Variational integrators in plasma physics:
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| Main Author: | |
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| Format: | Thesis Book |
| Language: | English |
| Published: |
2013
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| Subjects: | |
| Online Access: | Volltext https://nbn-resolving.org/urn:nbn:de:bvb:91-diss-20130712-1140308-0-7 Inhaltsverzeichnis |
| Physical Description: | XI, 206 S. graph. Darst. |
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Record in the Search Index
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| adam_text | IMAGE 1
CONTENTS
V
CONTENTS
ABSTRACT I
ZUSAMMENFASSUNG III
1 INTRODUCTION 1
1.1 GEOMETRIC DISCRETISATION 1
1.2 SYMPLECTICITY 2
1.3 VARIATIONAL INTEGRATORS 2
1.4 OUTLINE AND CONTRIBUTIONS 3
2 GEOMETRIC MECHANICS AND FIELD THEORY 5
2.1 GEOMETRIC FOUNDATIONS 6
2.1.1 SMOOTH MANIFOLDS 6
2.1.2 VECTOR FIELDS 7
2.1.3 INTEGRAL CURVES AND FLOWS 8
2.1.4 FIBRE BUNDLES 9
2.1.5 DIFFERENTIAL FORMS 11
2.1.6 PULLBACK 17
2.1.7 LIE DERIVATIVE 18
2.2 LAGRANGIAN DYNAMICS 19
2.2.1 HAMILTON S ACTION PRINCIPLE 20
2.2.2 DYNAMICS ON THE TANGENT BUNDLE 24
2.2.3 DYNAMICS ON THE JET BUNDLE 28
2.2.4 VARIATIONAL ROUTE TO THE CARTAN FORM 33
2.2.5 PRESERVATION OF THE SYMPLECTIC FORM 41
2.2.6 PRESERVATION OF THE MULTISYMPLECTIC FORM 45
2.2.7 EXTENDED LAGRANGIANS 45
2.3 NOETHER THEOREM 49
2.3.1 POINT TRANSFORMATIONS AND ONE PARAMETER GROUPS 49
2.3.2 NOETHER THEOREM FOR PARTICLE SYSTEMS 49
2.3.3 NOETHER THEOREM FOR FIELD THEORIES 50
2.3.4 NOETHER THEOREM FOR EXTENDED LAGRANGIANS 52
3 VARIATIONAL INTEGRATORS 53
3.1 DISCRETE PARTICLE DYNAMICS 53
3.1.1 DISCRETE ACTION PRINCIPLE 54
3.1.2 DISCRETE TANGENT SPACE 56
3.1.3 DISCRETE ONE- AND TWO-FORM * 58
HTTP://D-NB.INFO/1042589283
IMAGE 2
VI CONTENTS
3.1.4 PRESERVATION OF THE DISCRETE SYMPLECTIC FORM 59
3.1.5 COMPOSITION METHODS 59
3.1.6 DISCRETE NOETHER THEOREM 62
3.2 DISCRETE FIELD THEORY 65
3.2.1 DISCRETE JET SPACE 69
3.2.2 DISCRETE CARTAN FORM 71
3.2.3 DISCRETE MULTISYMPLECTIC FORM - 72
3.2.4 DISCRETE NOETHER THEOREM 73
3.2.5 DISCRETE MOMENTUM MAPS 75
3.3 EXAMPLE: THE ADVECTION EQUATION 76
3.3.1 EXTENDED LAGRANGIAN 76
3.3.2 VARIATIONAL INTEGRATOR 77
3.3.3 CONTINUOUS CONSERVATION LAWS 77
3.3.4 DISCRETE CONSERVATION LAWS 79
4 CHARGED PARTICLE MOTION 81
4.1 GUIDING CENTRE DYNAMICS 81
4.2 VARIATIONAL DISCRETISATION 82
4.2.1 TRAPEZOIDAL DISCRETISATION 83
4.2.2 MIDPOINT DISCRETISATION 85
4.2.3 HIGHER ORDER SCHEMES 86
4.3 PARTICLE MOTION IN THE POLOIDAL PLANE 87
4.3.1 TRAPEZOIDAL DISCRETISATION 88
4.3.2 MIDPOINT DISCRETISATION 89
4.3.3 NUMERICAL RESULTS 90
4.4 PARTICLE MOTION IN THE TOKAMAK 93
4.4.1 TRAPEZOIDAL DISCRETISATION 93
4.4.2 MIDPOINT DISCRETISATION 94
