Scaling of differential equations

• Main Authors: Langtangen, Hans Petter, Pedersen, Geir K
• Format: eBook Book
• Language: English
• Published: Cham Springer 2016; Springer Nature; Springer International Publishing AG; Springer Open
• Edition: 1
• Series: Simula SpringerBriefs on Computing
• Subjects: Differential equations; Mathematical models; Mathematics; Mathematics and Science; Nonfiction; Simulation and modeling
• ISBN: 9783319327259; 3319327259; 9783319327266; 3319327267
• DOI: 10.1007/978-3-319-32726-6

This work serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models.

Table of Contents:

• 4.6 Two-phase porous media flow -- References -- Index
• Intro -- Foreword -- Preface -- Contents -- 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 PhysicalQuantity: 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 -- 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 -- 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 -- 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