Computational partial differential equations: numerical methods and diffpack programming
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2003
|
| Edition: | 2. ed. |
| Series: | Texts in computational science and engineering
1 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVI, 855 S. graph. Darst. |
| ISBN: | 354043416X |
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| 100 | 1 | |a Langtangen, Hans Petter |d 1962- |e Verfasser |0 (DE-588)1019109599 |4 aut | |
| 245 | 1 | 0 | |a Computational partial differential equations |b numerical methods and diffpack programming |c Hans Petter Langtangen |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
| 300 | |a XXVI, 855 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Texts in computational science and engineering |v 1 | |
| 630 | 0 | 4 | |a Diffpack (Computer file) |
| 650 | 4 | |a Partielle Differentialgleichung - Numerisches Verfahren - C++ - Programmierumgebung | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Differential equations, Partial -- Numerical solutions -- Data processing | |
| 650 | 0 | 7 | |a C++ |0 (DE-588)4193909-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Programmierumgebung |0 (DE-588)4134837-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | 1 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
| 689 | 0 | 2 | |a C++ |0 (DE-588)4193909-8 |D s |
| 689 | 0 | 3 | |a Programmierumgebung |0 (DE-588)4134837-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a Texts in computational science and engineering |v 1 |w (DE-604)BV016971315 |9 1 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010006507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-010006507 | ||
Record in the Search Index
| _version_ | 1804129562672496640 |
|---|---|
| adam_text | Table
of
Contents
1
Getting Started
......................................... 1
1.1
The First Diffpack Encounter
.............................. 3
1.1.1
What is Diffpack?
................................. 3
1.1.2
A Simple
C++
Program
............................ 5
1.1.3
A Simple Diffpack Program
......................... 6
1.2
Overview of Application Examples
.......................... 7
1.2.1
Very Simple Introductory Program Examples
.......... 8
1.2.2
Finite Difference Simulators
......................... 9
1.2.3
Finite Element Simulators
.......................... 10
1.2.4
More Advanced Applications
........................ 11
1.3
Steady One-Dimensional Heat Conduction
................... 12
1.3.1
The Physical and Mathematical Model
............... 12
1.3.2
A Finite Difference Method
......................... 15
1.3.3
Implementation in Diffpack
......................... 18
1.3.4
Dissection of the Program
.......................... 22
1.3.5
Tridiagonal Matrices
............................... 23
1.3.6
Variable Coefficients
............................... 24
1.3.7
A Nonlinear Heat Conduction Problem
............... 28
1.4
Simulation of Waves
...................................... 32
1.4.1
Modeling Vibrations of a String
..................... 33
1.4.2
A Finite Difference Method
......................... 35
1.4.3
Implementation
.................................... 38
1.4.4
Visualizing the Results
............................. 41
1.4.5
Automating Simulation and Visualization in Scripts
.... 45
1.4.6
A 2D Wave Equation with Variable Wave Velocity
..... 52
1.4.7
A Model for Water Waves
.......................... 56
1.5
Projects
................................................. 62
1.5.1
A Uni-Directional Wave Equation
.................... 62
1.5.2
Centered Differences for a Boundary-Layer Problem
.... 63
1.5.3
Upwind Differences for a Boundary-Layer Problem
..... 64
1.6
About Programming with Objects
.......................... 65
1.6.1
Motivation for the Object Concept
................... 67
1.6.2
Example: Implementation of a Vector Class in
C++.... 73
1.6.3
Arrays in Diffpack
................................. 81
1.6.4
Example: Design of an ODE Solver Environment
...... 87
1.6.5
Abstractions for Grids and Fields
.................... 98
1.7
Coding the PDE Simulator as a Class
.......................101
1.7.1
Steady ID Heat Conduction Revisited
................102
1.7.2
Nonlinear ID Heat Conduction Revisited
.............107
1.7.3
Empirical Investigation of a Numerical Method
........109
XVI Table of
Contents
1.7.4 Simulation
of ID Waves Revisited
...................114
1.7.5
Simulation of 2D Waves Revisited
...................119
1.7.6
Transient Heat Conduction
.........................121
1.8
Projects
.................................................131
1.8.1
Transient Flow Between Moving Plates
...............131
1.8.2
Transient Channel Flow
............................133
1.8.3
Coupled Heat and Fluid Flow
.......................134
1.8.4
Difference Schemes for Transport Equations
...........136
1.8.5 3D
Sound Waves
..................................138
2
Introduction to Finite Element Discretization
......141
2.1
Weighted Residual Methods
...............................142
2.1.1
Basic Principles
