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Contents
- Table of contents
- 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

References

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H. P. Langtangen. A Primer on Scientific Programming with Python, fifth edition, Texts in Computational Science and Engineering, Springer, 2016.
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