3,033 research outputs found
Development of a Generalized Finite Difference Scheme for Convection-Diffusion Equation
The traditional finite difference method has an important limitation in practical applications, which is the requirement of a structured grid. The purpose of this thesis is to improve the finite difference scheme for application on complex domains. The analysis of the Finite Difference method is carried out for 1D model problems governed by the convection-diffusion equation. The Stencil Mapping method is developed for complex domains. One of the features of this new scheme is that the value at a node can be calculated by using only the neighbouring values on the 3-point stencil. This allows finite differencing for arbitrary nodal distribution in the mesh, and is developed for 2 nd -order and 4 th -order differencing schemes. The numerical solutions for typical boundary and initial value problems are compared with exact solutions. Local truncation error is introduced as an effective parameter to assess accuracy of the scheme. An adaptive meshing procedure is also presented
Simulation of stochastic reaction-diffusion processes on unstructured meshes
Stochastic chemical systems with diffusion are modeled with a
reaction-diffusion master equation. On a macroscopic level, the governing
equation is a reaction-diffusion equation for the averages of the chemical
species. On a mesoscopic level, the master equation for a well stirred chemical
system is combined with Brownian motion in space to obtain the
reaction-diffusion master equation. The space is covered by an unstructured
mesh and the diffusion coefficients on the mesoscale are obtained from a finite
element discretization of the Laplace operator on the macroscale. The resulting
method is a flexible hybrid algorithm in that the diffusion can be handled
either on the meso- or on the macroscale level. The accuracy and the efficiency
of the method are illustrated in three numerical examples inspired by molecular
biology
A 3D MHD model of astrophysical flows: algorithms, tests and parallelisation
In this paper we describe a numerical method designed for modelling different
kinds of astrophysical flows in three dimensions. Our method is a standard
explicit finite difference method employing the local shearing-box technique.
To model the features of astrophysical systems, which are usually
compressible, magnetised and turbulent, it is desirable to have high spatial
resolution and large domain size to model as many features as possible, on
various scales, within a particular system. In addition, the time-scales
involved are usually wide-ranging also requiring significant amounts of CPU
time.
These two limits (resolution and time-scales) enforce huge limits on
computational capabilities. The model we have developed therefore uses parallel
algorithms to increase the performance of standard serial methods. The aim of
this paper is to report the numerical methods we use and the techniques invoked
for parallelising the code. The justification of these methods is given by the
extensive tests presented herein.Comment: 17 pages with 21 GIF figures. Accepted for publication in A&
Derivation of SPH equations in a moving referential coordinate system
The conventional SPH method uses kernel interpolation to derive the spatial
semi-discretisation of the governing equations. These equations, derived using a
straight application of the kernel interpolation method, are not used in
practice. Instead the equations, commonly used in SPH codes, are heuristically
modified to enforce symmetry and local conservation properties. This paper
revisits the process of deriving these semi-discrete SPH equations. It is shown
that by using the assumption of a moving referential coordinate system and
moving control volume, instead of the fixed referential coordinate system and
fixed control volume used in the conventional SPH method, a set of new semi-
discrete equations can be rigorously derived. The new forms of semi-discrete
equations are similar to the SPH equations used in practice. It is shown through
numerical examples that the new rigorously derived equations give similar
results to those obtained using the conventional SPH equations
Simulation of cell movement through evolving environment: a fictitious domain approach
A numerical method for simulating the movement of unicellular organisms which respond to chemical signals is presented. Cells are modelled as objects of finite size while the extracellular space is described by reaction-diffusion partial differential equations. This modular simulation allows the implementation of different models at the different scales encountered in cell biology and couples them in one single framework. The global computational cost is contained thanks to the use of the fictitious domain method for finite elements, allowing the efficient solve of partial differential equations in moving domains. Finally, a mixed formulation is adopted in order to better monitor the flux of chemicals, specifically at the interface between the cells and the extracellular domain
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
Steady compressible vortex flows: the hollow-core vortex array
We examine the effects of compressiblity on the structure of a single row of hollowcore, constant-pressure vortices. The problem is formulated and solved in the hodograph plane. The transformation from the physical plane to the hodograph plane results in a linear problem that is solved numerically. The numerical solution is checked via a Rayleigh-Janzen expansion. It is observed that for an appropriate choice of the parameters M[infty infinity] = q[infty infinity]/c[infty infinity], and the speed ratio, a = q[infty infinity]/qv, where qv is the speed on the vortex boundary, transonic shock-free flow exists. Also, for a given fixed speed ratio, a, the vortices shrink in size and get closer as the Mach number at infinity, M[infty infinity], is increased. In the limit of an evacuated vortex core, we find that all such solutions exhibit cuspidal behaviour corresponding to the onset of limit lines
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