Conservative finite difference methods on general grids:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
1996
|
| Series: | Symbolic and numeric computation series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | WST: Conservative finite-difference methods on general grids |
| Physical Description: | 359 S. graph. Darst. Diskette (3,5") |
| ISBN: | 0849373751 |
Staff View
MARC
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| 100 | 1 | |a Šaškov, Michail J. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Conservative finite difference methods on general grids |c Mikhail Shashkov |
| 246 | 1 | 3 | |a Conservative finite-difference methods on general grids |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 1996 | |
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Record in the Search Index
| _version_ | 1804125498912014336 |
|---|---|
| adam_text | Contents
1 Introduction 1
1 Governing Equations 1
1.1 Elliptic Equations 1
1.2 Heat Equations 3
1.3 Equations of Gas Dynamics in Lagrangian Form . . 4
2 The Main Ideas of Finite Difference Algorithms 5
2.1 1 D Case 6
2.1.1 Grid Functions 7
2.1.2 Smooth and Non Smooth Grids 8
2.1.3 Different Types of Grid Functions 9
2.1.4 Finite Difference Operators,
Truncation Error 13
2.1.5 Two Approaches for the Investigation
of Truncation Errors 16
2.1.6 Approximation of Second Order
Derivatives 18
2.1.7 Finite Difference Schemes for the
1 D Poison Equation 19
2.1.8 Finite Difference Operators and the
Related Matrix 23
2.1.9 Finite Difference Schemes and Related
Matrix Problems 24
2.2 2 D case 26
2.2.1 Grid in 2 D 26
2.2.2 Grid Functions and Difference
Operators in 2 D 32
2.3 Methods for the Solution of the System of
Linear Algebraic Equations 37
2.3.1 Application of the Gauss Seidel Method
for a Model Problem 40
2.4 Methods for the Solution of the System of
Non Linear Equations 41
2 Method of Support Operators 47
1 Main Stages 48
2 The Elliptic Equations 49
3 Gas Dynamic Equations 51
4 System of Consistent Difference Operators in 1 D 53
4.1 Inner Product in Spaces of Discrete Functions
and Properties of Difference Operators 56
5 System of Consistent Discrete Operators in 2 D 57
3 The Elliptic Equations 65
1 Introduction 65
1.1 Continuum Elliptic Problems. Dirichlet
Boundary Conditions 65
1.1.1 The Properties of the Differential
Operator 66
1.1.2 The Properties of First Order Operators
and the Operator of the BVP 67
1.2 Continuum Elliptic Problems. Robin Boundary
Conditions 68
1.2.1 The Properties of the Differential
Operator 68
1.2.2 The Properties of First Order Operators
and the Operator of the BVP 69
2 One Dimensional Support Operators Algorithms 70
2.1 Nodal Discretisation of Scalar Functions and
Cell Centered Discretisation of Vector Functions . . 71
2.1.1 Spaces of Discrete Functions 72
2.1.2 The Prime Operator 73
2.1.3 The Derived Operator 73
2.1.4 Multiplication by a Matrix K 74
2.1.5 The Finite Difference Scheme for the
Dirichlet Boundary Value Problem 75
2.1.6 The Matrix Problem 77
2.1.7 The Finite Difference Scheme for the
Robin Boundary Problem 77
2.1.8 The Matrix Problem for Robin
Boundary Conditions 78
2.1.9 Approximation Properties 79
2.1.10 The Numerical Solution of Test
Problems 84
2.2 Cell Valued Discretisation of Scalar Functions
and Nodal Discretisation of Vector Functions .... 90
2.2.1 Spaces of Discrete Functions 94
2.2.2 The Prime Operator 95
2.2.3 The Derived Operator 96
2.2.4 Multiplication by a Matrix and the
Operator V 96
2.2.5 The Finite Difference Scheme for
Dirichlet Boundary Conditions 97
2.2.6 The Matrix Problem 98
2.2.7 The Finite Difference Scheme for
Robin Boundary Conditions 98
2.2.8 The Matrix Problem 99
2.2.9 Approximation Properties 99
