Increasing asymptotic stability of Crank-Nicolson method

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Increasing asymptotic stability of Crank-Nicolson
method
Alexei A. Medovikov Vyacheslav I. Lebedev
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Summary

• The Crank-Nicolson method has second order
  accuracy, but for stiff ODEs, the numerical
  solution has unexpected oscillatory behavior,
  which can be explained in term of its stability
  function
• Variable time steps by Crank-Nicolson method
  allow us to formulate optimization problem for
  roots of the stability function
• The solution of this problem is the rational
  Zolotarev function
• We present robust algorithm of step-size
  selection and numerical results of the
  optimization procedure

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The exact solution of the heat equation can be
found by the method of separation of variables
We expect to have similar properties from the
numerical solution
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Method of lines Lebedev, V. I. The equations and
convergence of a differential-difference method
(the method of lines). (Russian) Vestnik Moskov.
Univ. 10 (1955), no. 10, 47--57
To solve the ODE we use midpoint rule or
trapezoidal rule
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Stability function of the classical
Crank-Nicolson method (a)
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Initial value of the heat equation (a). Exact
solution of the heat equation (b). The solution
of the heat equation by Crank-Nicolson method
with 3 constant steps
(c), and solution by the optimal method
with the same sum of steps (d)
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Fourier coefficients
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Composition Methods
ODEs generate a map
Runge-Kutta method generates a map
Composition of maps generated by RK method is
composition method
Properties of RK map depend on division
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Stability function
Applying RK method to simple test problem lead to
function-multiplier, which is responsible for
stability of the method
Stability function of composition methods
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Composition of mid-point rules define new method
and appropriate choice of steps allows us to
improve properties of the stability function
in order to have
How to optimize m and
maximum average time-step
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Zolotarev rational function
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Theorem (Medovikov) Sum of steps of Zolotarev
function for the interval equals
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The Algorithm
Wachspress E.L. Extended application of
alternating direction implicit model problem
theory. SIAM J. Appl. Math. 11 (1963)
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Initial value of the heat equation (a). Exact
solution of the heat equation (b). The solution
of the heat equation by Crank-Nicolson method
with 3 constant steps
(c), and solution by the optimal method
with the same sum of steps (d)
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Embedded methods