Holdings: Numerical solutions of partial differential equations

Numerical solutions of partial differential equations:

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Bibliographic Details
Format:	Book
Language:	English
Published:	Basel [u.a.] Birkhäuser 2009
Series:	Advanced courses in mathematics CRM Barcelona
Subjects:	Partielle Differentialgleichung - Numerisches Verfahren Differential equations, Partial > Numerical solutions > Congresses Galerkin methods > Congresses Wavelets (Mathematics) > Congresses Numerisches Verfahren Partielle Differentialgleichung Konferenzschrift
Online Access:	Inhaltsverzeichnis Inhaltsverzeichnis
Physical Description:	VIII, 201 S. Ill., graph. Darst. 240 mm x 170 mm
ISBN:	9783764389390 9783764389406

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MARC


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adam_text	CONTENTS FOREWORD IX I WAVELETS AND PARTIAL DIFFERENTIAL EQUATIONS SILVIA BERTOLUZZA AND SILVIA FALLETTA 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 WHAT IS A WAVELET? 5 1.1 MULTIRESOLUTION ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 EXAMPLE I: THE HAAR BASIS . . . . . . . . . . . . . . . . . . 13 1.1.2 EXAMPLE II: B-SPLINES . . . . . . . . . . . . . . . . . . . . . 14 1.1.3 EXAMPLE III: DAUBECHIESS WAVELETS . . . . . . . . . . . . . 14 1.1.4 EXAMPLE IV: THE SCHAUDER BASIS . . . . . . . . . . . . . . . 15 1.2 BEYOND L 2 ( R ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 THE FUNDAMENTAL PROPERTY OF WAVELETS 23 2.1 THE CASE = R : THE FREQUENCY DOMAIN POINT OF VIEW VS. THE SPACE DOMAIN POINT OF VIEW . . . . . . . . . . . . . . . . . . . 24 2.2 THE GENERAL CASE: * DOMAIN OF R D . . . . . . . . . . . . . . . . . . 32 2.3 THE ISSUE OF BOUNDARY CONDITIONS . . . . . . . . . . . . . . . . . . 35 3 WAVELETS FOR PARTIAL DIFFERENTIAL EQUATIONS 37 3.1 WAVELET PRECONDITIONING . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 NONLINEAR WAVELET METHODS FOR THE SOLUTION OF PDES . . . . . . . . 41 3.2.1 NONLINEAR VS. LINEAR WAVELET APPROXIMATION . . . . . . . . 41 3.2.2 NONLINEAR SOLUTION OF PDES . . . . . . . . . . . . . . . . . . 44 3.3 WAVELET STABILISATION OF UNSTABLE PROBLEMS . . . . . . . . . . . . . 47 3.4 A-POSTERIORI ERROR ESTIMATES . . . . . . . . . . . . . . . . . . . . . 50 3.5 OPERATIONS ON INFINITE MATRICES AND VECTORS . . . . . . . . . . . . . 51 BIBLIOGRAPHY 55 VI CONTENTS II HIGH ORDER SHOCK-CAPTURING SCHEMES FOR BALANCE LAWS GIOVANNI RUSSO 59 1 INTRODUCTION 61 1.1 HYPERBOLIC SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.2 SIMPLE THREE-POINT SCHEMES . . . . . . . . . . . . . . . . . . . . . . 64 1.3 CONSERVATION FORM, JUMP CONDITIONS AND CONSERVATIVE SCHEMES . 68 1.4 CONSISTENCY AND CONVERGENCE . . . . . . . . . . . . . . . . . . . . . 72 1.5 CONSERVATION PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . 72 1.6 ENTROPY CONDITION . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 1.7 DISCRETE ENTROPY CONDITION . . . . . . . . . . . . . . . . . . . . . . 75 1.8 DISSIPATION, DISPERSION, AND THE MODIFIED EQUATION . . . . . . . . . 75 1.9 SECOND-ORDER METHODS AND DISPERSION . . . . . . . . . . . . . . . . 77 2 UPWIND SCHEME FOR SYSTEMS 83 2.1 THE RIEMANN PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.2 GODUNOV SCHEME . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3 THE NUMERICAL FLUX FUNCTION 89 3.1 HIGHER-ORDER EXTENSIONS OF THE GODUNOV METHOD . . . . . . . . . . 