................................... 142
2.1.2
Example: A ID
Poisson
Equation
.................... 144
2.1.3
Treatment of Boundary Conditions
................... 148
2.2
Time Dependent Problems
................................ 152
2.2.1
A Wave Equation
..................................153
2.2.2
A Heat Equation
..................................155
2.3
Finite Elements in One Space Dimension
....................157
2.3.1
Piecewise Polynomials
..............................157
2.3.2
Handling of Essential Boundary Conditions
...........160
2.3.3
Direct Computation of the Linear System
.............161
2.3.4
Element-by-Element Formulation
....................163
2.3.5
Extending the Concepts to Quadratic Elements
........171
2.3.6
Summary of the Element-by-Element Algorithm
.......174
2.4
Example: A ID Wave Equation
............................176
2.4.1
The Finite Element Equations
.......................176
2.4.2
Interpretation of the Discrete Equations
..............177
2.4.3
Accuracy and Stability
.............................179
2.5
Naive Implementation
.....................................183
2.6
Projects
.................................................189
2.6.1
Steady Heat Conduction with Cooling Law
...........189
2.6.2
Stationary Pipe Flow
...............................190
2.6.3
Transient Pipe Flow
................................192
2.6.4
Retardation of a Well-Bore
..........................193
2.7
Higher-Dimensional Finite Elements
........................195
2.7.1
The Bilinear Element and Generalizations
............196
2.7.2
The Linear Triangle
................................198
2.7.3
Example: A 2D Wave Equation
......................200
2.7.4
Other Two-Dimensional Element Types
..............203
2.7.5
Three-Dimensional Elements
........................204
2.8
Calculation of Derivatives
.................................206
2.8.1
Global Least-Squares Smoothing
.....................207
2.8.2
Flux Computations in Heterogeneous Media
...........208
2.9
Convection-Diffusion Equations
............................212
2.9.1
A One-Dimensional Model Problem
..................214
Table
of Contents
XVII
2.9.2
Multi-Dimensional Equations
........................216
2.9.3
Time-Dependent Problems
..........................218
2.10
Analysis of the Finite Element Method
......................220
2.10.1
Weak Formulations
................................221
2.10.2
Variational Problems
...............................223
2.10.3
Results for Continuous Problems
....................226
2.10.4
Results for Discrete Problems
.......................232
2.10.5
A Priori Error Estimates
...........................238
2.10.6
Numerical Experiments
.............................241
2.10.7
Adaptive Finite Element Methods
...................243
3
Programming of Finite Element Solvers
.............251
3.1
A Simple Program for the
Poisson
Equation
.................252
3.1.1
Discretization
.....................................252
3.1.2
Basic Parts of a Simulator Class
.....................253
3.2
Increasing the Flexibility
..................................261
3.2.1
A Generalized Model Problem
.......................262
3.2.2
Using the Menu System
............................262
3.2.3
Creating the Grid Object
...........................266
3.3
Some Visualization Tools
..................................268
3.3.1
Storing Fields for Later Visualization
.................269
3.3.2
Filtering Simres Data
..............................269
3.3.3
Visualizing Diffpack Data in Plotmtv
.................272
3.3.4
Visualizing Diffpack Data in Gnuplot
.................276
3.3.5
Visualizing Diffpack Data in
Matlab
..................277
3.3.6
Visualizing Diffpack Data in Vtk
....................277
3.3.7
Visualizing Diffpack Data in IRIS Explorer
............279
3.3.8
Plotting Fields along Lines
..........................280
3.4
Some Useful Diffpack Features
.............................281
3.4.1
The Menu System
.................................281
3.4.2
Multiple Loops
....................................285
3.4.3
Computing Numerical Errors
........................286
3.4.4
Functors
..........................................287
3.4.5
Computing Derivatives of Finite Element Fields
.......289
3.4.6
Specializing Code in Subclass Solvers
.................292
3.5
Introducing More Flexibility
...............................295
3.5.1
Setting Boundary Condition Information in the Grid
.. . 296
3.5.2
Line and Surface Integrals
..........................302
3.5.3
Simple Mesh Generation Tools
......................304
3.5.4
Grid Generation by Super Elements
..................311
3.5.5
Debugging
........................................316
3.5.6
Automatic Report Generation
.......................319
3.5.7
Specializing Code in Subclass Solvers
.................321
3.5.8
Overriding Menu Answers in the Program
............326
3.5.9
Estimating Convergence Rates
......................329
3.5.10
Axisymmetric Formulations and Cartesian 2D Code
.... 332
XVIII Table of
Contents
3.5.11
Summary.........................................
