104 research outputs found

### 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

### Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach

### Efficient computation of two-dimensional steady free-surface flows

We consider a family of steady free-surface flow problems in two dimensions,
concentrating on the effect of nonlinearity on the train of gravity waves that
appear downstream of a disturbance. By exploiting standard complex variable
techniques, these problems are formulated in terms of a coupled system of
Bernoulli's equation and an integral equation. When applying a numerical
collocation scheme, the Jacobian for the system is dense, as the integral
equation forces each of the algebraic equations to depend on each of the
unknowns. We present here a strategy for overcoming this challenge, which leads
to a numerical scheme that is much more efficient than what is normally
employed for these types of problems, allowing for many more grid points over
the free surface. In particular, we provide a simple recipe for constructing a
sparse approximation to the Jacobian that is used as a preconditioner in a
Jacobian-free Newton-Krylov method for solving the nonlinear system. We use
this approach to compute numerical results for a variety of prototype problems
including flows past pressure distributions, a surface-piercing object and
bottom topographies.Comment: 20 pages, 13 figures, under revie

### Drug diffusion from polymeric delivery devices: a problem with two moving boundaries

An existing model for solvent penetration and drug release from a spherically-shaped polymeric drug delivery device is revisited. The model has two moving boundaries, one that describes the interface between the glassy and rubbery states of polymer, and another that defines the interface between the polymer ball and the pool of solvent. The model is extended so that the nonlinear diffusion coefficient of drug explicitly depends on the concentration of solvent, and the resulting equations are solved numerically using a front-fixing transformation together with a finite difference spatial discretisation and the method of lines. We present evidence that our scheme is much more accurate than a previous scheme. Asymptotic results in the small-time limit are presented, which show how the use of a kinetic law as a boundary condition on the innermost moving boundary dictates qualitative behaviour, the scalings being very different to the similar moving boundary problem that arises from modelling the melting of an ice ball. The implication is that the model considered here exhibits what is referred to as ``non-Fickian'' or Case II diffusion which, together with the initially constant rate of drug release, has certain appeal from a pharmaceutical perspective

### Spectrograms of ship wakes: identifying linear and nonlinear wave signals

A spectrogram is a useful way of using short-time discrete Fourier transforms
to visualise surface height measurements taken of ship wakes in real world
conditions. For a steadily moving ship that leaves behind small-amplitude
waves, the spectrogram is known to have two clear linear components, a
sliding-frequency mode caused by the divergent waves and a constant-frequency
mode for the transverse waves. However, recent observations of high speed ferry
data have identified additional components of the spectrograms that are not yet
explained. We use computer simulations of linear and nonlinear ship wave
patterns and apply time-frequency analysis to generate spectrograms for an
idealised ship. We clarify the role of the linear dispersion relation and ship
speed on the two linear components. We use a simple weakly nonlinear theory to
identify higher order effects in a spectrogram and, while the high speed ferry
data is very noisy, we propose that certain additional features in the
experimental data are caused by nonlinearity. Finally, we provide a possible
explanation for a further discrepancy between the high speed ferry spectrograms
and linear theory by accounting for ship acceleration.Comment: 21 pages, 10 figures, submitte

### Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern
formations which have been the subject of intense focus among the physics and
applied mathematics community for many years. In recent times, much attention
has been devoted to devising strategies for controlling such patterns and
reducing the growth of the interfacial fingers. We continue this research by
reporting on numerical simulations, based on the level set method, of a
generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is
altered. First, we investigate how imposing constant and time-dependent
injection rates in a Hele-Shaw cell that is either standard, tapered or
rotating can be used to reduce the development of viscous fingering when an
inviscid fluid is injected into a viscous fluid over a finite time period. We
perform a series of numerical experiments comparing the effectiveness of each
strategy to determine how these non-standard Hele-Shaw configurations influence
the morphological features of the inviscid-viscous fluid interface. Tapering
plates in either converging or diverging directions leads to reduced metrics of
viscous fingering at the final time when compared to the standard parallel
configuration, especially with carefully chosen injection rates; for the
rotating plate case, the effect is even more dramatic, with sufficiently large
rotation rates completely stabilising the interface. Next, we illustrate how
the number of non-splitting fingers can be controlled by injecting the inviscid
fluid at a time-dependent rate while increasing the gap between the plates.
Simulations compare well with previous experimental results for various
injection rates and geometric configurations. Further, we demonstrate how the
fully nonlinear dynamics of the problem affect the number of fingers that
emerge and how well this number agrees with predictions from linear stability
analysis

