Scaling of Differential Equations /

Scaling of Differential Equations /

Differential equations; Simulation and modeling.

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Bibliographic Details
Main Authors: Pedersen, Geir (Author), Petter Langtangen, Hans (Author)
Format: Electronic eBook
Language:English
Published: Cham : Springer, 2016.
Cham : Springer Open, 2016.
Series:SIMULA SPRINGERBRIEFS ON COMPUTING. ; 2
Online Access:CONNECT
Table of Contents:
  • Machine generated contents note: 1. Dimensions and units
  • 1.1. Fundamental concepts
  • 1.1.1. Base units and dimensions
  • 1.1.2. Dimensions of common physical quantities
  • 1.1.3. The Buckingham Pi theorem
  • 1.1.4. Absolute errors, relative errors, and units
  • 1.1.5. Units and computers
  • 1.1.6. Unit systems
  • 1.1.7. Example on challenges arising from unit systems
  • 1.1.8. Physical Quantity: a tool for computing with units
  • 1.2. Parampool: user interfaces with automatic unit conversion
  • 1.2.1. Pool of parameters
  • 1.2.2. Fetching pool data for computing
  • 1.2.3. Reading command-line options
  • 1.2.4. Setting default values in a file
  • 1.2.5. Specifying multiple values of input parameters
  • 1.2.6. Generating a graphical user interface
  • 2. Ordinary differential equation models
  • 2.1. Exponential decay problems
  • 2.1.1. Fundamental ideas of scaling
  • 2.1.2. The basic model problem
  • 2.1.3. The technical steps of the scaling procedure
  • Note continued: 2.1.4. Making software for utilizing the scaled model
  • 2.1.5. Scaling a generalized problem
  • 2.1.6. Variable coefficients
  • 2.1.7. Scaling a cooling problem with constant temperature in the surroundings
  • 2.1.8. Scaling a cooling problem with time-dependent surroundings
  • 2.1.9. Scaling a nonlinear ODE
  • 2.1.10. SIR ODE system for spreading of diseases
  • 2.1.11. SIRV model with finite immunity
  • 2.1.12. Michaelis-Menten kinetics for biochemical reactions
  • 2.2. Vibration problems
  • 2.2.1. Undamped vibrations without forcing
  • 2.2.2. Undamped vibrations with constant forcing
  • 2.2.3. Undamped vibrations with time-dependent forcing
  • 2.2.4. Damped vibrations with forcing
  • 2.2.5. Oscillating electric circuits
  • 3. Basic partial differential equation models
  • 3.1. The wave equation
  • 3.1.1. Homogeneous Dirichlet conditions in 1D
  • 3.1.2. Implementation of the scaled wave equation
  • 3.1.3. Time-dependent Dirichlet condition
  • Note continued: 3.1.4. Velocity initial condition
  • 3.1.5. Variable wave velocity and forcing
  • 3.1.6. Damped wave equation
  • 3.1.7.A three-dimensional wave equation problem
  • 3.2. The diffusion equation
  • 3.2.1. Homogeneous 1D diffusion equation
  • 3.2.2. Generalized diffusion PDE
  • 3.2.3. Jump boundary condition
  • 3.2.4. Oscillating Dirichlet condition
  • 3.3. Reaction-diffusion equations
  • 3.3.1. Fisher's equation
  • 3.3.2. Nonlinear reaction-diffusion PDE
  • 3.4. The convection-diffusion equation
  • 3.4.1. Convection-diffusion without a force term
  • 3.4.2. Stationary PDE
  • 3.4.3. Convection-diffusion with a source term
  • 4. Advanced partial differential equation models
  • 4.1. The equations of linear elasticity
  • 4.1.1. The general time-dependent elasticity problem
  • 4.1.2. Dimensionless stress tensor
  • 4.1.3. When can the acceleration term be neglected?
  • 4.1.4. The stationary elasticity problem
  • 4.1.5. Quasi-static thermo-elasticity
  • 4.2. The Navier-Stokes equations
  • Note continued: 4.2.1. The momentum equation without body forces
  • 4.2.2. Scaling of time for low Reynolds numbers
  • 4.2.3. Shear stress as pressure scale
  • 4.2.4. Gravity force and the Froude number
  • 4.2.5. Oscillating boundary conditions and the Strouhal number
  • 4.2.6. Cavitation and the Euler number
  • 4.2.7. Free surface conditions and the Weber number
  • 4.3. Thermal convection
  • 4.3.1. Forced convection
  • 4.3.2. Free convection
  • 4.3.3. The Grashof, Prandtl, and Eckert numbers
  • 4.3.4. Heat transfer at boundaries and the Nusselt and Biot numbers
  • 4.4.Compressible gas dynamics
  • 4.4.1. The Euler equations of gas dynamics
  • 4.4.2. General isentropic flow
  • 4.4.3. The acoustic approximation for sound waves
  • 4.5. Water surface waves driven by gravity
  • 4.5.1. The mathematical model
  • 4.5.2. Scaling
  • 4.5.3. Waves in deep water
  • 4.5.4. Long waves in shallow water
  • 4.6. Two-phase porous media flow
  • References.

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