334
3.6
Step-by-Step
Development of a Diffpack Solver
...............335
3.6.1
Physical and Mathematical Problem
................. 335
3.6.2
Editing and Writing Source Code
.................... 337
3.6.3
A Simplified Test Case
............................. 345
3.6.4
Creating the Grid
.................................. 350
3.6.5
Running Some Initial 2D Simulations
................. 354
3.6.6
Running Real Simulations
.......................... 360
3.7
Adaptive Grids
.......................................... 362
3.7.1
Grid Classes with Local Mesh Refinements
............364
3.7.2
How to Extend an Existing Simulator
................365
3.7.3
Organization of Refinement Criteria
..................367
3.7.4
Grid Refinements as a Preprocessor
..................372
3.7.5
Example: Corner-Flow Singularity
...................374
3.7.6
User-Defined Refinement Criteria
....................376
3.7.7
Transient Problems
................................376
3.8
Projects
.................................................380
3.8.1
Flow in an Open Inclined Channel
...................380
3.8.2
Stress Concentration due to Geometric Imperfections
.. . 382
3.8.3
A Poisson
Problem with Pure Neumann Conditions
.... 383
3.8.4
Lifting Airfoil
.....................................385
3.9
A Convection-Diffusion Solver
..............................387
3.10
A Heat Equation Solver
...................................391
3.10.1
Discretization
.....................................391
3.10.2
Implementation
....................................392
3.11
A More Flexible Heat Equation Solver
......................398
3.11.1
About the Model Problem and the Simulator
..........399
3.11.2
Variable Time Step Size
............................400
3.11.3
Applying a Transient Solver to a Stationary PDE
......401
3.11.4
Thermal Conditions During Welding
.................401
3.12
Visualization of Time-Dependent Fields
.....................402
3.12.1
Filtering Time-Dependent Simres Data
...............402
3.12.2
Storing Fields at Selected Time Points
...............403
3.12.3
Time Series at Selected Spatial Points
................404
3.12.4
Using ImageMagick Tools
...........................405
3.12.5
Animation Using Plotmtv
...........................407
3.12.6
Animation Using Vtk
..............................408
3.12.7
Animation Using
Matlab
...........................409
3.12.8
Real-Time Visualization
............................409
3.12.9
Handling Simulation and Visualization from a Script
... 411
3.12.10
Heat Transfer Exercises
.............................415
3.13
A Transient Heat Transfer Application
......................417
3.13.1
The Mathematical and Physical Model
...............417
3.13.2
Implementation
....................................419
3.13.3
Testing and Debugging the Initial State
..............425
Table
of Contents
XIX
3.13.4
Creating the Grid
..................................426
3.13.5
Running Time-Dependent Simulations
................428
3.13.6
A Scripting Interface for Automating Simulations
......433
3.14
Projects
.................................................443
3.14.1
Transient Heat Transfer in a Two-Material Structure
.. 443
3.14.2
Transient Flow with Non-Circular Cross Section
......444
3.14.3
Transient Groundwater Flow
.......................446
3.15
Efficient Solution of the Wave Equation
.....................447
3.15.1
Discretization
.....................................447
3.15.2
Implementation
....................................448
3.15.3
Extensions of the Model Problem
....................452
3.15.4
Flexible Representation of Variable Coefficients
........453
4
Nonlinear Problems
....................................459
4.1
Discretization and Solution of Nonlinear PDEs
...............459
4.1.1
Finite Difference Discretization
......................459
4.1.2
Finite Element Discretization
.......................460
4.1.3
The Group Finite Element Method
..................462
4.1.4
Successive Substitutions
............................464
4.1.5
Newton-Raphson s Method
..........................464
4.1.6
A Transient Nonlinear Heat Conduction Problem
......470
4.1.7
Iteration Methods at the PDE Level
.................473
4.1.8
Continuation Methods
..............................476
4.2
Software Tools for Nonlinear Finite Element Problems
........477
4.2.1
A Solver for a Nonlinear Heat Equation
..............477
4.2.2
Extending the Solver
...............................483