2.3 Numerical Solution for Test Problems 103
3 Two Dimensional Support Operators Algorithms 108
3.1 Nodal Discretisation of Scalar Functions and
Cell Valued Discretisation of Vector Functions . . . 108
3.1.1 The Finite Difference Scheme for the
Dirichlet Boundary Problem 121
3.1.2 The Finite Difference Scheme for the
Robin Boundary Problem 123
3.2 Cell Valued Discretisation of Scalar Functions
and Nodal Discretisation of Vector Functions .... 127
3.2.1 The Finite Difference Scheme for the
Dirichlet Boundary Problem 134
3.2.2 The Finite Difference Scheme for the
Robin Boundary Problem 136
3.3 The Numerical Solution of Test Problems 140
3.3.1 The Dirichlet Boundary Value Problem . . 140
3.3.2 The Robin Boundary Value Problem .... 146
4 Conclusion 147
4 The Heat Equation 149
1 Introduction 149
1.1 The Conservation Law 149
1.2 Time Discretisation 150
1.3 Explicit and Implicit Finite Difference Schemes . . . 151
1.4 Stability of the Finite Difference Scheme for the
Heat Equation 154
1.5 The Positivity Preserving Methods 159
1.6 The Method of Lines 160
2 Finite Difference Schemes for the Heat Equation in 1 D . . 161
2.1 Introduction 161
2.2 Nodal Discretisation for Scalar Functions and
Cell Valued Discretisation for Vector Functions . . . 162
2.2.1 Stability of the Finite Difference Scheme and
Results of the Solution for Test Problems . 165
2.3 Cell Valued Discretisation of Scalar Functions and
Nodal Discretisation of Vector Functions 173
2.4 Conservation Laws and
the Iteration Process 176
3 Finite Difference Schemes for the Heat Equation in 2 D . . 180
3.1 Stability Conditions in 2 D 180
3.2 Symmetry Preserving Finite Difference Schemes
and Iteration Methods 181
3.3 Numerical Examples 186
5 Lagrangian Gas Dynamics 193
1 Fundamentals of Gas Dynamics 195
1.1 The Integral Form of Gas Dynamics Equations ... 201
1.2 Integral Equations for the One Dimensional Case . . 202
1.3 Differential Equations of Gas Dynamics in
Lagrangian Form 206
1.4 The Differential Equations in 1 D Lagrangian
Mass Coordinates 207
1.5 Statements for Gas Dynamics Problems in
Lagrange Variables 209
1.6 Different Forms of Energy Equations 210
1.7 Acoustic Equations 212
1.8 Reference Information 214
1.9 The Characteristic Form of Gas Dynamics
Equations 216
1.10 Riemann s Invariants 217
1.11 Discontinuous Solutions 220
1.11.1 Contact Discontinuity 223
1.11.2 Shock Waves. The Hugoniot Adiabat ... 224
1.11.3 Shock Waves in an Ideal Gas 224
1.11.4 Zemplen Theorem 227
1.11.5 Approximation of the Strong Wave . . . . 228
1.11.6 Structure of the Shock Front 229
2 Conservation Laws and Properties of Operators 233
3 Finite Difference Algorithms in 1 D 237
3.1 Discretisation in 1 D 237
3.2 Discrete Operators in 1 D 238
3.2.1 The Prime Operator 238
3.2.2 The Derived Operator 239
3.2.3 Boundary Conditions and
Discretisations 239
3.3 Semi Discrete Finite Difference Schemes in 1 D . . . 240
3.4 Fully Discrete, Explicit, Computational
Algorithms 242
3.5 Fully Discrete, Implicit, Computational
Algorithm 248
3.6 Stability Conditions 254
3.6.1 General Remarks 254
3.6.2 One Dimensional Transport Equations . . . 255
3.6.3 Finite Difference Schemes for the
One Dimensional Transport Equation . . . 255
3.6.4 Stability Conditions for 1 D Acoustic
Equations 259
3.7 Homogeneous Finite Difference Schemes.
Artificial Viscosity 260
3.8 Artificial Viscosity in 1 D 263
3.9 The Numerical Example 265
4 The Finite Difference Algorithm in 2 D 268
4.1 Discretisation in 2 D 268
4.2 Discrete Operators in 2 D 269
4.2.1 The Prime Operator 269
4.2.2 The Derived Operator 271
4.2.3 Boundary Conditions and
Discretisations 272
4.3 The Semi Discrete Finite Difference Scheme
in 2 D 272
4.4 The Finite Difference Algorithm in 2 D 276
4.4.1 The Fully Discrete, Explicit,
