Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2009. / ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternative
method for the numerical solution of PDEs. Effective methods for the
numerical inversion are based on the approximation of the Bromwich integral.
In this thesis, a numerical study is undertaken to compare the efficiency of
the Laplace inversion method with more conventional time integrator methods.
Particularly, we consider the method-of-lines based on MATLAB’s ODE15s
and the Crank-Nicolson method.
Our studies include an introductory chapter on the Laplace inversion method.
Then we proceed with spectral methods for the space discretization where we
introduce the interpolation polynomial and the concept of a differentiation
matrix to approximate derivatives of a function. Next, formulas of the numerical
differentiation formulas (NDFs) implemented in ODE15s, as well as the
well-known second order Crank-Nicolson method, are derived. In the Laplace
method, to compute the Bromwich integral, we use the trapezoidal rule over
a hyperbolic contour. Enhancement to the computational efficiency of these
methods include the LU as well as the Hessenberg decompositions.
In order to compare the three methods, we consider two criteria: The
number of linear system solves per unit of accuracy and the CPU time per
unit of accuracy. The numerical results demonstrate that the new method,
i.e., the Laplace inversion method, is accurate to an exponential order of convergence
compared to the linear convergence rate of the ODE15s and the
Crank-Nicolson methods. This exponential convergence leads to high accuracy
with only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and the
Crank-Nicolson method as the results show.
Finally, we apply with satisfactory results the inversion method to the axial
dispersion model and the heat equation in two dimensions. / AFRIKAANSE OPSOMMING: In die afgelope paar jaar het die Laplace omkeringsmetode na vore getree
as ’n lewensvatbare alternatiewe metode vir die numeriese oplossing van
PDVs. Effektiewe metodes vir die numeriese omkering word gebasseer op die
benadering van die Bromwich integraal.
In hierdie tesis word ’n numeriese studie onderneem om die effektiwiteit
van die Laplace omkeringsmetode te vergelyk met meer konvensionele tydintegrasie
metodes. Ons ondersoek spesifiek die metode-van-lyne, gebasseer
op MATLAB se ODE15s en die Crank-Nicolson metode.
Ons studies sluit in ’n inleidende hoofstuk oor die Laplace omkeringsmetode.
Dan gaan ons voort met spektraalmetodes vir die ruimtelike diskretisasie,
waar ons die interpolasie polinoom invoer sowel as die konsep van ’n
differensiasie-matriks waarmee afgeleides van ’n funksie benader kan word.
Daarna word formules vir die numeriese differensiasie formules (NDFs) ingebou
in ODE15s herlei, sowel as die welbekende tweede orde Crank-Nicolson
metode. Om die Bromwich integraal te benader in die Laplace metode, gebruik
ons die trapesiumreël oor ’n hiperboliese kontoer. Die berekeningskoste
van al hierdie metodes word verbeter met die LU sowel as die Hessenberg
ontbindings.
Ten einde die drie metodes te vergelyk beskou ons twee kriteria: Die aantal
lineêre stelsels wat moet opgelos word per eenheid van akkuraatheid, en
die sentrale prosesseringstyd per eenheid van akkuraatheid. Die numeriese resultate demonstreer dat die nuwe metode, d.i. die Laplace omkeringsmetode,
akkuraat is tot ’n eksponensiële orde van konvergensie in vergelyking tot
die lineêre konvergensie van ODE15s en die Crank-Nicolson metodes. Die
eksponensiële konvergensie lei na hoë akkuraatheid met slegs ’n klein aantal
oplossings van die lineêre stelsel. Netso, in terme van berekeningskoste is die
Laplace omkeringsmetode meer effektief as ODE15s en die Crank-Nicolson
metode.
Laastens pas ons die omkeringsmetode toe op die aksiale dispersiemodel
sowel as die hittevergelyking in twee dimensies, met bevredigende resultate.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/2735 |
| Date | 12 1900 |
| Creators | Ngounda, Edgard |
| Contributors | Weideman, J. A. C., University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Rights | University of Stellenbosch |
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