4.3
Projects
.................................................487
4.3.1
Operator Splitting for a Reaction-Diffusion Model
.....487
4.3.2
Compressible Potential Flow
........................489
5
Solid Mechanics Applications
.........................493
5.1
Linear Thermo-Elasticity
..................................494
5.1.1
The Physical and Mathematical Model
...............494
5.1.2
A Finite Element Method
...........................497
5.1.3
Engineering Finite Element Notation
.................499
5.1.4
Implementation
....................................502
5.1.5
Examples
.........................................510
5.1.6
Elastic Vibrations
..................................518
5.2
Elasto-Viscoplasticity
.....................................522
5.2.1
Basic Physical Features of Elasto-Viscoplasticity
.......522
5.2.2
A Three-Dimensional Elasto-Viscoplastic Model
.......525
5.2.3
Simplification; a Forward Scheme in Time
............529
5.2.4
Numerical Handling of Yield Criteria
.................530
5.2.5
Implementation
....................................533
5.2.6
Examples
.........................................534
XX
Table
of Contents
6
Fluid Mechanics Applications
........................539
6.1
Convection-Diffusion Equations
............................539
6.1.1
The Physical and Mathematical Model
...............539
6.1.2
A Finite Element Method
...........................541
6.1.3
Incorporation of Nonlinearities
......................541
6.1.4
Software Tools
....................................542
6.1.5
Melting and Solidification
...........................544
6.2
Shallow Water Equations
..................................546
6.2.1
The Physical and Mathematical Model
...............547
6.2.2
Finite Difference Methods on Staggered Grids
.........549
6.2.3
Implementation
....................................552
6.2.4
Nonlinear and Dispersive Terms
.....................556
6.2.5
Finite Element Methods
............................558
6.3
An Implicit Finite Element Navier-Stokes Solver
.............562
6.3.1
The Physical and Mathematical Model
...............562
6.3.2
A Finite Element Method
...........................564
6.3.3
Solution of the Nonlinear Systems
...................566
6.3.4
Implementation
....................................568
6.4
A Classical Finite Difference Navier-Stokes Solver
............576
6.4.1
Operator Splitting
.................................576
6.4.2
Finite Differences on
3D
Staggered Grids
.............578
6.4.3
A Multigrid Solver for the Pressure Equation
..........582
6.4.4
Implementation
....................................582
6.5
A Fast Finite Element Navier-Stokes Solver
..................586
6.5.1
Operator Splitting and Finite Element Discretization
... 586
6.5.2
An Optimized Implementation
......................589
6.6
Projects
.................................................591
6.6.1
Analysis of Discrete Shallow Water Waves
............591
6.6.2
Approximating the Navier-Stokes Equations by a Laplace
Equation
.........................................593
7
Coupled Problems
......................................595
7.1
Fluid-Structure Interaction; Squeeze-Film Damping
...........595
7.1.1
The Physical and Mathematical Model
...............595
7.1.2
Numerical Methods
................................601
7.1.3
Implementation
....................................603
7.2
Fluid Flow and Heat Conduction in Pipes
...................611
7.2.1
The Physical and Mathematical Model
...............611
7.2.2
Numerical Methods
................................615
7.2.3
Implementation
....................................619
7.3
Projects
.................................................626
7.3.1
Transient Spherical-Symmetric Thermo-Elasticity
......626
7.3.2
Transient 2D/3D Thermo-Elasticity
..................629
7.3.3
Convective-Diffusive Transport in Viscous Flow
........630
7.3.4
Chemically Reacting Fluid
..........................631
Table
of Contents
XXI
A Mathematical Topics
...................................633
A.I Scaling and Dimensionless Variables
........................633
A.2
Indiciai
Notation
.........................................641
A.3 Compact Notation for Difference Equations
..................643
A.
4
Stability and Accuracy of Difference Approximations
..........646
A.
4.1
Typical Solutions of Simple Prototype PDEs
..........646
A.4.2 Physical Significance of Parameters in the Solution
.....649
A.