Computational Algorithm 276
4.5 Implicit Finite Difference Scheme 284
4.5.1 The Method of Parallel Chords 292
4.5.2 Boundary Conditions 293
4.5.3 Properties of the System of Linear
Equations 298
4.5.4 Some Properties of Algorithm 301
4.6 Artificial Viscosity in 2 D 302
4.7 The Numerical Example 304
6 Conclusion 311
A Fortran Code Directory 313
1 The Structure of Directories on the Disk 313
2 Programs for Elliptic Equations 315
2.1 Programs for 1 D Equations 316
2.1.1 Dirichlet Boundary Conditions 316
2.1.2 Robin Boundary Conditions 318
2.2 Programs for 2 D Equations 319
2.2.1 Dirichlet Boundary Conditions 319
2.2.2 Robin Boundary Conditions 322
3 Programs for Heat Equations 325
3.1 Programs for 1 D Equations 325
3.1.1 Dirichlet Boundary Conditions 325
3.1.2 Robin Boundary Conditions 327
3.2 Programs for 2 D Equations 330
3.2.1 Dirichlet Boundary Conditions 330
3.2.2 Robin Boundary Conditions 332
4 Programs for Gas Dynamics Equations 336
4.1 Programs for 1 D Equations 336
4.1.1 Explicit Finite Difference Scheme 336
4.1.2 Implicit Finite Difference Scheme 336
4.2 Programs for 2 D Equations 337
4.2.1 Explicit Finite Difference Scheme 337
4.2.2 Implicit Finite Difference Scheme 338
Bibliography 341
Index 355
|
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| id | DE-604.BV011007423 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:02:31Z |
| institution | BVB |
| isbn | 0849373751 |
| language | English |
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| physical | 359 S. graph. Darst. Diskette (3,5") |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Symbolic and numeric computation series |
| spelling | Šaškov, Michail J. Verfasser aut Conservative finite difference methods on general grids Mikhail Shashkov Conservative finite-difference methods on general grids Boca Raton [u.a.] CRC Press 1996 359 S. graph. Darst. Diskette (3,5") txt rdacontent n rdamedia nc rdacarrier Symbolic and numeric computation series WST: Conservative finite-difference methods on general grids Gitter Mathematik (DE-588)4157375-4 gnd rswk-swf Finite-Differenzen-Methode (DE-588)4194626-1 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Finite-Differenzen-Methode (DE-588)4194626-1 s DE-604 Gitter Mathematik (DE-588)4157375-4 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007369977&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Šaškov, Michail J. Conservative finite difference methods on general grids Gitter Mathematik (DE-588)4157375-4 gnd Finite-Differenzen-Methode (DE-588)4194626-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4157375-4 (DE-588)4194626-1 (DE-588)4044779-0 |
| title | Conservative finite difference methods on general grids |
| title_alt | Conservative finite-difference methods on general grids |
| title_auth | Conservative finite difference methods on general grids |
| title_exact_search | Conservative finite difference methods on general grids |
| title_full | Conservative finite difference methods on general grids Mikhail Shashkov |
| title_fullStr | Conservative finite difference methods on general grids Mikhail Shashkov |
| title_full_unstemmed | Conservative finite difference methods on general grids Mikhail Shashkov |
| title_short | Conservative finite difference methods on general grids |
| title_sort | conservative finite difference methods on general grids |
| topic | Gitter Mathematik (DE-588)4157375-4 gnd Finite-Differenzen-Methode (DE-588)4194626-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Gitter Mathematik Finite-Differenzen-Methode Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007369977&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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