Computational Physics 1 (Fundamental Numerical Methods)

1. Introduction
‒ Numbers on computers
‒ Sources of errors
2. Integration
‒ Riemann definition
‒ Trapezoid rule, Simpson rule
‒ Gauss integration
‒ Adaptive stepsize
3. Differentiation
‒ Forward Difference, Central Difference
‒ Higher Orders
4. Root Finding
‒ Bisection
‒ Newton-Raphson
5. Linear Algebra
‒ Gauss elimination with back-substitution
‒ LU-Decomposition
‒ Singular Value Decomposition
6. Multidimensional Newton-Raphson
7. Data fitting / Inter-/Extrapolation
‒ Lagrange interpolation, Splines
‒ Least squares fit
‒ Linear least squares, non-linear chi^2
8. Ordinary Differendial Equations
‒ Classification of DEs
‒ Euler algorithm
‒ Midpoint algorithm
‒ Runge-Kutta
‒ Applications: Planetary motion, etc.
‒ Initial/boundary value problems
‒ ODE eigenvalues
9. Partial Differential Equations
‒ Elliptic PDEs
‒ Parabolic PDEs
‒ Hyperbolic PDEs

After successful completion of this module, students are able to

• understand and implement (in various programming languages) basic numerical methods such as integration, (multidimensional) root finding, data interpolation and fitting.
• classify and numerically solve all ordinary and simple partial differential equations.
• construct a proper DE description for a given physical problem.

No preconditions in addition to the requirements for the Master’s program in Physics.

This module consists of a lecture and an exercise class.

In the lecture, the contents are first explained on a theoretical level on an electronic whiteboard (the slides can be downloaded from the lecturer's web site immediately after the lectures). Then, the algorithms are implemented in the computer algebra system Mathematica to study the practical applicability of the concept. Whenever possible, the students are asked for input during this process, and if a suggested approach fails (e.g. due to numerical instabilities), the causes are discussed and alternatives are presented.

Exercise sheets (which frequently include reproduction of the results of the lecture) are first worked on individually by the students and then discussed in small groups (exercises) under supervision.

Students are invited to a guided tour to the Leibniz Supercomputing Centre (LRZ). Additionally an introduction to programming is offered. This lecture is directed especially and especially recommended to students without programming experience.

Presentations on an electronic Whiteboard, demonstrations in Mathematica, C and Python ; exercise sheets. Accompanying web page: http://users.ph.tum.de/srecksie/lehre

• R.H. Landau, M.J. Páez and C.C. Bordeianu: Computational Physics: Problem Solving with Computers, Wiley-Vch, (2007)
• G.P. Lepage: Lattice QCD for Novices, arXiv, (2005), http://arxiv.org/abs/hep-lat/0506036

There will be a written exam of 90 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using calculation problems and comprehension questions.

For example an assignment in the exam might be:

• Use the trapezoidal rule to calculate the integral of a function and show that the result is exact for linear functions.
• What is the order of magnitude of the error for non-linear functions (in terms of the step size and derivatives of the function to be integrated)?
• Taking into account the algorithmic error and the roundoff error, what is the optimum number of integration steps?

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.