Numerical Time Integration on Sparse Grids (PhDD thesis)
| Date: 15 April 2011, 17:59
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Introduction
Thee field of applied mathematics consists of the part of mathematics that can be
usefull in solving real life-problems. Since surprisingly often even the most exotic
mathematicall techniques can be applied on real-life problems, applied mathemat
icss is still a very broad term. In fact, most mathematical sub-fields can be classified
bothh under applied mathematics and pure mathematics.
Numerical mathematics is a field that would rarely be seen as purely theoretical
mathematics,, since its applications are so obvious. The term numerical mathemat
icss is not always clear to laymen; they sometimes wonder if not all mathematics is
numerical.. Numerical mathematics focuses on mathematical methods that can be
implementedd as computer programs, which then solve the problem under consid
eration. .
Even before the advent of computers numerical mathematics existed, but then as a
farr more academic subject. It is only since the computer has become mainstream,
thatt numerical mathematics has become a highly applied discipline. Due to the
ingenuityy of modern numerical algorithms and the computational power of mod
ernn computers, highly complex problems can be solved that could not be solved
withoutt numerical mathematics.
An important sub-field of numerical mathematics concerns the solution of ordi
naryy differential equations (ODEs) and partial differential equations (PDEs). Many
reall world problems can be formulated in terms of systems of differential equa
tions.. These can often only be solved by means of numerical mathematics. Espe
ciallyy engineering and physics provide numerous problems formulated in terms of
differentiall equations, but in other disciplines differential equations are frequently
encounteredd as well.
Innthis thesis the focus lies on time-dependent PDEs. To solve these equations
wee apply the method of lines. This implies that first the spatial derivatives are
approximatedd by finite differences, i.e., they are discretized, yielding ODEs in time.
Then a time stepping method is applied to integrate the resulting semi-discrete
problemm in time.
We focus on problems with spatial variables, but the methods presented are
equallyy well applicable to problems with other independent variables. For in
stance,, in option pricing models one encounters the Black-Scholes equation [22].
This equation has the form of an advection-diffusion-reaction equation when one
interpretss the value of the underlying asset as a spatial variable.
Thee focus of this thesis lies mostly on systems of PDEs of the advection-diffusion
type.. These systems are frequently encountered in applications. They, for instance,
playy a prominent role in the mathematical modelling of pollution of atmospheric
air,, surface water and groundwater. Advanced models are three-dimensional in
space.. Their 3D nature and the necessity of modelling transport over long time
spanss requires very efficient algorithms and implementations of algorithms.
Inn the past, much research has been done on developing efficient solvers, notably
advectionn schemes, tailored integrators for stiff systems of ordinary differential
equationss and other time stepping techniques. This has already led to significant
progress.. However, for advanced 3D modelling, computer capacity (computing
timee and memory) still is a severe limiting factor.
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