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E-raamat: Scaling of Differential Equations

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Scaling of Differential EquationsSuurem pilt
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The book serves both as a reference forvarious scaled models with corresponding dimensionless numbers, and as aresource for learning the art of scaling.A special feature of the book is the emphasis on how to create softwarefor scaled models, based on existing software for unscaled models.Scaling (or non-dimensionalization) is amathematical technique that greatly simplifies the setting of input parameters innumerical simulations. Moreover, scaling enhances the understanding of howdifferent physical processes interact in a differential equation model.Compared to the existing literature, where the topic of scaling is frequentlyencountered, but very often in only a brief and shallow setting, the presentbook gives much more thorough explanations of how to reason about finding theright scales. This process is highly problem dependent, and therefore the bookfeatures a lot of worked examples, from very simple ODEs to systems of PDEs,especially from fluid mechanics.The tex

t is easily accessible andexample-driven. The first part on ODEs fits even a lower undergraduate level,while the most advanced multiphysics fluid mechanics examples target thegraduate level. The scientific literature is full of scaled models, but in mostof the cases, the scales are just stated without thorough mathematicalreasoning. This book explains how the scales are found mathematically.This book will be a valuable read for anyonedoing numerical simulations based on ordinary or partial differential equations.

Preface.- 1 Dimensions and Units.- 2 Ordinary Differential Equations Models.- 3 Basic Partial Differential Equations Models.- Advanced Partial Differential Equations Models.- References.- Index.

Muu info

This is an open access book, the electronic versions are freely accessible online.
Foreword v
Preface vii
1 Dimensions and units
1(16)
1.1 Fundamental concepts
1(8)
1.1.1 Base units and dimensions
1(1)
1.1.2 Dimensions of common physical quantities
2(1)
1.1.3 The Buckingham Pi theorem
3(2)
1.1.4 Absolute errors, relative errors, and units
5(1)
1.1.5 Units and computers
5(1)
1.1.6 Unit systems
5(1)
1.1.7 Example on challenges arising from unit systems
6(1)
1.1.8 Physical Quantity: a tool for computing with units
7(2)
1.2 Parampool: user interfaces with automatic unit conversion
9(8)
1.2.1 Pool of parameters
10(1)
1.2.2 Fetching pool data for computing
11(1)
1.2.3 Reading command-line options
11(1)
1.2.4 Setting default values in a file
12(1)
1.2.5 Specifying multiple values of input parameters
13(1)
1.2.6 Generating a graphical user interface
14(3)
2 Ordinary differential equation models
17(52)
2.1 Exponential decay problems
17(32)
2.1.1 Fundamental ideas of scaling
17(1)
2.1.2 The basic model problem
18(1)
2.1.3 The technical steps of the scaling procedure
19(2)
2.1.4 Making software for utilizing the scaled model
21(4)
2.1.5 Scaling a generalized problem
25(6)
2.1.6 Variable coefficients
31(1)
2.1.7 Scaling a cooling problem with constant temperature in the surroundings
32(1)
2.1.8 Scaling a cooling problem with time-dependent surroundings
33(4)
2.1.9 Scaling a nonlinear ODE
37(2)
2.1.10 SIR ODE system for spreading of diseases
39(2)
2.1.11 SIRV model with finite immunity
41(1)
2.1.12 Michaelis-Menten kinetics for biochemical reactions
42(7)
2.2 Vibration problems
49(20)
2.2.1 Undamped vibrations without forcing
49(4)
2.2.2 Undamped vibrations with constant forcing
53(1)
2.2.3 Undamped vibrations with time-dependent forcing
53(8)
2.2.4 Damped vibrations with forcing
61(6)
2.2.5 Oscillating electric circuits
67(2)
3 Basic partial differential equation models
69(30)
3.1 The wave equation
69(12)
3.1.1 Homogeneous Dirichlet conditions in 1D
69(2)
3.1.2 Implementation of the scaled wave equation
71(1)
3.1.3 Time-dependent Dirichlet condition
72(3)
3.1.4 Velocity initial condition
75(2)
3.1.5 Variable wave velocity and forcing
77(3)
3.1.6 Damped wave equation
80(1)
3.1.7 A three-dimensional wave equation problem
81(1)
3.2 The diffusion equation
81(8)
3.2.1 Homogeneous 1D diffusion equation
82(1)
3.2.2 Generalized diffusion PDE
83(2)
3.2.3 Jump boundary condition
85(1)
3.2.4 Oscillating Dirichlet condition
86(3)
3.3 Reaction-diffusion equations
89(3)
3.3.1 Fisher's equation
89(2)
3.3.2 Nonlinear reaction-diffusion PDE
91(1)
3.4 The convection-diffusion equation
92(7)
3.4.1 Convection-diffusion without a force term
92(3)
3.4.2 Stationary PDE
95(2)
3.4.3 Convection-diffusion with a source term
97(2)
4 Advanced partial differential equation models
99(38)
4.1 The equations of linear elasticity
99(7)
4.1.1 The general time-dependent elasticity problem
99(2)
4.1.2 Dimensionless stress tensor
101(1)
4.1.3 When can the acceleration term be neglected?
101(2)
4.1.4 The stationary elasticity problem
103(2)
4.1.5 Quasi-static thermo-elasticity
105(1)
4.2 The Navier-Stokes equations
106(7)
4.2.1 The momentum equation without body forces
107(2)
4.2.2 Scaling of time for low Reynolds numbers
109(1)
4.2.3 Shear stress as pressure scale
110(1)
4.2.4 Gravity force and the Froude number
110(1)
4.2.5 Oscillating boundary conditions and the Strouhal number
110(1)
4.2.6 Cavitation and the Euler number
111(1)
4.2.7 Free surface conditions and the Weber number
112(1)
4.3 Thermal convection
113(8)
4.3.1 Forced convection
113(1)
4.3.2 Free convection
114(3)
4.3.3 The Grashof, Prandtl, and Eckert numbers
117(3)
4.3.4 Heat transfer at boundaries and the Nusselt and Biot numbers
120(1)
4.4 Compressible gas dynamics
121(5)
4.4.1 The Euler equations of gas dynamics
121(2)
4.4.2 General isentropic flow
123(1)
4.4.3 The acoustic approximation for sound waves
124(2)
4.5 Water surface waves driven by gravity
126(4)
4.5.1 The mathematical model
126(1)
4.5.2 Scaling
127(1)
4.5.3 Waves in deep water
128(1)
4.5.4 Long waves in shallow water
129(1)
4.6 Two-phase porous media flow
130(7)
References
135(2)
Index 137
Hans Petter Langtangen is a professor of computer science at the University of Oslo. He has formerly been a professor of mechanics and is now the director of a Norwegian Center of Excellence: "Center for Biomedical Computing", at Simula Research Laboratory. Langtangen has published over 100 scientific publications and written several books, including papers and the bestseller TCSE 6 "A Primer on Scientific Programming with Python", now in its 5th edition. He has also developed open source and commercial software systems for computational sciences.

Geir K. Pedersen is a professor of mechanics at the Department of Mathematics, University of Oslo. He has a life-long experience in fluid dynamics and mathematical modeling. Pedersen has published articles on wave theory, numerical modeling, perturbation techniques, tsunamis, hydrodynamic stability and experimental fluid dynamics.     
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