Scaling of Differential Equations
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Scaling of Differential Equations
Preface
Dimensions and units
Fundamental concepts
Base units and dimensions
Dimensions of common physical quantities
Prefixes for units
The Buckingham Pi theorem
Absolute errors, relative errors, and units
Units and computers
Unit systems
Example on challenges arising from unit systems
PhysicalQuantity: a tool for computing with units
Parampool: user interfaces with automatic unit conversion
Pool of parameters
Fetching pool data for computing
Reading command-line options
Setting default values in a file
Specifying multiple values of input parameters
Generating a graphical user interface
Ordinary differential equation models
Exponential decay problems
Fundamental ideas of scaling
The basic model problem
Example: Population dynamics
Example: Decay of pressure with altitude
The technical steps of the scaling procedure
Step 1: Identify independent and dependent variables
Step 2: Make independent and dependent variables dimensionless
Step 3: Derive the model involving only dimensionless variables
Step 4: Make each term dimensionless
Step 5: Estimate the scales
Making software for utilizing the scaled model
Software for the original unscaled problem
Software for the scaled problem
Implementation with joblib
Scaling a generalized problem
Exact solution
Theory
Software
Variable coefficients
Scaling a cooling problem with constant temperature in the surroundings
Exact solution
Scaling
Software
Alternative scaling
Scaling a cooling problem with time-dependent surroundings
Exact solution
Scaling
Software
Discussion of the time scale
Scaling a nonlinear ODE
Scaling
Alternative scaling
SIR ODE system for spreading of diseases
Scaling
Software
Alternative scaling
SIRV model with finite immunity
Michaelis-Menten kinetics for biochemical reactions
Classical analysis
Dimensionless ODE system
Determining scales
Conservation equations
Analysis of the scaled system
Vibration problems
Undamped vibrations without forcing
The first technical steps of scaling
The exact solution
Discussion of the displacement scale
Discussion of the time scale
The dimensionless solution
Alternative displacement scale
About frequency and dimensions
Undamped vibrations with constant forcing
Undamped vibrations with time-dependent forcing
Investigating scales via analytical solutions
The displacement and time scales
Finding the displacement scale from the differential equation
Scaling with free vibrations as time scale
Software
Choice of
\(u_c\)
close to resonance
Unit size of all terms in the ODE
Choice of
\(u_c\)
when
\(\psi\gg\omega\)
Displacement scale based on
\(I\)
Damped vibrations with forcing
The exact solution
Choosing scales
Choice of
\(u_c\)
at resonance
Choice of
\(u_c\)
when
\(\omega\gg\psi\)
Choice of
\(u_c\)
when
\(\omega\ll\psi\)
Software
Oscillating electric circuits
Exercises
Exercise 2.1: Perform unit conversion
Problem 2.2: Scale a simple formula
Exercise 2.3: Perform alternative scalings
Problem 2.4: A nonlinear ODE for vertical motion with air resistance
Exercise 2.5: Solve a decay ODE with discontinuous coefficient
Exercise 2.6: Implement a scaled model for cooling
Problem 2.7: Decay ODE with discontinuous coefficients
Exercise 2.8: Alternative scalings of a cooling model
Exercise 2.9: Projectile motion
Problem 2.10: A predator-prey model
Problem 2.11: A model for competing species
Problem 2.12: Find the period of sinusoidal signals
Remarks
Problem 2.13: Oscillating mass with sliding friction
Problem 2.14: Pendulum equations
Exercise 2.15: ODEs for a binary star
Problem 2.16: Duffing’s equation
Problem 2.17: Vertical motion in a varying gravity field
Problem 2.18: A simplified Schroedinger equation
Remarks
Basic partial differential equation models
The wave equation
Homogeneous Dirichlet conditions in 1D
Implementation of the scaled wave equation
Waves on a string
Detecting an already computed case
Time-dependent Dirichlet condition
Scaling
Software
Specific case
Velocity initial condition
Analytical insight
Scaling
Nonzero initial shape
Variable wave velocity and forcing
Non-dimensionalization
Choosing the time scale
Choosing the spatial scale
Scaling the velocity initial condition
Damped wave equation
A three-dimensional wave equation problem
The diffusion equation
Homogeneous 1D diffusion equation
Choosing the time scale
Analytical insight
Choosing other scales
Generalized diffusion PDE
Jump boundary condition
Oscillating Dirichlet condition
Scaling issues
Exact solution
Time and length scales
The scaled problem
Simulations
Reaction-diffusion equations
Fisher’s equation
Balance of all terms
Fixed length scale
Nonlinear reaction-diffusion PDE
The convection-diffusion equation
Convection-diffusion without a force term
Stationary PDE
Convection-diffusion with a source term
Exercises
Problem 3.1: Stationary Couette flow
Remarks
Exercise 3.2: Couette-Poiseuille flow
Exercise 3.3: Pulsatile pipeflow
Exercise 3.4: The linear cable equation
Exercise 3.5: Heat conduction with discontinuous initial condition
Remarks
Problem 3.6: Scaling a welding problem
Advanced partial differential equation models
The equations of linear elasticity
The general time-dependent elasticity problem
Software
Dimensionless stress tensor
When can the acceleration term be neglected?
