Efficient simulation of unsaturated flow using exponential time integration

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ(A) where φ(z) = [exp(z) - 1]/z on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ(A)v can be approximated by Krylov subspace methods that require only matrix-vector products with A. We prove that despite the use of this approximation the scheme remains second order. Furthermore, we provide a practical variable-stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation. Numerical experiments performed on two-dimensional test problems demonstrate that this implementation outperforms second-order, variable-stepsize implementations of the backward differentiation formulae.

Inexact uniformization method for computing transient distributions of Markov chains

The uniformization method (also known as randomization) is a numerically stable algorithm for computing transient distributions of a continuous time Markov chain. When the solution is needed after a long run or when the convergence is slow, the uniformization method involves a large number of matrix-vector products. Despite this, the method remains very popular due to its ease of implementation and its reliability in many practical circumstances. Because calculating the matrix-vector product is the most time-consuming part of the method, overall efficiency in solving large-scale problems can be significantly enhanced if the matrix-vector product is made more economical. In this paper, we incorporate a new relaxation strategy into the uniformization method to compute the matrix-vector products only approximately. We analyze the error introduced by these inexact matrix-vector products and discuss strategies for refining the accuracy of the relaxation while reducing the execution cost. Numerical experiments drawn from computer systems and biological systems are given to show that significant computational savings are achieved in practical applications.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

A variable-stepsize Jacobian-free exponential integrator for simulating transport in heterogeneous porous media : application to wood drying

A Jacobian-free variable-stepsize method is developed for the numerical integration of the large, stiff systems of differential equations encountered when simulating transport in heterogeneous porous media. Our method utilises the exponential Rosenbrock-Euler method, which is explicit in nature and requires a matrix-vector product involving the exponential of the Jacobian matrix at each step of the integration process. These products can be approximated using Krylov subspace methods, which permit a large integration stepsize to be utilised without having to precondition the iterations. This means that our method is truly "Jacobian-free" - the Jacobian need never be formed or factored during the simulation. We assess the performance of the new algorithm for simulating the drying of softwood. Numerical experiments conducted for both low and high temperature drying demonstrates that the new approach outperforms (in terms of accuracy and efficiency) existing simulation codes that utilise the backward Euler method via a preconditioned Newton-Krylov strategy.

Stability and convergence of a finite volume method for the space fractional advection-dispersion equation

We consider the space fractional advection–dispersion equation, which is obtained from the classical advection–diffusion equation by replacing the spatial derivatives with a generalised derivative of fractional order. We derive a finite volume method that utilises fractionally-shifted Grünwald formulae for the discretisation of the fractional derivative, to numerically solve the equation on a finite domain with homogeneous Dirichlet boundary conditions. We prove that the method is stable and convergent when coupled with an implicit timestepping strategy. Results of numerical experiments are presented that support the theoretical analysis.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

Unsteady natural convection within a porous enclosure of sinusoidal corrugated side walls

Numerically investigation of free convection within a porous cavity with differential heating has been performed using modified corrugated side walls. Sinusoidal hot left and cold right walls are assumed to receive sudden differentially heating where top and bottom walls are insulated. Air is considered as working fluid and is quiescent, initially. Numerical experiments reveal 3 distinct stages of developing pattern including initial stage, oscillatory intermediate and finally steady state condition. Implicit Finite Volume Method with TDMA solver is used to solve the governing equations. This study has been performed for the Rayleigh numbers ranging from 100 to 10,000. Outcomes have been reported in terms of isotherms, streamline, velocity and temperature plots and average Nusselt number for various Ra, corrugation frequency and corrugation amplitude. The effects of sudden differential heating and its resultant transient behavior on fluid flow and heat transfer characteristics have been shown for the range of governing parameters. The present results show that the transient phenomena are enormously influenced by the variation of the Rayleigh Number with corrugation amplitude and frequency.

