# 401-0674-00L Numerical Methods for Partial Differential Equations

Semester | Frühjahrssemester 2017 |

Dozierende | S. Mishra |

Periodizität | jährlich wiederkehrende Veranstaltung |

Lehrsprache | Englisch |

Kommentar | Not meant for BSc/MSc students of mathematics. |

Kurzbeschreibung | Derivation, properties, and implementation of fundamental numerical methods for a few key partial differential equations: Poisson equation, heat equation, transport equation, conservation laws. Implementation of the algorithms in C++ |

Lernziel | Main skills to be acquired in this course: * Ability to implement advanced numerical methods for the solution of partial differential equations efficiently * Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations * Ability to select and assess numerical methods in light of the predictions of theory * Ability to identify features of a PDE (= partial differential equation) based model that are relevant for the selection and performance of a numerical algorithm * Ability to understand research publications on theoretical and practical aspects of numerical methods for partial differential equations. * Skills in the efficient implementation of numerical methods for PDEs. This course is neither a course on the mathematical foundations and numerical analysis of methods nor an course that merely teaches recipes and how to apply software packages. |

Inhalt | 1. General introduction to PDEs and their solutions. 2. 1-D Poisson equation -- Green's function, maximum principles, finite difference schemes. 3. Finite elements for the Poisson equation -- Variational formulation, basic finite element formulation with piecewise linear elements, error analysis, different boundary conditions. Efficient implementation of the finite element method. FEM for general elliptic PDEs, Higher-order finite elements. 4. Parabolic PDEs: exact solution formulas for the heat equation, energy method, maximum principles, Finite difference schemes for the heat equation with explicit, implicit and Crank-Nicolson schemes, error analysis. 5. Linear Transport equations -- method of characteristics, central and upwind finite difference schemes. 6. Scalar conservation laws -- Shocks, rarefactions, solutions to the Riemann problem, weak solutions, entropy conditions, Godunov type schemes, high-resolution schemes. |

Skript | Hand-written notes and script will be made available. |

Literatur | Lecture notes and references mentioned in the lecture notes. |

Voraussetzungen / Besonderes | Mastery of basic calculus and linear algebra is taken for granted. Familiarity with fundamental numerical methods (solution methods for linear systems of equations, interpolation, approximation, numerical quadrature, numerical integration of ODEs) is essential. Important: Coding skills in MATLAB and C++ are essential. Homework asssignments involve substantial coding in C++. |