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4. Ordinary Differential Equations
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Introduction to Computational Physics
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II. Differential Equations Everywhere
The laws and relations of physics are often formulated in terms of DEs. Usually, analytical solutions are hard to come by, therefore numerical integration schemes are wanted:
ORDINARY DIFFERENTIAL EQUATIONS
PARTIAL DIFFERENTIAL EQUATIONS
Subsections
4. Ordinary Differential Equations (ODE)
4.1 Initial Value Problems of First Order
4.1.1 Euler-Cauchy Algorithm
4.1.2 Stability and Accuracy of Difference Schemes
4.1.3 Explicit Methods
4.1.4 Implicit Methods
4.1.5 Predictor-Corrector Method
4.1.6 Runge-Kutta Method
4.2 Initial Value Problems of Second Order
4.2.1 Verlet Method
4.2.2 Predictor-Corrector Method for 2nd order ODE
4.2.3 Nordsieck Formulation of the PC Method
4.2.4 Runge-Kutta Method for 2nd order ODE
4.2.5 Symplectic Algorithms
4.2.6 Numerov's Method
4.3 Boundary Value Problems
4.3.1 Shooting Method
4.3.2 Relaxation Method
5. Partial Differential Equations (PDE)
5.1 Initial Value Problems I: Conservative-hyperbolic DE
5.1.1 FTCS Scheme; Stability Analysis
5.1.2 Lax Scheme
5.1.3 Leapfrog Scheme (LF)
5.1.4 Lax-Wendroff Scheme (LW)
5.1.5 Lax and Lax-Wendroff in Two Dimensions
5.1.6 Resumé: Conservative-hyperbolic DE
5.2 Initial Value Problems II: Conservative-parabolic DE
5.2.1 FTCS Scheme for Parabolic DE
5.2.2 Implicit Scheme of First Order
5.2.3 Crank-Nicholson Scheme (CN)
5.2.4 Dufort-Frankel Scheme (DF)
5.2.5 Resumé: Conservative-parabolic DE
5.3 Boundary Value Problems: Elliptic DE
5.3.1 Relaxation and Multigrid Techniques
5.3.2 ADI Method for the Potential Equation
5.3.3 Fourier Transform Method (FT)
5.3.4 Cyclic Reduction (CR)
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001
PARTIAL DIFFERENTIAL EQUATIONS