89 3.2 THE SCALAR EQUATION AND MONOTONE FLUXES . . . . . . . . . . . . . 90 4 NONLINEAR RECONSTRUCTION AND HIGH-ORDER SCHEMES 97 4.1 HIGH-ORDER FINITE-VOLUME SCHEMES . . . . . . . . . . . . . . . . . . 97 4.2 ESSENTIALLY NON-OSCILLATORY RECONSTRUCTION (ENO) . . . . . . . . . 99 4.3 WEIGHTED ENO RECONSTRUCTION (WENO) . . . . . . . . . . . . . . . 102 4.4 CONSERVATIVE FINITE-DIFFERENCE SCHEMES . . . . . . . . . . . . . . . . 104 4.5 TIME INTEGRATION: RUNGE-KUTTA METHODS . . . . . . . . . . . . . . . 106 4.6 SSP SCHEMES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.7 EXTENSION TO MORE DIMENSIONS . . . . . . . . . . . . . . . . . . . . 108 5 CENTRAL SCHEMES 109 5.1 NESSYAHU-TADMOR SECOND-ORDER SCHEME . . . . . . . . . . . . . . . 110 5.2 DESCRIPTION OF CRK SCHEMES . . . . . . . . . . . . . . . . . . . . . 112 5.3 A SECOND-ORDER SCHEME: CRK2 . . . . . . . . . . . . . . . . . . . . 114 5.4 HIGHER-ORDER SCHEMES: CRK3, CRK4, CRK5 . . . . . . . . . . . . 114 5.5 NUMERICAL TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.6 SYSTEMS OF EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.7 COMPONENTWISE APPLICATION . . . . . . . . . . . . . . . . . . . . . . 120 5.8 PROJECTION ALONG CHARACTERISTIC DIRECTIONS . . . . . . . . . . . . . . 123 CONTENTS VII 6 SYSTEMS WITH STIFF SOURCE 125 6.1 SYSTEMS OF BALANCE LAWS . . . . . . . . . . . . . . . . . . . . . . . 126 6.2 IMEX RUNGE-KUTTA SCHEMES . . . . . . . . . . . . . . . . . . . . . 128 6.3 HYPERBOLIC SYSTEMS WITH RELAXATION . . . . . . . . . . . . . . . . . 130 6.3.1 ZERO RELAXATION LIMIT . . . . . . . . . . . . . . . . . . . . . 130 6.3.2 ASYMPTOTIC PROPERTIES OF IMEX SCHEMES . . . . . . . . . . 132 6.4 NUMERICAL TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.4.1 BROADWELL MODEL . . . . . . . . . . . . . . . . . . . . . . . . 134 6.4.2 SHALLOW WATER . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.4.3 TRAFFIC FLOWS . . . . . . . . . . . . . . . . . . . . . . . . . . 137 APPENDIX: BUTCHER TABLEAU OF IMEX-RK 141 BIBLIOGRAPHY 143 III DISCONTINUOUS GALERKIN METHODS: GENERAL APPROACH AND STABILITY CHI-WANG SHU 149 PREFACE 151 1 INTRODUCTION 153 2 TIME DISCRETIZATION 155 3 DISCONTINUOUS GALERKIN METHOD FOR CONSERVATION LAWS 157 3.1 TWO-DIMENSIONAL STEADY-STATE LINEAR EQUATIONS . . . . . . . . . . 157 3.2 ONE-DIMENSIONAL TIME-DEPENDENT CONSERVATION LAWS . . . . . . . . 160 3.2.1 CELL ENTROPY INEQUALITY AND L 2 -STABILITY . . . . . . . . . . . 162 3.2.2 LIMITERS AND TOTAL VARIATION STABILITY . . . . . . . . . . . . 164 3.2.3 ERROR ESTIMATES FOR SMOOTH SOLUTIONS . . . . . . . . . . . . 168 3.3 COMMENTS FOR MULTI-DIMENSIONAL CASES . . . . . . . . . . . . . . . 170 4 DISCONTINUOUS GALERKIN METHOD FOR CONVECTION-DIFFUSION EQUATIONS 175 4.1 LDG SCHEME FORMULATION . . . . . . . . . . . . . . . . . . . . . . . 176 4.2 STABILITY ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.3 ERROR ESTIMATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.4 MULTI-DIMENSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5 DISCONTINUOUS GALERKIN METHOD FOR PDES CONTAINING HIGHER-ORDER SPATIAL DERIVATIVES 183 5.1 LDG SCHEME FOR THE KDV EQUATIONS . . . . . . . . . . . . . . . . . 183 5.1.1 STABILITY ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . 185 5.1.2 ERROR ESTIMATES . . . . . . . . . . . . . . . . . . . . . . . . . 187 VIII CONTENTS 5.2 LDG SCHEMES FOR OTHER HIGHER-ORDER PDES . . . . . . . . . . . . 