4.3
Analytical Dispersion Relations
......................650
A.
4.4
Solution of Discrete Equations
.......................653
A.4.5 Numerical Dispersion Relations
......................656
A.
4.6
Convergence
......................................657
A.4.7 Stability
..........................................658
A.4.8 Accuracy
.........................................659
A.4.9 Truncation Error
..................................662
A.4.10 Traditional
von
Neumann Stability Analysis
...........665
A.4.11 Examples: Analysis of the Heat Equation
.............666
A.5 Exploring the Nature of Some PDEs
........................670
A.5.1 A Hyperbolic Equation
.............................670
A.5.2 An Elliptic Equation
...............................674
A.5.3 A Parabolic Equation
..............................674
A.5.4 The Laplace Equation Solved by a Wave Simulator
.....677
A.5.5 Well-Posed Problems
...............................680
В
Diffpack Topics
.........................................681
B.I Brief Overview of Important Diffpack Classes
................681
B.2 Diffpack-Related Operating System Interaction
...............686
B.2.1 Unix
.............................................686
B.2.2 Windows
.........................................689
B.3 Combining Diffpack with Other Types of Software
............695
B.3.1 Calling Other Software Packages from Diffpack
........695
B.3.
2
Calling Diffpack from Other Types of Software
........699
B.4 Basic Diffpack Features
...................................700
B.4.1 Diffpack Man Pages
................................700
B.4.2 Standard Command-Line Options
....................701
B.4.3 Generalized Input and Output
.......................703
B.4.4 Automatic Verification of a Code
....................707
B.5 Visualization Support
.....................................708
B.5.1 Curves
...........................................709
B.5.2 Scalar and Vector Fields
............................714
B.6 Details on Finite Element Programming
.....................717
B.6.1 Basic Functions for Finite Element Assembly
..........717
B.6.2 Using Functors for the Integrands
....................720
B.6.3 Integrating Quantities over the Grid or the Boundary
.. 721
B.6.4 Class Relations in the Finite Element Engine
..........724
B.7 Optimizing Diffpack Codes
................................725
B.7.1 Avoiding Repeated Matrix Factorizations
.............727
XXII Table of
Contents
В.
7.2
Avoiding Repeated Assembly of Linear Systems
.......730
B.7.3 Optimizing the Assembly Process
....................739
B.7.4 Optimizing Array Indexing
..........................744
С
Iterative Methods for Sparse Linear Systems
......751
C.I Classical Iterative Methods
................................752
C.I.I A General Framework
..............................753
C.1.2 Jacobi, Gauss-Seidel,
SOR,
and SSOR Iteration
.......754
C.2 Conjugate Gradient-Like Iterative Methods
..................760
C.2.1 Galerkin and Least-Squares Methods
.................761
C.2.
2
Summary of the Algorithms
.........................765
C.2.3 A Framework Based on the Error
....................767
C.3 Preconditioning
..........................................768
C.3.1 Motivation and Basic Principles
.....................768
C.3.
2
Classical Iterative Methods as Preconditioners
.........770
C.3.3 Incomplete Factorization Preconditioners
.............771
C.4 Multigrid and Domain Decomposition Methods
..............773
C.4.1 Domain Decomposition
.............................774
C.4.2 Multigrid Methods
.................................775
D
Software Tools for Solving Linear Systems
..........781
D.I Storing and Initializing Linear Systems
......................781
D.I.I Vector and Matrix Formats
.........................782
D.I.2
Detailed Matrix Examples
..........................788
D.1.3 Representation of Linear Systems
....................794
D.2 Programming with Linear Solvers
..........................797
D.2.1 Gaussian Elimination
..............................797
D.2.2 A Simple Demo Program
...........................798
D.2.3 A
3D
Poisson
Equation Solver
.......................800
D.3 Classical Iterative Methods
................................804
D.4 Conjugate Gradient-like Methods
...........................805
D.4.1 Symmetric Systems
................................805
D.4.
2
Nonsymmetric Systems
.............................806
D.5 Preconditioning Strategies
.................................807
D.6 Convergence History and Stopping Criteria
..................811
D.7 Example: Implicit Methods for Transient Diffusion
............821
D.8 High-Level Stencil Programming of Finite Difference Schemes
. . 823
D.8.1 Finite Difference Stencils
...........................823
D.8.
