2,007 research outputs found
Numerical investigation of the parabolic mixed-derivative diffusion equation via alternating direction implicit methods
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Master of Science, May 14, 2013.In this dissertation, we investigate the parabolic mixed derivative diffusion equation modeling
the viscous and viscoelastic effects in a non-Newtonian viscoelastic fluid. The model is
analytically considered using Fourier and Laplace transformations. The main focus of the
dissertation, however, is the implementation of the Peaceman-Rachford Alternating Direction
Implicit method. The one-dimensional parabolic mixed derivative diffusion equation
is extended to a two-dimensional analog. In order to do this, the two-dimensional analog
is solved using a Crank-Nicholson method and implemented according to the Peaceman-
Rachford ADI method. The behaviour of the solution of the viscoelastic fluid model is
analysed by investigating the effects of inertia and diffusion as well as the viscous behaviour,
subject to the viscosity and viscoelasticity parameters. The two-dimensional parabolic diffusion
equation is then implemented with a high-order method to unveil more accurate
solutions. An error analysis is executed to show the accuracy differences between the numerical
solutions of the general ADI and high-order compact methods. Each of the methods
implemented in this dissertation are investigated via the von-Neumann stability analysis to
prove stability under certain conditions
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing
We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation
High order operator splitting methods based on an integral deferred correction framework
Integral deferred correction (IDC) methods have been shown to be an efficient
way to achieve arbitrary high order accuracy and possess good stability
properties. In this paper, we construct high order operator splitting schemes
using the IDC procedure to solve initial value problems (IVPs). We present
analysis to show that the IDC methods can correct for both the splitting and
numerical errors, lifting the order of accuracy by with each correction,
where is the order of accuracy of the method used to solve the correction
equation. We further apply this framework to solve partial differential
equations (PDEs). Numerical examples in two dimensions of linear and nonlinear
initial-boundary value problems are presented to demonstrate the performance of
the proposed IDC approach.Comment: 33 pages, 22 figure
A compressible solution of the Navier-Stokes equations for turbulent flow about an airfoil
A compressible time dependent solution of the Navier-Stokes equations including a transition turbulence model is obtained for the isolated airfoil flow field problem. The equations are solved by a consistently split linearized block implicit scheme. A nonorthogonal body-fitted coordinate system is used which has maximum resolution near the airfoil surface and in the region of the airfoil leading edge. The transition turbulence model is based upon the turbulence kinetic energy equation and predicts regions of laminar, transitional, and turbulent flow. Mean flow field and turbulence field results are presented for an NACA 0012 airfoil at zero and nonzero incidence angles of Reynolds number up to one million and low subsonic Mach numbers
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