Spring School on the Introduction to Numerical Modelling of Differential Equations

Precourse October 23 -- 25, 2019

Ole Klein (HD), Jan Philipp Thiele (LUH), Robin Görmer (LUH)

Language :  Englisch

  • Class 1: Scientific modelling and scientific computing (high level overview), best practices for software development in scientific computing.


  • Class 2: Introduction to C++: functions, variables, arithmetic, recursion, loops.


  • Class 3: Objects, classes and containers, introduction to git.


  • Class 4: Floating point numbers, differential condition analysis, (forward) rounding error analysis, stability.

  • Class 5: Truncation error, cancellation, error estimates and asymptotics, difference quotients and swinging pendulum as simple ODE model.


  • Class 6: Pipe network / electric circuit to motivate large linear systems, Poisson equation as limit, extension to heterogeneous media.


Classes 1-3 are based on [8] and our programming course slides, 4-6 are basically the first few lectures of [9]. 


School October 28 -- 31, 2019

Lectures:   Peter Bastian (HD), Thomas Wick (LUH)

Exercises: Ole Klein, Jan Philipp Thiele, Robin Görmer

Language :  Englisch

  • Class 1: Introduction to numerical modeling (what is scientific computing, examples of differential equations, population models, predator-prey models, N-body problem (energy conservation, Hamiltonian Systems), chemical networks, electrical networks, ...)


         Numerical methods for ODEs


  • Class 2: A brief classification of differential equations (first order, second order, linear/nonlinear, differential algebraic systems).

          Model problem u' = f(t,u)


  • Class 3: Derivation and analysis of three well-known numerical schemes: forward Euler, backward Euler, trapezoidal rule; higher-order methods: Taylor and Runge-Kutta methods.


  • Class 4: Brief numerical analysis: discretization error, stability, convergence for initial-value problems. In case, there is time, we deliver proofs for general Runge-Kutta methods.

         Practical demonstrations: Numerical simulations and discussions to discretization errors, stability, and convergence. For instance using a simple                      realization of a population model.


  • Class 5: Galerkin methods for ODEs (i.e., methods for stiff ODEs), time step adaptivity,

         Lit. [3], i.e., Rannacher, 2017, chapter 7 - in german

         Lit. [1], chapter 10.6: Galerkin; chapter 10.7: error analysis and time step control)


        Numerical methods for PDEs


  • Class 6: Introduction to partial differential equations (PDEs), Modelling with partial differential equations: conservation laws, type classification of 2nd order linear PDEs.


  • Class 7: Variational formulations, Lax-Milgram, basics in functional analysis.


  • Class 8: Conforming finite element method, Pk finite element space, Lagrange basis, linear system, basic error analysis, interpolation error.


  • Class 9: Finite elements on a practical level, assembling the linear system, iterative solution of linear systems.


 Spring School on the Introduction on Numerical Modelling of Differential Equations - UNALM 2019