Nanoscale plasma chemistry enables fast, size-selective nanotube nucleation

The possibility of fast, narrow-size/chirality nucleation of thin single-walled carbon nanotubes (SWCNTs) at low, device-tolerant process temperatures in a plasma-enhanced chemical vapor deposition (CVD) is demonstrated using multiphase, multiscale numerical experiments. These effects are due to the unique nanoscale reactive plasma chemistry (NRPC) on the surfaces and within Au catalyst nanoparticles. The computed three-dimensional process parameter maps link the nanotube incubation times and the relative differences between the incubation times of SWCNTs of different sizes/chiralities to the main plasma- and precursor gas-specific parameters and explain recent experimental observations. It is shown that the unique NRPC leads not only to much faster nucleation of thin nanotubes at much lower process temperatures, but also to better selectivity between the incubation times of SWCNTs with different sizes and chiralities, compared to thermal CVD. These results are used to propose a time-programmed kinetic approach based on fast-responding plasmas which control the size-selective, narrow-chirality nucleation and growth of thin SWCNTs. This approach is generic and can be used for other nanostructure and materials systems.

Pulsed i-PVD of dielectric nanodot arrays using a nanoporous template

The results of multi-scale numerical simulations of pulsed i-PVD template-assisted nanofabrication of ZnO nanodot arrays on a silicon substrate are presented. The ratios and spatial distributions of the ion fluxes deposited on the lateral and bottom surfaces of the nanopores are computed as a function of the external bias and plasma parameters. The results show that the pulsed bias plays a significant role in the ion current distribution inside the nanopores. The results of numerical experiments of this work suggest that by finely adjusting the pulse voltage, the pulse duration and the duty cycle of the external pulsed bias, the nanopore clogging can be successfully avoided during the deposition and the shapes of the deposited ZnO nanodots can be effectively controlled. A figure is presented.

Deterministic shape control in plasma-aided nanotip assembly

The possibility of deterministic plasma-assisted reshaping of capped cylindrical seed nanotips by manipulating the plasma parameter-dependent sheath width is shown. Multiscale hybrid gas phase/solid surface numerical experiments reveal that under the wide-sheath conditions the nanotips widen at the base and when the sheath is narrow, they sharpen up. By combining the wide- and narrow-sheath stages in a single process, it turns out possible to synthesize wide-base nanotips with long- and narrow-apex spikes, ideal for electron microemitter applications. This plasma-based approach is generic and can be applied to a larger number of multipurpose nanoassemblies. © 2005 American Institute of Physics.

Ion-assisted functional monolayer coating of nanorod arrays in hydrogen plasmas

Uniformity of postprocessing of large-area, dense nanostructure arrays is currently one of the greatest challenges in nanoscience and nanofabrication. One of the major issues is to achieve a high level of control in specie fluxes to specific surface areas of the nanostructures. As suggested by the numerical experiments in this work, this goal can be achieved by manipulating microscopic ion fluxes by varying the plasma sheath and nanorod array parameters. The dynamics of ion-assisted deposition of functional monolayer coatings onto two-dimensional carbon nanorod arrays in a hydrogen plasma is simulated by using a multiscale hybrid numerical simulation. The numerical results show evidence of a strong correlation between the aspect ratios and nanopattern positioning of the nanorods, plasma sheath width, and densities and distributions of microscopic ion fluxes. When the spacing between the nanorods and/or their aspect ratios are larger, and/or the plasma sheath is wider, the density of microscopic ion current flowing to each of the individual nanorods increases, thus reducing the time required to apply a functional monolayer coating down to 11 s for a 7-μm-wide sheath, and to 5 s for a 50-μm-wide sheath. The computed monolayer coating development time is consistent with previous experimental reports on plasma-assisted functionalization of related carbon nanostructures [B. N. Khare et al., Appl. Phys. Lett. 81, 5237 (2002)]. The results are generic in that they can be applied to a broader range of plasma-based processes and nanostructures, and contribute to the development of deterministic strategies of postprocessing and functionalization of various nanoarrays for nanoelectronic, biomedical, and other emerging applications.