S waves
P waves
Time-varying load
The stationary elasticity problem
Scaling of the PDE
Remark on the characteristic displacement
Scaling of displacement boundary conditions
Scaling of traction boundary conditions
Quasi-static thermo-elasticity
The Navier-Stokes equations
The momentum equation without body forces
Scaling
Dimensonless PDEs and the Reynolds number
Scaling of time for low Reynolds numbers
Shear stress as pressure scale
Gravity force and the Froude number
Oscillating boundary conditions and the Strouhal number
Cavitation and the Euler number
Free surface conditions and the Weber number
Thermal convection
Forced convection
Free convection
Governing equations
Heating by viscous effects
Relation between density and temperature
The Boussinesq approximation
Scaling
The Grashof, Prandtl, and Eckert numbers
Interpretations of the Grashof number
Heat transfer at boundaries and the Nusselt and Biot numbers
Compressible gas dynamics
The Euler equations of gas dynamics
General isentropic flow
Elimination of the pressure
The acoustic approximation for sound waves
Wave nature of isentropic flow with small perturbations
Basic scaling for small wave perturbations
Water surface waves driven by gravity
The mathematical model
Scaling
Waves in deep water
Long waves in shallow water
Two-phase porous media flow
The bidomain model in electrophysiology
The mathematical model
Scaling
An alternative
\(I_{\rm{ion}}\)
Exercises
Exercise 4.1: Comparison of vibration models for elastic structures
Exercise 4.2: A model for quasi-static poro-elasticity
Problem 4.3: Starting Couette flow
Problem 4.4: Channel flow
Remarks
References
Page
Index
A
|
B
|
C
|
D
|
E
|
F
|
G
|
J
|
L
|
M
|
N
|
P
|
Q
|
R
|
S
|
T
|
U
|
V
|
W
A
angular frequency
assert
B
base unit
Bernoulli's equation
bidomain equations
Biot number
Buckingham Pi theorem
C
characteristic time
creeping flow
D
dimension of physical quantities
dimensionless number
,
[1]
,
[2]
,
[3]
,
[4]
,
[5]
,
[6]
dimensionless variable
,
[1]
E
e-folding time
Eckert number
Euler number
exponential decay
F
forced convection
free convection
frequency
frequency, angular
Froude number
G
graphical web interface
Grashof number
J
joblib
,
[1]
L
length
logistic equation
Lotka-Volterra, competing species model
Lotka-Volterra, predator-prey model
low Reynolds number flow
M
Mach number
mass
memoize function
multiple software runs
N
Navier-Stokes equations
non-dimensionalization
Nusselt number
P
parampool
Peclet number
,
[1]
,
[2]
period (of oscillations)
phase shift
PhysicalQuantity
Pi theorem
Q
quality factor Q
R
radians
Reynolds number
,
[1]
,
[2]
,
[3]
,
[4]
S
scaling
Stokes problem
Stokes' flow
Strouhal number
T
thermo-elasticity
time
U
units
British
US
conversion
software
V
vortex shedding
W
web interface (Parampool)
Weber number