Differential equations

Since physical laws are commonly expressed as differential equations, their numerical solution is arguably the most important subject in computational physics and forms the topic of this chapter. As is common practice, the discussion is divided into ordinary differential equations and partial differential equations. Unlike analytical methods, the discussion will be largely applicable to both linear and nonlinear differential equations and homogeneous and non-homogeneous differential equations. The ordinary differential equations section starts with a discussion of Euler, Runge-Kutta and leap-frog methods. Improvements using adaptive step size algorithms are mentioned. Python libraries for solving the initial value problem are introduced. Boundary value problems using the shooting method and a Python library are illustrated. The more advanced methods of pseudo-spectral, asymptotic iteration, and Frobenius methods are introduced. The section on ordinary differential equations ends with the concept of streamlines which are useful for displaying vector fields. Solving partial differential equations using grid methods for the Laplace equation, diffusion equation, wave equation, continuity equation and Schrodinger equation is covered. Finally, an application of von Neumann stability analysis to these partial differential equations is discussed.