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MS-E1652 - Computational methods for differential equations, 09.09.2019-23.10.2019

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Syllabus

Materials

  • Materials

    Materials

    The (hand-written) lecture notes can be found below. Most of the lectures roughly follow

    D. F. Griffiths and D. J. Higham, Numerical Methods for Ordinary Differential Equations, Springer.

    • September 9

      Introduction of initial value problems for ordinary differential equations. Existence and uniqueness. Continuous dependence on initial data.

      • Lecture 1Lecture 1
        • lecture1.pdflecture1.pdf579.7KB
    • September 11

      Euler's method: rationale, local truncation error (LTE), global error, weaknesses. (Four Matlab scripts/functions, designed for demonstrating the weaknesses of the Euler's method, are also enclosed.)

      • Lecture 2Lecture 2
        • ff.mff.m49 bytes
        • lect2a.mlect2a.m1.7KB
        • lect2b.mlect2b.m921 bytes
        • lect2c.mlect2c.m1.8KB
        • lecture2.pdflecture2.pdf436.1KB
    • September 16

      Linear multistep methods (LMMs): examples, construction and consitency.

      • Lecture 3Lecture 3
        • lecture3.pdflecture3.pdf1.6MB
    • September 18

      Linear multistep methods: zero-stability and (global) convergence.

      • Lecture 4Lecture 4
        • lecture4.pdflecture4.pdf407.7KB
    • September 23

      Linear multistep methods: stiff problems, absolute stability, region of absolute stability. (Two Matlab scripts for comparing the explicit and implicit Euler's methods for stiff problems are enclosed.)

      • Lecture 5Lecture 5
        • lect5a.mlect5a.m1.2KB
        • lect5b.mlect5b.m1.2KB
        • lecture5.pdflecture5.pdf443.7KB
    • September 25

      Linear multistep methods: A-stability, absolute stability for systems (n>1), stiff systems.

      • Lecture 6Lecture 6
        • lecture6.pdflecture6.pdf585.6KB
    • September 30

      Runge-Kutta methods: definition, examples, order conditions.

      • Lecture 7Lecture 7
        • lecture7.pdflecture7.pdf567.9KB
    • October 2

      Runge-Kutta methods: absolute stability, stability function.

      • Lecture 8Lecture 8
        • lecture8.pdflecture8.pdf465.9KB
    • October 7

      Runge-Kutta methods: absolute stability for systems (n>1).

      • Lecture 9Lecture 9
        • lecture9.pdflecture9.pdf522.6KB
    • October 9

      Parabolic PDEs: (Spatial) discretization of a simple model problem, stiffness of the resulting initial value problem.

      • Lecture 10Lecture 10
        • lecture10.pdflecture10.pdf516.6KB
    • October 14

      Parabolic PDEs: Crank-Nicolson method and its properties. Hyperbolic PDEs: conservation of energy.

      • Lecture 11Lecture 11
        • lecture11.pdflecture11.pdf516.4KB
    • October 16

      Hyperbolic PDEs: discretization and conservation of "discrete energy".

      • Lecture 12Lecture 12
        • lecture12.pdflecture12.pdf471.3KB
    • icon for activity Discretizing differential equations File PDF document

      Some additional reading related to the topic of the course: the lecture notes "Discretizing differential equations" by Timo Eirola and Olavi Nevanlinna.

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