Australian Mathematical Society (AustMS): E-Journals
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Simulation of vortex shedding flows using high-order fractional step methods
Full text linkUnsteady flow past a square cylinder is simulated using a fractional step method to advance the Navier-Stokes equations in time. The fractional step method is a single step method whereby the momentum equations are solved using an explicit/implicit scheme and an approximate pressure field to yield an estimate of the velocity. This velocity is then projected onto a divergence free field using a pressure correction obtained by the solution of a Poisson pressure correction equation. The integration then proceeds to the next time step. Results were obtained using a Crank-Nicolson scheme and hybrid second and third order Adams-Bashforth/Adams-Moulton schemes and second order in time behaviour is verified for velocities for a developed flow over a square cylinder. Results will be presented comparing the accuracy and efficiency of these schemes with unsteady flows of this type, as well as detailing some of the pitfalls that can be encountered with this approach
Inversion of a generalised Hilbert transform
Full text linkAn integral transform Hy is defined which reduces to the ordinary Hilbert transform H0 when y=0, and is useful in some hydrodynamic applications. Although Hy does not seem to be explicitly invertible for y≠0 (in contrast to H0-1=-H0), it is readily invertible numerically for y less than a certain precision-dependent bound
Similarity, attraction and initial conditions in an example of nonlinear diffusion
Full text linkSimilarity solutions play an important role in many fields of science.
The recent book of Barenblatt (1996) discusses many
examples.
Often, outstanding unresolved issues are whether a similarity solution
is dynamically attractive, and if it is, to what particular solution
does the system evolve.
By recasting the dynamic problem in a form to which centre manifold
theory may be applied, based upon a transformation by
Wayne (1994), we may resolve these issues in many cases.
For definiteness we illustrate the principles by discussing the
application of centre manifold theory to a particular nonlinear
diffusion problem arising in filtration.
Theory constructs the similarity solution, confirms its relevance, and
determines the correct solution for any compact initial condition.
The techniques and results we discuss are applicable to a wide range
of similarity problems
Sigmoidal Transformations and the Trapezoidal Rule
Full text linkA sigmoidal transformation is a one-to-one mapping of the compact interval
[0,1] onto itself whose graph is S-shaped. After giving a formal
definition, various mappings already given in the literature are reviewed in
the light of the definition. At least one new transformation is introduced
and criteria given for generating transformations having special properties.
The use of these transformations in using the trapezoidal rule to evaluate
the integral of f(x) over the limits [0,1]
is then considered and asymptotic estimates of
the truncation errors are obtained under different conditions. The paper
concludes with some numerical examples
Effect of a moving boundary on the deformation of a poro-elastic cylinder
Full text linkThe deformation of a poro-elastic cylinder due to radial fluid flow is considered. This has application to modelling arterial flow and certain filtration processes. A diffusion equation for the dilatation with unusual integral boundary conditions is derived for two typical boundary conditions. Asymptotic solutions, to the linearised equations for small times, are found using Maple and shown to be remarkably accurate even for relatively large times. Since the position of the boundaries changes with time, the fully nonlinear system is solved numerically as a moving boundary value problem. Solutions for the dilatation and displacement are found and comparisons made between the standard linearised and full moving boundary problems with a nonlinear, strain dependent permeability. It is shown that inclusion of the correct position of the moving boundaries has a comparable effect to inclusion of a nonlinear permeability on the deformation of the cylinder
Sparse inverse and characteristic polynomial of generalized arrow matrix
Full text linkA generalized arrow matrix of order n with m non-zero rows and columns is presented. If a simple condition holds, the inverse of this matrix is also an arrow matrix of the same form. We then derive a simple expression for its characteristic polynomial
The Euler-Maclaurin formula revisited
Full text linkThe Euler-Maclaurin summation formula for the approximate evaluation of I = \int01f(x) dx comprises a sum of the form (1/m)\sumj=0m-1f((j+t?)/m), where 0?t? ? 1, a second sum whose terms involve the difference between the derivatives of f at the end-points 0 and 1 and a truncation error term expressed as an integral. By introducing an appropriate change of variable of integration using a sigmoidal transformation of order r?1, (other authors call it a periodizing transformation) it is possible to express I as a sum of m terms involving the new integrand with the second sum being zero. We show that for all functions in a certain weighted Sobolev space, the truncation error is of order O(1/mn1) , for some integer n1 which depends on r. In principle we may choose n1 to be arbitrarily large thereby giving a good rate of convergence to zero of the truncation error. This analysis is then extended to Cauchy principal value and certain Hadamard finite-part integrals over (0,1). In each case, the truncation error is O(1/mn1). This result should prove particularly useful in the context of the approximate solution of integral equations although such discussion is beyond the scope of this paper
The L-curve in regularisation of optimal control computation
No full textWe define the L-curve for a regularised solution of an optimal
control problem, and give a method for computing and locating an
optimal regularisation parameter, corresponding to the L-curve's
corner . The regularisation parameter corresponding to this
corner yields a smoother control function, with only a slight change
in the value of the objective. Implementation in the optimal control
software MISER3, and computations of test examples are discussed
Application of wavefront coordinates to acoustic ray tracing
Full text linkAcoustic wave propagation is considered by transforming the equations of inviscid compressible flow to a coordinate system defined by the wavefront geometry. These equations are linearised and equations for the trajectory of rays are derived in the high frequency limit. The formulation in terms of the new coordinates facilitates a rapid derivation of an expression for the transmission loss associated with propagation along a ray. The form of the equations permits easy and robust calculation of sound propagation through media characterised by a non-uniform sound speed and demonstrates the utility of the coordinate system defined by the natural geometry of the wavefront