of Phase-Field Type and Applications

A general system of abstract nonlinear parabolic equations deriving from phase-field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite differences approximation scheme and existence and uniqueness theorems are proved. Then, applications are presented to two physical situations: first, a standard heat diffusion problem is briefly discussed mainly as a justification of the choices of the abstract setting. Second, a more interesting transmission problem between two different adjoining fluids is studied under very general compatibility hypotheses and the abstract results are adapted to provide an existence and uniqueness theorem also in this case. Moreover, through convex analysis techniques, a mathematical study is performed of the subdifferential-type relations linking the phase fields and the temperatures to the internal energies of the fluids.

Numerical analysis of a stabilized finite element method for tracer injection simulations

To solve a system of partial differential equations that models tracer injection processes in oil reservoirs we use a post-processing technique to compute an accurate approximation for the velocity field and a stabilized finite element method (SUPG) for the spatial approximation of the tracer concentration combined with a backward finite difference discretization in time. Stability, convergence and error estimates are achieved. Numerical results for homogeneous and heterogeneous media are presented illustrating the performance of the proposed methodology.

Numerical simulation of infiltration, evaporation and shallow groundwater levels with the Richards equation

Analysis of water and solute movement in unsaturated–saturated soil systems would greatly benefit from an accurate and efficient numerical solution of the Richards equation. Recently the mass balance problem has been solved by proper evaluation of the water capacity term. However, the Darcy fluxes as calculated by various numerical schemes still deviate significantly due to differences in nodal spacing and spatial averaging of the hydraulic conductivity K. This paper discusses a versatile, implicit, backward finite difference scheme which is relatively easy to implement. Special attention is given to the selection of a head or flux controlled top boundary condition during the iterative solution of the Richards equation. The stability of the scheme is shown for extreme events of infiltration, evaporation and rapidly fluctuating, shallow groundwater levels in case of two strongly non-linear soils. For nodal distances of 5cm, arithmetic means of K overestimate the soil water fluxes, while geometric means of K underestimate these fluxes. At smaller nodal distances, arithmetic means of K converge faster to the theoretical solution than geometric means. In case of nodal distances of 1cm and arithmetic averages of K, errors due to numerical discretization are small compared to errors due to hysteresis and horizontal spatial variability of the soil hydraulic functions.

Computational Electrochemistry. Simulations of Homogeneous Chemical Reactions in the Confluence Reactor and Channel Flow Cell

The finite element method (FEM) is employed to simulate steady-state convection diffusion problems involving electrode processes and coupled homogeneous kinetic reactions in the channel cell and the confluence reactor. Initial FEM simulations are reported for EC, ECE, and DISP reactions in the channel cell, and the results are shown to agree with previous art. Calculations reveal the FEM extends the kinetic range accessible for such mechanisms beyond those reported previously using the backward implicit finite difference method. The FEM is then applied to simulate, for the first time, the quantitative effects of coupled homogeneous reactivity on the voltammetric response of the confluence reactor. Specifically, a working surface is presented for the application of the confluence reactor to the investigation of the CE type reactions. This surface provides a quantitative relationship between the chemical reaction rate, volume flow rate, and inlet concentrations using this new device. In all simulations performed, the FEM is found to be a highly efficient and accurate alternative to the finite difference method when applied to hydrodynamic voltammetric measurements.

A posteriori error estimation and adaptivity for degenerate parabolic problems

Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.

Parameter estimation in nonlinear age-dependent population dynamics

This paper studies a parameter estimation problem for the Gurtin-MacCamy equation, which is a nonlinear model of age-dependent population dynamics. In estimating parameters such as the death rate and the fertility which depend on the age and the total population from a set of fully discrete observations of the population, we use a backward finite-difference scheme. The function-space parameter estimation convergence (FSPEC) of this scheme is proved and numerical simulations are performed.

Time-domain numerical solution of the KZK equation for modeling sound beams radiated by rectangular sources

A time-domain numerical solution of the KZK equation for modeling finite-amplitude sound beams radiated by rectangular sources similar to those used in diagnostic ultrasound is presented. The KZK equation accounts for the effects of diffraction within the parabolic approximation, nonlinearity, and thermoviscous absorption, on the propagation of sound beams. The algorithm is based on that used by Lee and Hamilton [J. Acoust. Soc. Am. 97, 906–917 (1995)] to model circular sources. The diffraction integral has the form of a two-dimensional diffusion equation. The implicit backward finite difference (IBFD) scheme adapted for two dimensions is used in the near field, where the solution tends to be oscillatory. For the remainder of the field, the alternate direction implicit (ADI) method is used for higher accuracy with larger steps than those used in IBFD. The simulation model may be used to predict the FDA regulatory measurements of medical diagnostic scanners.