190 5.2.1 BI-HARMONIC EQUATIONS . . . . . . . . . . . . . . . . . . . . . 190 5.2.2 FIFTH-ORDER CONVECTION-DISPERSION EQUATIONS . . . . . . . . 191 5.2.3 THE K ( M, N ) EQUATIONS . . . . . . . . . . . . . . . . . . . . 191 5.2.4 THE KDV-BURGERS-TYPE (KDVB) EQUATIONS . . . . . . . . . 191 5.2.5 THE FIFTH-ORDER KDV-TYPE EQUATIONS . . . . . . . . . . . . 192 5.2.6 THE FULLY NONLINEAR K ( N, N, N ) EQUATIONS . . . . . . . . . . 192 5.2.7 THE NONLINEAR SCHR¨ ODINGER (NLS) EQUATION . . . . . . . . . 192 5.2.8 THE KADOMTSEV-PETVIASHVILI (KP) EQUATIONS . . . . . . . . 193 5.2.9 THE ZAKHAROV-KUZNETSOV (ZK) EQUATION . . . . . . . . . . . 193 5.2.10 THE KURAMOTO-SIVASHINSKY-TYPE EQUATIONS . . . . . . . . . 193 5.2.11 THE ITO-TYPE COUPLED KDV EQUATIONS . . . . . . . . . . . . 194 5.2.12 THE CAMASSA-HOLM (CH) EQUATION . . . . . . . . . . . . . . 194 5.2.13 THE CAHN-HILLIARD EQUATION . . . . . . . . . . . . . . . . . . 194 BIBLIOGRAPHY 197
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spelling	Numerical solutions of partial differential equations Silvia Bertoluzza ... Basel [u.a.] Birkhäuser 2009 VIII, 201 S. Ill., graph. Darst. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier Advanced courses in mathematics CRM Barcelona Partielle Differentialgleichung - Numerisches Verfahren Differential equations, Partial Numerical solutions Congresses Galerkin methods Congresses Wavelets (Mathematics) Congresses Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Bertoluzza, Silvia Sonstige (DE-588)1020162023 oth http://d-nb.info/990037037/04 Inhaltsverzeichnis SWBplus Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017072256&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Numerical solutions of partial differential equations Partielle Differentialgleichung - Numerisches Verfahren Differential equations, Partial Numerical solutions Congresses Galerkin methods Congresses Wavelets (Mathematics) Congresses Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd
subject_GND	(DE-588)4128130-5 (DE-588)4044779-0 (DE-588)1071861417
title	Numerical solutions of partial differential equations
title_auth	Numerical solutions of partial differential equations
title_exact_search	Numerical solutions of partial differential equations
title_full	Numerical solutions of partial differential equations Silvia Bertoluzza ...
title_fullStr	Numerical solutions of partial differential equations Silvia Bertoluzza ...
title_full_unstemmed	Numerical solutions of partial differential equations Silvia Bertoluzza ...
title_short	Numerical solutions of partial differential equations
title_sort	numerical solutions of partial differential equations
topic	Partielle Differentialgleichung - Numerisches Verfahren Differential equations, Partial Numerical solutions Congresses Galerkin methods Congresses Wavelets (Mathematics) Congresses Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd
topic_facet	Partielle Differentialgleichung - Numerisches Verfahren Differential equations, Partial Numerical solutions Congresses Galerkin methods Congresses Wavelets (Mathematics) Congresses Numerisches Verfahren Partielle Differentialgleichung Konferenzschrift
url	http://d-nb.info/990037037/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017072256&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT bertoluzzasilvia numericalsolutionsofpartialdifferentialequations

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