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
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Class 1: Scientific modelling and scientific computing (high level overview), best practices for software development in scientific computing.
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Class 2: Introduction to C++: functions, variables, arithmetic, recursion, loops.
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Class 3: Objects, classes and containers, introduction to git.
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Class 4: Floating point numbers, differential condition analysis, (forward) rounding error analysis, stability.
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Class 5: Truncation error, cancellation, error estimates and asymptotics, difference quotients and swinging pendulum as simple ODE model.
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Class 6: Pipe network / electric circuit to motivate large linear systems, Poisson equation as limit, extension to heterogeneous media.
Information:
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
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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
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Class 2: A brief classification of differential equations (first order, second order, linear/nonlinear, differential algebraic systems).
Model problem u' = f(t,u)
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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.
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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.
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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
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Class 6: Introduction to partial differential equations (PDEs), Modelling with partial differential equations: conservation laws, type classification of 2nd order linear PDEs.
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Class 7: Variational formulations, Lax-Milgram, basics in functional analysis.
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Class 8: Conforming finite element method, Pk finite element space, Lagrange basis, linear system, basic error analysis, interpolation error.
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Class 9: Finite elements on a practical level, assembling the linear system, iterative solution of linear systems.
Sponsored by
Spring School on the Introduction on Numerical Modelling of Differential Equations - UNALM 2019
