Keywords:
Mathematics
;
Numerical analysis
;
Computer mathematics
;
Computer software
;
Mathematics
;
Computational Science and Engineering
;
Numeric Computing
;
Mathematical Software
;
Finite difference methods
;
Programming
;
Python
;
Verification
;
Numerical methods
;
Differential equations
Description / Table of Contents:
1. Vibration ODEs ---
1.1 Finite Difference Discretization ---
1.2 Implementation ---
1.3 Visualization of Long Time Simulations ---
1.4 Analysis of the Numerical Scheme ---
1.5 Alternative Schemes Based on 1st-Order Equations ---
1.6 Energy Considerations ---
1.7 The Euler-Cromer Method ---
1.8 Staggered Mesh ---
1.9 Exercises and Problems ---
1.10 Generalization: Damping, Nonlinearities, and Excitation ---
1.11 Exercises and Problems ---
1.12 Applications of Vibration Models ---
1.13 Exercises ---
2. Wave Equations ---
2.1 Simulation of Waves on a String ---
2.2 Verification ---
2.3 Implementation ---
2.4 Vectorization ---
2.5 Exercises ---
2.6 Generalization: Reflecting Boundaries ---
2.7 Generalization: Variable Wave Velocity ---
2.8 Building a General 1D Wave Equation Solver ---
2.9 Exercises ---
2.10 Analysis of the Difference Equations ---
2.11 Finite Difference Methods for 2D and 3D Wave Equations ---
2.12 Implementation ---
2.13 Exercises ---
2.14 Applications of Wave Equations ---
2.15 Exercises ---
3. Diffusion Equations ---
3.1 An Explicit Method for the 1D Diffusion Equation ---
3.2 Implicit Methods for the 1D Diffusion Equation ---
3.3 Analysis of Schemes for the Diffusion Equation ---
3.4 Exercises ---
3.5 Diffusion in Heterogeneous Media ---
3.6 Diffusion in 2D ---
3.7 Random Walk ---
3.8 Applications ---
3.9 Exercises ---
4. Advection-Dominated Equations ---
4.1 One-Dimensional Time-Dependent Advection Equations ---
4.2 One-Dimensional Stationary Advection-Diffusion Equation ---
4.3 Time-dependent Convection-Diffusion Equations ---
4.4 Applications of Advection Equations ---
4.5 Exercises ---
5. Nonlinear Problems ---
5.1 Introduction of Basic Concepts ---
5.2 Systems of Nonlinear Algebraic Equations ---
5.3 Linearization at the Differential Equation Level ---
5.4 1D Stationary Nonlinear Differential Equations ---
5.5 Multi-Dimensional Nonlinear PDE Problems ---
5.6 Operator Splitting Methods ---
5.7 Exercises
Pages:
Online-Ressource (XXIII, 507 pages)
,
150 illustrations
ISBN:
9783319554563
URL:
https://link.springer.com/book/10.1007/978-3-319-55456-3
Language:
English
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