Comparison of different time discretization methods explicitimplicit

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Comparison of different time discretization
methodsexplicit-implicit

• SIWIR-Seminar, 12.01.2009, Jana Hutter

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Contents

• Definition Initial value problem, Time
  discretization
• Intuitive try Euler-procedure
• Explicit one-step procedures
• Implicit one-step procedures
• Example Crank-Nicolson procedure
• Comparision with regard to convergence and
  stability

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Initial value problem

• Initial value problem for a system of ordinary
  differential equations of first degree

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Time Discretization
We define a grid in the following way
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Intuitive and easy try
Now all values y can be calculated
successively -gt This is the explicit Euler
procedure (Polygonzug-Verfahren) (Which looks
good on the first look, but .)
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Evaluation of a procedure I

• Errors made by the approximation from one step in
  time to the next one should be very small
• Consistence
• The local errors shouldnt be accumulated
  Stability

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Evaluation of a procedure II

• y exact solution of the initial value problem
• yh approximation on the grid
• Convergence
• Consistence

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Back to the explicit Euler procedure

• It can be shown that Euler has a convergence and
  consistence degree of order 1, which is really
  bad!
• -gt Back on the road for better procedures

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Next try general explicit one-step procedures
Example Runge-Kutta methods
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Results
Degree h2
Degree h
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Problem Stiff ODEs I

• general problem with explicit methods
• - perhaps higher degree of convergence than
  Euler (Heun h2)
• -but big problems with so called stiff
  differential equations (Real values of negative
  Eigenvalues very different)

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Problem Stiff ODEs II

• Example
• Solution
• Explicit/Implicit Euler method
• -gtBlackboard
• Better
• Methods which offer qualitative right solution
  independent from
• the value of h (so called A-Stability)

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Example
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Implicit one-step procedures
Difference to explicit methods To calculate
yj1, not only the already known values
y0,y1,,yi are used, but also the (unknown) value
yi1! -gt Ín every step a linear equation system
has to be solved Example implicit Euler-method
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Implicit Examples
The implicit Euler method can be written in the
following way (with Lamda0)
With Lamda0.5, we get the implicit
Crank-Nicolson method
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Overview

• Implicit methods
• Examples
• implicit Euler, Cranc-Nicolson
• -Solving of a linear equations system
• stable

• Explicit methods
• Examples
• explicit Euler, Heun
• easy to calculate
• - Not A-stable