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Boriss Lastdrager, "Numerical Time Integration on Sparse Grids (PhDD thesis)"

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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