4.4.3 DISCRETE NOETHER THEOREM 95
4.4.4 NUMERICAL RESULTS 95
4.5 VARIATIONAL PIC SCHEME 96
4.5.1 TOTAL LAGRANGIAN AND EULER-LAGRANGE EQUATIONS 101
4.5.2 VARIATIONAL INTEGRATOR 102
4. A CALCULATION OF TRANSIT AND BOUNCE TIMES 103
4.B JACOBIANS 104
4.C DERIVATIVES . 106
5 KINETIC THEORY 109
5.1 . THE VLASOV-POISSON AND VLASOV-MAXWELL SYSTEMS 109
5.1.1 THE VLASOV-MAXWELL SYSTEM 110
5.1.2 THE VLASOV-POISSON SYSTEM 110
5.1.3 CONSERVATION PROPERTIES 112
5.2 REVIEW OF ACTION PRINCIPLES 113
5.2.1 PARAMETRISATION OF THE DISTRIBUTION FUNCTION 113
5.2.2 CONSTRAINED VARIATIONS 115
5.2.3 EULER-POINCARE REDUCTION 116
IMAGE 3
CONTENTS VII
5.2.4 LIE ACTION PRINCIPLES 118
5.3 VARIATIONAL DISCRETISATION 118
5.3.1 EXTENDED LAGRANGIAN 118
5.3.2 VARIATIONAL INTEGRATOR 119
5.3.3 LINEARISED LAGRANGIAN 123
5.4 VELOCITY SPACE COLLISION OPERATOR 124
5.4.1 CONTINUOUS COLLISION OPERATOR 125
5.4.2 DISCRETE COLLISION OPERATOR 127
5.5 NUMERICAL EXAMPLES 128
5.5.1 SIMULATION CODE 128
5.5.2 DIAGNOSTICS 129
5.5.3 LANDAU DAMPING 130
5.5.4 TWOSTREAM INSTABILITY 133
5.5.5 JEANS INSTABILITY 133
6 MAGNETOHYDRODYNAMICS 149
6.1 INCOMPRESSIBLE IDEAL MHD 149
6.1.1 LIE DERIVATIVE FORMULATION 151
6.1.2 POTENTIAL FORMULATION IN TWO DIMENSIONS 152
6.2 VARIATIONAL DISCRETISATION 153
6.2.1 STAGGERED GRID 154
6.2.2 NAVIER-STOKES EQUATION 156
6.2.3 INDUCTION EQUATION 158
6.2.4 VARIATIONAL INTEGRATOR 159
6.3 NUMERICAL EXAMPLES 160
6.3.1 DIAGNOSTICS 160
6.3.2 ALFVEN WAVES 161
6.3.3 LOOP ADVECTION 162
6.3.4 ORSZAG-TANG VORTEX 163
6.3.5 CURRENT SHEATH 163
7 SUMMARY AND OUTLOOK 175
7.1 RESULTS 175
7.1.1 THEORY 175
7.1.2 PARTICLE DYNAMICS 175
7.1.3 KINETIC THEORY 176
7.1.4 FLUID DYNAMICS 177
7.1.5 SEMI-DISCRETISATIONS 177
7.2 FUTURE WORK 177
7.2.1 THEORY 178
7.2.2 VLASOV-POISSON AND VLASOV-MAXWELL 178
7.2.3 IDEAL AND REDUCED MHD 180
A MIXED SPECTRAL-VARIATIONAL SCHEMES 183
IMAGE 4
VIII CONTENTS
A.L THE VORTICITY EQUATION IN 2D 183
A.2 CONSERVATION LAWS 184
A.3 EXTENDED LAGRANGIAN 185
A.4 VARIATIONAL INTEGRATOR 185
A.5 THE VLASOV-POISSON-SYSTEM 186
B DISCRETISATION OF BRACKETS 187
B.L CANONICAL POISSON BRACKETS 187
B.L.L DISCRETE POISSON BRACKETS ON A RECTANGULAR MESH 188
B.1.2 ARAKAWA S DISCRETISATION 189
B.1.3 DISCRETE POISSON BRACKETS ON A TRIANGULAR MESH 190
B.2 NAMBU THREE BRACKETS 192
B.2.1 DISCRETE NAMBU BRACKETS 193
B.2.2 APPLICATION TO GYROKINETICS 193
B.3 LIE-POISSON AND NAMBU FIELD BRACKETS 194
B.3.1 NONCANONICAL HAMILTONIAN FIELD THEORY . . 194
B.3.2 LIE-POISSON AND NAMBU BRACKETS IN THE VLASOV EQUATION 194 B.3.3
DISCRETISATION OF NAMBU FIELD BRACKETS 195
BIBLIOGRAPHY 197
|
| any_adam_object | 1 |
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| author_facet | Kraus, Michael |
