# 401-0435-00L Computational Methods for Engineering Applications II

Semester | Autumn Semester 2015 |

Lecturers | S. Mishra |

Periodicity | yearly recurring course |

Language of instruction | English |

Abstract | The course gives an introduction to the numerical methods for the solution of ordinary and partial differential equations that play a central role in engineering applications. Both basic theoretical concepts and implementation techniques necessary to understand and master the methods will be addressed. |

Objective | At the end of the course the students should be able to: - implement numerical methods for the solution of ODEs (= ordinary differential equations); - identify features of a PDE (= partial differential equation) based model that are relevant for the selection and performance of a numerical algorithm; - implement the finite difference, finite element and finite volume method for the solution of simple PDEs using C++; - read engineering research papers on numerical methods for ODEs or PDEs. |

Content | Initial value problems for ODE: review of basic theory for ODEs, Forward and Backward Euler methods, Taylor series methods, Runge-Kutta methods, multi-step methods, predictor-corrector methods, basic stability and consistency analysis, numerical solution of stiff ODEs. Two-point boundary value problems: Green's function representation of solutions, Maximum principle, finite difference schemes, stability analysis. Elliptic equations: Laplace's equation in one and two space dimensions, finite element methods, implementation of finite elements, error analysis. Parabolic equations: Heat equation, Fourier series representation, maximum principles, Finite difference schemes, Forward (backward) Euler, Crank-Nicolson method, stability analysis. Hyperbolic equations: Linear advection equation, method of characteristics, upwind schemes and their stability. Burgers equation, scalar conservation laws, shocks and rarefactions, Riemann problems, Godunov type schemes, TVD property. |

Lecture notes | Script will be provided. |

Literature | Chapters of the following book provide supplementary reading and are not meant as course material: - A. Tveito and R. Winther, Introduction to Partial Differential Equations. A Computational Approach, Springer, 2005. |

Prerequisites / Notice | (Suggested) Prerequisites: Analysis I-III (for D-MAVT), Linear Algebra, CMEA I, basic familiarity with programming in C++. |