NUMERICAL METHODS OF PHYSICS

Several physical problems of interest are proposed, which provide examples of standard mathematical problems. These, in turn, provide examples of standard numerical techniques, which are sometimes also implemented by means of computer codes. At the various levels of interest (physical, mathematical, numerical and, where available, in programming), different sources of “approximation” are presented and critically examined.

Approximation and interpolation of functions. Polynomial interpolation. Interpolation formulas according to Lagrange and to Newton.

Numerical error: point and global error. Scalar products and norms in functional spaces.

Orthonormal bases in linear spaces. Classical orthogonal polynomials. Legendre, Hermite, Laguerre, Chebyshev polynomials. Generating function. Rodriguez formula (for the Hermite polynomials). Multipole expansion. Central fields generated by a mass or charge distribution. Development of functions in Legendre polynomials.

Numerical derivation and integration. Newton-Cotes formulas. Trapeze method. Simpson method. Adaptive integration (hint). Monte Carlo method (hint).

Zero finding. Bisection method. Convergence order. Secant method. Newton-Raphson method. The Babylonian algorithm and other examples.

Ordinary differential equations (ODE). Physical examples. Picard-Lindelöf theorem. Lipschitz condition. Picard method. Continuous dependence on initial data values. Euler method. Local and global truncation error. Heun, implicit Euler, and Runge-Kutta methods. Symplectic methods.

Numerical solution of the Schrödinger equation: Numerov method. Case of the harmonic oscillator, of the potential well, and other confining potentials.

Systems of linear equations: direct and iterative methods. Cramer method. Laplace algorithm for the determinant of a matrix. Computational complexity: polynomial and non-polynomial. Stirling formula. Factorial of a number in terms of Euler's Gamma function. Saddle-point approximation for the numerical estimate of integrals. Gauss-Jordan method and its computational complexity. Power sums. LU factorization method. Iterative methods. Convergence criterion. Matrix norms. Jacobi and Gauss-Seidel methods. Sparse and dense matrices. Successive over-relaxation.

Eigenvalues and eigenvectors (reminder). Spectral representation. Relevance of symmetries in physics. Graphs: adjacency matrix. Google and Perron-Frobenius theorem (hint). Applications to Internet and the search engines. Power method. Normal modes of a one-dimensional chain of harmonic oscillators: periodic case. Analytical solution for a homogeneous chain. Bands. Continuum limit. Long wavelength limit. Sound velocity. Acoustic modes. Normal modes of a one-dimensional chain of harmonic oscillators: quasi-periodic case. Quasicrystals. Numerical solution for the Fibonacci chain.

Partial differential equations (PDE). PDE of physical relevance. Classification (hint). Local and global formulation of a physical law. Examples of PDE from physics: Maxwell equations. General problem of the electrostatics. Poisson equation: Dirichlet and von Neumann problems. Poisson equation: variational derivation. Discretization of the Poisson equation and of the electrostatic energy. Richardson method. Convergence criterion. Liebmann method.

Discrete Fourier Transforms (DFT). Numerical solution of the Poisson equation via DFT.

S. E. Koonin, D. C. Meredith, Computational physics (Addison-Wesley, Redwood, 1990).

G. Naldi, L. Pareschi, G. Russo, Introduzione al calcolo scientifico (McGraw-Hill, Milano, 2001).

J. F. Epperson, Introduzione all'analisi numerica (McGraw-Hill, Milano, 2003).