Generalized Galerkin approximations of elastic waves with absorbing boundary conditions

For the propagation of elastic waves in unbounded domains, absorbing boundary conditions (ABCs) at the fictitious numerical boundaries have been proposed. In this paper we focus on both first- and second-order ABCs in the framework of variational (weak) approximations, like those stemming from Galerkin method (or its variants) for finite element or spectral approximations [1]. In particular, we recover first order conditions as natural (or Neumann) conditions, whereas we propose a penalty residual method for the treatment of second order ABCs. The time discretization is based on implicit backward finite differences, whereas we use spectral Legendre collocation methods set in a variational form for the spatial discretization (treatment of finite element or spectral element approximations is completely similar). Numerical experiments exhibit that the present formulation of second-order ABCs improves the one based on first-order ABCs with regard to both the reduction of the total energy in the computational domain, and the Fourier spectrum of the displacement field at selected points of the elastic medium. A stability analysis is developed for the variational problem in the continuous case both for first- and second-order ABCs. A suitable treatment of ABCs at corners is also proposed.

Pricing the competing risks of mortgage default and prepayment in stochastic metropolitan economies

Outlines previous research relevant to the risks involved in residential mortgages and suggests some reasons for the gap between theory and market practice. Develops a model which adds household income, ability to pay problems and mortgage underwriting constraints to the standard pricing models, using a combination of Monte Carlo simulation and the backward finite difference method to apply it to data on house prices, income and interest rates for 62 US metropolitan areas. Discusses the results which suggest that prepayment risk dominates default risk in all except very low growth housing markets. Adds that increasing loan‐to‐value levels decrease loan values in low growth markets, slightly increase them in high growth/low volatility markets (due to decline in prepayment risk), but have little impact on high growth/high volatility markets (because they are offset by changes in default and prepayment costs). Considers the practical implications of the findings, e.g. for portfolio managers.

Non-linear dynamics of a two phase flow system in an evaporator: The effects of (i) a time varying pressure drop (ii) an axially varying heat flux

In this paper we study the phenomena of density wave oscillations (DWO) in a vertical heated channel. The homogeneous equilibrium model is used to simulate the flow in the two-phase region. The equations are solved numerically using a `shooting-method' technique. This in its turn employs an implicit backward finite difference scheme. The scheme can incorporate the movement of the interface. It is very elegant and does not involve storage of variables in large N× N matrices. This scheme is sufficiently general and can be used to simulate the dynamic behaviour when: (i) the heat flux imposed at the surface is non-constant, i.e. exhibits an axial variation; and (ii) the imposed pressure drop is varied periodically at a fixed frequency. A possible explanation for the conflicting reports of the effect of a periodic variation in heat flux is provided using a linear stability analysis and the D-partition method. The interaction of the natural frequency of the DWO and the fixed forcing frequency of the imposed pressure drop gives rise to various phenomena viz relaxation oscillations, sub-harmonic oscillations, quasi-periodic and chaotic solutions. To aid the experimentalist describe this infinite-dimensional system on the basis of his experimental results we discuss the characterisation using only the velocity time series data. This is done employing the method of delay coordinate embedding. The phase portraits, stroboscopic map and correlation dimension of the actual attractor are compared with that of the reconstructed attractor from the velocity time series.

Experimental data analysis

Reaction rate determination from experimental data is generally an essential part of evaluating enzyme or microorganism growth kinetics and the effects on them. Commonly used methods include forward, centered, or backward finite difference equations using two or more data points. Another commonly applied method for determining rates is least-square regression techniques, and when the sought function is unknown, polynomials are often applied to represent the data. The cubic spline functions presented in this article represent a versatile method of evaluating rates. The advantage in using this method is that experimental error may be largely accounted for by the incorporation of a smoothing step of the experimental data without force-fitting of the data. It also works well when data are unevenly spaced (often the case for experiments running over long periods of time). The functions are easily manipulated, and the algorithm can be written concisely for computer programming. The development of spline functions to determine derivatives as well as integrals is presented.