### Saffman-Taylor fingers with kinetic undercooling

The mathematical model of a steadily propagating Saffman-Taylor finger in a
Hele-Shaw channel has applications to two-dimensional interacting streamer
discharges which are aligned in a periodic array. In the streamer context, the
relevant regularisation on the interface is not provided by surface tension,
but instead has been postulated to involve a mechanism equivalent to kinetic
undercooling, which acts to penalise high velocities and prevent blow-up of the
unregularised solution. Previous asymptotic results for the Hele-Shaw finger
problem with kinetic undercooling suggest that for a given value of the kinetic
undercooling parameter, there is a discrete set of possible finger shapes, each
analytic at the nose and occupying a different fraction of the channel width.
In the limit in which the kinetic undercooling parameter vanishes, the fraction
for each family approaches 1/2, suggesting that this 'selection' of 1/2 by
kinetic undercooling is qualitatively similar to the well-known analogue with
surface tension. We treat the numerical problem of computing these
Saffman-Taylor fingers with kinetic undercooling, which turns out to be more
subtle than the analogue with surface tension, since kinetic undercooling
permits finger shapes which are corner-free but not analytic. We provide
numerical evidence for the selection mechanism by setting up a problem with
both kinetic undercooling and surface tension, and numerically taking the limit
that the surface tension vanishes.Comment: 10 pages, 6 figures, accepted for publication by Physical Review

### Time-frequency analysis of ship wave patterns in shallow water: modelling and experiments

A spectrogram of a ship wake is a heat map that visualises the time-dependent
frequency spectrum of surface height measurements taken at a single point as
the ship travels by. Spectrograms are easy to compute and, if properly
interpreted, have the potential to provide crucial information about various
properties of the ship in question. Here we use geometrical arguments and
analysis of an idealised mathematical model to identify features of
spectrograms, concentrating on the effects of a finite-depth channel. Our
results depend heavily on whether the flow regime is subcritical or
supercritical. To support our theoretical predictions, we compare with data
taken from experiments we conducted in a model test basin using a variety of
realistic ship hulls. Finally, we note that vessels with a high aspect ratio
appear to produce spectrogram data that contains periodic patterns. We can
reproduce this behaviour in our mathematical model by using a so-called
two-point wavemaker. These results highlight the role of wave interference
effects in spectrograms of ship wakes.Comment: 14 pages, 7 figure

### What is the apparent angle of a Kelvin ship wave pattern?

While the half-angle which encloses a Kelvin ship wave pattern is commonly
accepted to be 19.47 degrees, recent observations and calculations for
sufficiently fast-moving ships suggest that the apparent wake angle decreases
with ship speed. One explanation for this decrease in angle relies on the
assumption that a ship cannot generate wavelengths much greater than its hull
length. An alternative interpretation is that the wave pattern that is observed
in practice is defined by the location of the highest peaks; for wakes created
by sufficiently fast-moving objects, these highest peaks no longer lie on the
outermost divergent waves, resulting in a smaller apparent angle. In this
paper, we focus on the problems of free surface flow past a single submerged
point source and past a submerged source doublet. In the linear version of
these problems, we measure the apparent wake angle formed by the highest peaks,
and observe the following three regimes: a small Froude number pattern, in
which the divergent waves are not visible; standard wave patterns for which the
maximum peaks occur on the outermost divergent waves; and a third regime in
which the highest peaks form a V-shape with an angle much less than the Kelvin
angle. For nonlinear flows, we demonstrate that nonlinearity has the effect of
increasing the apparent wake angle so that some highly nonlinear solutions have
apparent wake angles that are greater than Kelvin's angle. For large Froude
numbers, the effect on apparent wake angle can be more dramatic, with the
possibility of strong nonlinearity shifting the wave pattern from the third
regime to the second. We expect our nonlinear results will translate to other
more complicated flow configurations, such as flow due to a steadily moving
closed body such as a submarine.Comment: 19 pages, 9 figures, accepted for publication by Journal of Fluid
Mechanic

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