2
Basic Structure of a Stencil-Based Simulator
..........825
D.8.3 Defining the Stencils
...............................828
Bibliography
...............................................833
Index
........................................................843
|
| any_adam_object | 1 |
| author | Langtangen, Hans Petter 1962- |
| author_GND | (DE-588)1019109599 |
| author_facet | Langtangen, Hans Petter 1962- |
| author_role | aut |
| author_sort | Langtangen, Hans Petter 1962- |
| author_variant | h p l hp hpl |
| building | Verbundindex |
| bvnumber | BV014783472 |
| callnumber-first | Q - Science |
| callnumber-label | QA377 |
| callnumber-raw | QA377.L355 2003 |
| callnumber-search | QA377.L355 2003 |
| callnumber-sort | QA 3377 L355 42003 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 540 SK 920 |
| ctrlnum | (OCoLC)248651870 (DE-599)BVBBV014783472 |
| dewey-full | 515/.35321 515.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 21 515.353 |
| dewey-search | 515/.353 21 515.353 |
| dewey-sort | 3515 3353 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV014783472 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:07:07Z |
| institution | BVB |
| isbn | 354043416X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010006507 |
| oclc_num | 248651870 |
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| owner_facet | DE-703 DE-29T DE-355 DE-BY-UBR DE-1051 DE-521 DE-384 DE-523 DE-20 DE-83 DE-11 DE-B768 |
| physical | XXVI, 855 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Texts in computational science and engineering |
| series2 | Texts in computational science and engineering |
| spelling | Langtangen, Hans Petter 1962- Verfasser (DE-588)1019109599 aut Computational partial differential equations numerical methods and diffpack programming Hans Petter Langtangen 2. ed. Berlin [u.a.] Springer 2003 XXVI, 855 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in computational science and engineering 1 Diffpack (Computer file) Partielle Differentialgleichung - Numerisches Verfahren - C++ - Programmierumgebung Datenverarbeitung Differential equations, Partial -- Numerical solutions -- Data processing C++ (DE-588)4193909-8 gnd rswk-swf Programmierumgebung (DE-588)4134837-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s C++ (DE-588)4193909-8 s Programmierumgebung (DE-588)4134837-0 s DE-604 Texts in computational science and engineering 1 (DE-604)BV016971315 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010006507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Langtangen, Hans Petter 1962- Computational partial differential equations numerical methods and diffpack programming Texts in computational science and engineering Diffpack (Computer file) Partielle Differentialgleichung - Numerisches Verfahren - C++ - Programmierumgebung Datenverarbeitung Differential equations, Partial -- Numerical solutions -- Data processing C++ (DE-588)4193909-8 gnd Programmierumgebung (DE-588)4134837-0 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4193909-8 (DE-588)4134837-0 (DE-588)4044779-0 (DE-588)4128130-5 |
| title | Computational partial differential equations numerical methods and diffpack programming |
| title_auth | Computational partial differential equations numerical methods and diffpack programming |
| title_exact_search | Computational partial differential equations numerical methods and diffpack programming |
| title_full | Computational partial differential equations numerical methods and diffpack programming Hans Petter Langtangen |
| title_fullStr | Computational partial differential equations numerical methods and diffpack programming Hans Petter Langtangen |
| title_full_unstemmed | Computational partial differential equations numerical methods and diffpack programming Hans Petter Langtangen |
| title_short | Computational partial differential equations |
| title_sort | computational partial differential equations numerical methods and diffpack programming |
| title_sub | numerical methods and diffpack programming |
| topic | Diffpack (Computer file) Partielle Differentialgleichung - Numerisches Verfahren - C++ - Programmierumgebung Datenverarbeitung Differential equations, Partial -- Numerical solutions -- Data processing C++ (DE-588)4193909-8 gnd Programmierumgebung (DE-588)4134837-0 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Diffpack (Computer file) Partielle Differentialgleichung - Numerisches Verfahren - C++ - Programmierumgebung Datenverarbeitung Differential equations, Partial -- Numerical solutions -- Data processing C++ Programmierumgebung Partielle Differentialgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010006507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV016971315 |
| work_keys_str_mv | AT langtangenhanspetter computationalpartialdifferentialequationsnumericalmethodsanddiffpackprogramming |