NUMERICAL SOLUTIONS OF SECOND ORDER IMPLICIT NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS

The existing literature usually assumes that second order ordinary differential equations can be put in first order form, and this assumption is the starting point of most treatments of ordinary differential equations. This paper examines numerical schemes for solving second order implicitnon-linear differential equations. Based on the literature review, three specific methods have been selected here. Two of them are based on series expansions: Picard iteration using Chebyshev series and Incremental Harmonic Balance (IHB), in which the non-linear differential equation is transformed into a set of algebraic ones that are solved iteratively. The third method is based on a fourth order (Houbolt's scheme) and an eighth order backward finite difference method (FDM). Each method is presented first, and then applied to specific examples. It is shown: (1) that the Picard method is not valid for solving implicit equations containing large non-linear inertial terms; (2) that the IHB method yields accurate periodic solutions, together with the frequency of oscillation, and the dynamical stability of the system may be assessed very easily; and (3) that both Houbolt's and the eighth order scheme can be used to compute time histories of initial value problems, if the time step is properly chosen. Finally, it is also illustrated how the combination of IHB and FDM can be a powerful tool for the analysis of non-linear vibration problems defined by implicit differential equations (including also explicit ones), since bifurcation diagrams of stable and unstable periodic solutions can be computed easily with IHB, while periodic and non-periodic stable oscillations may be obtained with FDM.

Numerical method for a class of parabolic problems in composite media

AbstractThis paper deals with a numerical method, viz. a Rothe Galerkin finite element method, for a class of parabolic problems in composite media with jump conditions for the unknown at the interfaces of the subdomains, retaining the continuity of its conormal derivative. Crucial to our approach is a nonstandard variational formulation in a product Sobolev space setting. We begin with the discretization in time by the Rothe method, which, in essence, involves a backward finite difference scheme and which provides a constructive method for proving existence of a (unique) variational exact solution under weak conditions for the data. We are left with a recurrent system of elliptic problems at each subsequent time point which are solved numerically by a Galerkin finite element method. The resulting algorithm may be implemented by adapting standard codes for parabolic problems in one‐component domains, when suitably taking into account the specific transmission conditions at the internal boundaries. The emphasis of the paper is on convergence results and error estimates for both the fully discrete and the semi‐discrete approximation of the exact solution. The effectiveness of the approach is illustrated by means of a 1D and a 2D example, the analytical solution of which is known. © 1993 John Wiley & Sons, Inc.

Unsteady transonic wing flow computations using field-boundary element methods

An unsteady integral equation (or called field-panel, field-boundary element) scheme for solving the full-potential equation for transonic unsteady wing flows has been developed. The unsteady full-potential equation has been written in a moving frame of reference, in the form of the Poisson's equation. Compressibility and unsteadiness have been treated as non-homogeneity. The integral equation solution in terms of velocity field is obtained by the Green's theorem. The solution consists of a wing surface (boundary elements) integral term of vorticity distribution, a wake surface (boundary elements) integral term of free-vortex sheet and a volume (field-elements) integral term of compressibility and unsteadiness over a small limited domain around the wing. Numerical solutions are obtained by a time-marching, iterative procedure. Time-derivative term is calculated by a second-order backward finite-difference scheme. To be consistent with the mixed-nature of flows, the Murman-Cole type-difference scheme is used to compute the derivatives of the density. The present scheme is applied to flows around a rectangular wing at transonic speed undergoing acceleration motion and transient pitching motion, respectively. The time history of wing surface pressure distributions has been presented.

Three-point backward finite-difference method for solving a system of mixed hyperbolic—parabolic partial differential equations

A three-point backward finite-difference method has been derived for a system of mixed hyperbolic—parabolic (convection—diffusion) partial differential equations (mixed PDEs). The method resorts to the three-point backward differencing to approximate the first-order temporal and spatial derivatives, thereby leading to second-order temporal and spatial accuracy and a substantial reduction in numerical oscillations and diffusion. The resultant finite-difference equations are solved with the tridiagonal matrix method at each time step. For a system of mixed PDEs with coupled nonlinear reaction terms, a two-step expansion technique has been derived to linearize the finite-difference equations and uncouple the PDEs. The accuracy of the expansion is of third-order. Consequently, each PDE can be solved independently with the tridiagonal matrix method. Moreover, the present method can be extended to a system of mixed PDEs coupled with ordinary differential equations and/or algebraic equations.

