Table Of ContentNUMERICALL TIME INTEGRATION ON SPARSE GRIDS
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NUMERICALL TIME INTEGRATION ON SPARSE GRIDS
BORISS LASTDRAGER
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NUMERICALL TIME INTEGRATION ON SPARSE GRIDS
ACADEMISCHH PROEFSCHRIFT
TERR VERKRIJGING VAN DE GRAAD VAN DOCTOR
AANN DE UNIVERSITEIT VAN AMSTERDAM
OPP GEZAG VAN DE RECTOR MAGNIFICUS
PROF.. MR. P.F. VAN DER HEIJDEN TEN OVERSTAAN
VANN EEN DOOR HET COLLEGE VOOR PROMOTIES
INGESTELDEE COMMISSIE, IN HET OPENBAAR TE
VERDEDIGENN IN DE AULA DER UNIVERSITEIT OP
WOENSDAGG 18 SEPTEMBER 2002 TE 10:00 UUR.
DOOR R
BORISS LASTDRAGER
GEBORENN TE WINSCHOTEN
promotiecommissie e
promotorr prof. dr. J.G. Verwer
co-promotorr prof. dr. ir. B. Koren
overigee leden prof. dr. RW. Hemker
dr.. W. Hoffmann
prof.. dr. PJ. van der Houwen
prof.. dr. ir. P. Wesseling
prof.. dr. M. van Veldhuizen
prof.. dr. M.N. Spijker
FaculteitFaculteit der Natuurwetenschappen, Wiskunde en Informatica
Thee work in this thesis was made possible by financial support from the Dutch
Organizationn for Scientific Research NWO (Nederlandse Organisatie voor Weten-
schappelijkk Onderzoek) under project number 613-02-036.
Timee Integration on Sparse Grids
Boriss Lastdrager
PhDD thesis University of Amsterdam
ISBNN 90-6196-511-X
"PROGRAMMINGG TODAY IS A RACE
BETWEENN SOFTWARE ENGINEERS STRIVING TO BUILD BIGGER AND BETTER
IDIOT-PROOFF PROGRAMS, AND THE UNIVERSE TRYING TO PRODUCE BIGGER AND
BETTERR IDIOTS. SO FAR, THE UNIVERSE IS WINNING."
RICHH COOK
PREFACE E
Thiss PhD-thesis is based on four years of research that I have carried out at the
Centerr for Mathematics and Computer science (CWI) in Amsterdam, the Nether-
lands,, during the period 1998-2002. This research has resulted in four publications
whichh are listed below, together with the corresponding chapters of this thesis. The
chapterss can be read independently.
Chapterr 2 B. Lastdrager, B. Koren,
ErrorError Analysis for Function Representation
byby the Sparse-Grid Combination Technique,
Reportt MAS-R9823, CWI (1998)
Chapterr 3 B. Lastdrager, B. Koren, J.G. Verwer,
TheThe Sparse-Grid Combination Technique
AppliedApplied to Time-Dependent Advection Problems,
Appliedd Numerical Mathematics, Vol. 38, pp. 377-401 (2001)
Alsoo appeared in reduced form in
LectureLecture Notes in Computational Science and Engineering,
Vol.. 14, Multigrid Methods IV, Springer-Verlag (2000)
Chapterr 4 B. Lastdrager, B. Koren, J.G. Verwer,
SolutionSolution of Time-Dependent Advection-Diffusion Problems
withwith the Sparse-Grid Combination Technique and a Rosenbrock Solver,
Journall of Computational Methods in Applied Mathematics,
Vol.. 1, pp. 86-98(2001)
Chapterr 5 B. Lastdrager,
NumericalNumerical Solution of Mixed Gradient-Diffusion Equations
ModellingModelling Axon Growth,
Reportt MAS-R0203, CWI (2002)
Priorr to my PhD research I did research on theoretical physics for my Masters'
thesiss which led to the following publication
B.. Lastdrager, A. Tip, J. Verhoeven, Theory ofCherenkov and Transition Radiation from
LayeredLayered Structures, Physical Review E, Vol. 61, pp. 5767-5778 (2000)
andd patent
J.. Verhoeven, B. Lastdrager, A. Tip, D.K.G. de Boer, New Source-Optics Combination
forfor EUV Lithography, Philips patent ID-number 600353, WK24.886 (1997)
CONTENTS S
Prefacee l
1.. Introduction 1
1.11 Sparse grid combination technique 2
1.22 Mixed gradient-diffusion equations 5
1.33 Thesis outline 5
2.. Function Representation 9
2.11 Introduction 10
2.1.11 Sparse-grid techniques 10
2.1.22 The combination technique 11
2.22 Error accumulation 12
2.2.11 Introduction 12
2.2.22 The [5,5,0] combination scheme 13
2.2.33 The [1,1,-1] combination scheme 15
2.2.44 The [0,0,1] combination scheme 17
2.2.55 Discussion 17
2.33 Local errors 17
2.3.11 Piecewise-constant interpolation 19
2.3.22 Piecewise bi-linear interpolation 24
2.3.33 A numerical test 27
2.3.44 Discussion 27
2.44 Extension to three dimensions 30
2.4.11 Piecewise-constant interpolation 31
2.4.22 Piecewise tri-linear interpolation 32
2.4.33 The semi-sparse grid 33
2.4.44 A numerical test 35
2.4.55 Discussion 36
2.55 Discontinuous functions 37
2.5.11 The [ , ,0] piecewise-constant scheme 41
2 2
2.5.22 The [1,1, -1] piecewise-constant scheme 42
2.5.33 The [0,0,1] piecewise-constant scheme 44
2.5.44 A numerical test 45
2.5.55 Discussion 45
2.66 Conclusions 46