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| dewey-ones | 510 - Mathematics 530 - Physics |
| dewey-raw | 510 530 |
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| discipline | Physik Mathematik |
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| spelling | Kraus, Michael Verfasser aut Variational integrators in plasma physics Michael Kraus 2013 XI, 206 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier München, Techn. Univ., Diss., 2013 Plasmaphysik (DE-588)4046259-6 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Impulsübertragung (DE-588)4161448-3 gnd rswk-swf Transportprozess (DE-588)4185932-7 gnd rswk-swf Erhaltungssatz (DE-588)4131214-4 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Plasmaphysik (DE-588)4046259-6 s Impulsübertragung (DE-588)4161448-3 s Transportprozess (DE-588)4185932-7 s Erhaltungssatz (DE-588)4131214-4 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Erscheint auch als Online-Ausgabe urn:nbn:de:bvb:91-diss-20130712-1140308-0-7 http://mediatum.ub.tum.de/node?id=1140308 Verlag kostenfrei Volltext https://nbn-resolving.org/urn:nbn:de:bvb:91-diss-20130712-1140308-0-7 Resolving-System DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026175200&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kraus, Michael Variational integrators in plasma physics Plasmaphysik (DE-588)4046259-6 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Impulsübertragung (DE-588)4161448-3 gnd Transportprozess (DE-588)4185932-7 gnd Erhaltungssatz (DE-588)4131214-4 gnd |
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| title | Variational integrators in plasma physics |
| title_auth | Variational integrators in plasma physics |
| title_exact_search | Variational integrators in plasma physics |
| title_full | Variational integrators in plasma physics Michael Kraus |
| title_fullStr | Variational integrators in plasma physics Michael Kraus |
| title_full_unstemmed | Variational integrators in plasma physics Michael Kraus |
| title_short | Variational integrators in plasma physics |
| title_sort | variational integrators in plasma physics |
| topic | Plasmaphysik (DE-588)4046259-6 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Impulsübertragung (DE-588)4161448-3 gnd Transportprozess (DE-588)4185932-7 gnd Erhaltungssatz (DE-588)4131214-4 gnd |
| topic_facet | Plasmaphysik Numerisches Verfahren Impulsübertragung Transportprozess Erhaltungssatz Hochschulschrift |
| url | http://mediatum.ub.tum.de/node?id=1140308 https://nbn-resolving.org/urn:nbn:de:bvb:91-diss-20130712-1140308-0-7 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026175200&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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