Modeling and simulation of bioremediation of contaminated soil

AbstractA mathematical model has been developed for in situ biodegradation of contaminants in a soil bed. The model equations comprise three convection‐dispersion partial differential equations and one ordinary differential equation. Dimensional analysis of the model equations has been performed, and solution of these equations has been conducted by the newly‐developed three‐point backward finite difference method. The effects of insufficient oxygen supply, growth of biomass and resistance to contaminant migration on the rate of biodegradation have been examined by numerically simulating the dynamic behavior of in situ biodegradation processes. The results of numerical simulation indicate that the rate of biodegradation of contaminants in soil may be constrained not only by insufficient oxygen supply, but also by resistance to contaminant migration within the pore network. The effect of recycling the unreacted contaminants from the bottom of the bed to the top has been examined through simulation, showing that biodegradation takes place mainly in the upper part of the bed.

Hydrodynamic voltammetry at an interdigitated electrode array in a flow channel: Part II. Chemical reaction succeeding electron transfer

The amperometric response is investigated theoretically for an electron transfer reaction followed by a first-order, homogeneous chemical reaction at an interdigitated microelectrode array, consisting of multiple pairs of microelectrodes held at alternating applied potentials on one wall of a flow channel under steady-state flow conditions. Concentration profiles in the channel, amperometric response, signal-to-noise ratios for the detector and optimum electrode design are discussed, on the basis of backward implicit finite difference calculations. The agreement of the simulation with previous results for a single pair of electrodes under comparable conditions is very good. Both absolute and relative enhancements increase significantly with the number of generator-detector pairs, enabling investigations of much faster chemical reactions than feasible with a single pair of electrodes of the same total geometric area.

Simulation of the effects of forest cover, and its removal, on subsurface water

A distributed, physically based mathematical model was developed to simulate the effects of forestry operations on both soil water and streamflow. It combines Galerkin's finite‐element method with an implicit backward finite‐difference scheme to solve the Darcy‐Richards equation of porous media flow, and uses sink concepts to simulate water use by trees. The validity of the model was checked by simulation of one‐dimensional vertical infiltration into a soil column. Five forest configurations, imposed on a hillslope, were simulated: fully treed, upper slope clear‐cut, lower slope clear‐cut, patch‐cut, and entire slope clear‐cut. The results, displayed as total potential and soil water fields, were qualitatively compatible with accepted hydrological concepts. A field study was simulated and the results compared with field measurements. Explanations are offered for discrepancies between the simulation results and the field measurements. The model's primary usefulness rests in its capability to simulate soil water movement and redistribution, on a distributed basis, in response to the processes of infiltration, evaporation, and transpiration.

Multicomponent distillation calculation by a successive relaxation method using a finite difference approximation.

A successive relaxation algorithm is proposed for the solution of multicomponent distillation problems, where the working equation has less round-off error than the Euler equation. By applying the weighting average of forward and backward finite difference to the derivative of liquid mole fraction with respect to time, a numerical integration formula of higher accuracy is obtained. Although the resultant equations have implicit form, they are successfully solved by a successive relaxation algorithm after making reasonable assumptions and combining these equations with the summation equation of liquid compositions. The discussion extends to the method of determining a time increment and a weighting factor based on the results of various numerical examples, so that the proposed algorithm becomes more useful.

Hydrodynamic voltammetry at an interdigitated electrode array in a flow channel: Part I. Numerical simulation

This paper describes the numerical simulation of convective diffusion at an interdigitated electrode array, consisting of multiple pairs of microelectrodes held at alternating applied potentials on one wall of a flow channel. The downstream microelectrode of each pair detects species generated at the upstream microelectrode. Concentration profiles in the channel, amperometric response, and signal-to-noise ratios for the detector electrodes are calculated. The simple backward implicit finite difference (BIFD) simulation approach is applicable for a wide range of channel conditions. The upper number of electrode pairs treatable is limited only by computational time. The agreement of the simulation with previous results for a single pair of electrodes under comparable conditions is very good. Substantial improvements in signal-to-noise ratio are predicted for the multi-electrode interdigitated electrode array relative to a single generator-detector pair of equal overall area. Electrode dimensions are discussed for optimum signal/noise ratio. Relative enhancement increases significantly with the number of generator-detector pairs.