Integration of 3-body encounter.

1
Integration of 3-body encounter.
Figure taken from http//grape.c.u-tokyo.ac.jp/m
akino
2
4. Integration of Ordinary Differential Equations
(ODE).
We consider ODE with one variable, Most of
results for this single ODE can be applicable for
the above n-coupled ODE.
3
A quick look for
Existence, uniqueness, and stability of solution
for ODE,
4
Eulers method Consider the initial value
problem,
5
Error analysis for Eulers method
6
Stability of Eulers method
7
Taylors method
8
Exc 4-1) has a solution x(t) t ( 1 ln t).
Apply Eulers method changing the h 1/2n , n1
to 8, and estimate (a) the absolute error yi -
wi at t 6, and (b) the error ratio for the
successive h 1/2n at the same instant t 6,
namely,
Exc 4-2) Perform the same analysis as 4-1) using
the second order and fourth order Taylor
methods. Because computations of higher
derivatives is cumbersome, higher-order formula
which involves evaluation f(t,y) only is more
convenient.
9
Summary of the key concept on numerical method
for ODE.

• Local truncation error ti The amount that
  the solution of ODE fails to
• satisfy the the finite difference equation.

ex.) One-step method.

• Global discretization error

Definitions (Consistency, Convergence and
Stability)
10
One-step method.
One-step method is consistent if
Theorem (Convergence and stability of the
one-step method.)
Remark Above theorem says the one-step method
is consistent ) convergent. It can be proved
under the same conditions, the one-step method is
convergent , consistent.
11

• One-step method.
• 2nd order Runge-Kutta method.

Determine constants a1, a2, d2, D2, so that f
becomes O(h2) approximation of the O(h2) Taylor
method.

• Modified Euler method. (Half step Euler
  Midpoint integration.)

• Heun method. (One step Euler Trapezoidal
  integration.)

12

• Optimal RK2 method.

• Classical 4th order Runge-Kutta method.

Exc 4-3) Using an algebraic computing software,
show that the local truncation error of
Classical 4th-order Runge-Kutta method is O(h4).
13

• General s-stage Runge-Kutta method.

14
Remarks on General s-stage Runge-Kutta method.
(1) For Explicit s-stage formula, it is not known
in general what order O(hp) of formula one can
construct for each level s.
(2) For Implicit s-stage formula, it is known
that O(h2s) formula can be construct for each
level s.
Formulas with properties (1) Small local
truncation error, (2) Coefficients to be rational
numbers, (3) many zeros in aj,l , are more
practical.
Exc 4-4) Using the idea of Gauss-Legendre
integration formula, derive 2-stage Runge-Kutta
formula with 4th order accuracy.
15
Error control Estimate the local truncation
error and make it smaller than a certain
threshold value by changing step size.

• How to estimate the error ?
• 1) Just to make h ! h/2. This is fine, but
  inefficient.
• 2) Embedded formula.

http//www.unige.ch/hairer/software.html for
RK 8th order formula.
16
Linear (m-step) Multistep method.
Adams method aj 0, for j 2, .. ,
m Derivation Integrating the both side of ODE,

• Implicit linear multistep method is derived
  from interpolating polynomial of order m.

• Explicit linear multistep method from
  interpolating polynomial of order m-1

Substitute the form f(t,y(t)) p(t) R(t) in
the integral form of ODE, and integrate it to
calculate bj .
17

• Explicit scheme is also called
  Adams-Bashforth, implicit Adams-Moulton.
• Explicit scheme may be efficient since the
  f(ti,wi) of earlier steps are used.
• The starting values for w0 (initial value),
  w1, , wm-1 are required for the
• m-step method.
• These are calculated from one-step method of
  the same order.
• For the implicit method, value at ti1 , wi1
  , is calculated from an
• algebraic equation. It is iteratively solved
  using wi1 of an explicit
• multistep method of the same order as an
  initial guess.
• Usually this iteration is done by direct
  substitution, and only one or two
• iteration is made. This procedure is called
  predictor corrector schemes.
• For this scheme, the 4th-order formula is the
  most popular.

18

• In the Adams method, Newton form of polynomial
  interpolation formula is
• used for changing the step size as well as the
  starting formula.
• (Krogh type formula.)
• The local truncation error estimation is often
  made by the difference
• between predictor value and corrector value,
  which can be used for control
• the step size h.

Exc4-5) Assuming uniform discretization in t
domain, derive linear 2-step, 3-step 4-step
explicit formulas and 1-step, 2-step, 3-step
implicit formulas. Compare the coefficients of
error terms between implicit and explicit
formulas of the same order.
Exc4-6) Report on Krogh type formula.
Exc4-7) Report on the ODE solver that uses
Richardson extrapolation.
19
Order of General Linear (m-step) Multistep
method.
Conditions for the coefficients aj, and bJ to
satisfy for having the local truncation error ti
O(hp) are written
Exc4-8) Derive these conditions. hint) Substitute
dy/dt f(t,y) to the equation for ti above, and
expand y and y around t ti1-m .
20
Consistency Convergence, and Stability of General
Linear Multistep method.
Note the starting values w1, , wm-1 are assumed
to be converge.
Consistency (ti ! 0, as h ! 0) is satisfied if
the local truncation error ti is a at least
O(h).
21
Stability Consider the case with f(t,y) 0.
Definition (Root condition) The linear multistep
method satisfies the Root condition if the zeros
of the associated characteristic polynomial
satisfy 1. 2.
Theorem (Stability) A linear multistep method is
stable if and only if it satisfies the root
condition.
Definition A stable multistep method is said to
be strongly stable if l 1 is the only zero of
P(l) with l 1, and to be weakly stable
otherwise.
Remark A linear multistep method that satisfies
the consistency condition will always have at
least one zero with l 1.
22
Convergence
1.
2.
3.
Theorem (Convergence) A linear multistep method
is convergent, if and only if it is both
consistent and stable. .
23

• Absolute stability and stiff equations.
• Test problem
• This problem models a system of linear ODEs, in
  which case l represents an eigenvalue of the
  Jacobian associated with the r.h.s.
• When the asymptotic character of a numerical
  approximation wn, n ! 1 matches that of exact
  solution y(t), t ! 1, the numerical method is
  said to be absolute stable.

One-step method Consider the mth order Taylor
methods.
Definition The region of absolute stability for
a one-step method is the set
24
Multistep method Consider the linear m-step
multistep method.
Definition The region of absolute stability for
a multistep method is the set
Exc 4-9) Derive the roots bk for 2-step
Adams-Bashforth method and 2-step Adams Moulton
method, and show the region of absolute stability
on the complex plane.
25
Stiffness ratio Suppose lk is the set of
characteristic expoennts associated with a
particular ODE, stiffness ratio is defined by
Methods to efficiently compute a solution of
stiff ODE are required to have regions of
absolute stability as large as possible.
Possibly whole z lt 0 plane.
Definition (A-stable, Dahlquist) A numerical
method is A-stable if it is absolutely stable for
all hl such that Re hl lt 0.

• Some known results
• Explicit Runge-Kutta methods are not A-stable.
• Explicit linear multistep methods are not
  A-stable.
• Order of A-stable implicit multistep methods is
  less than 2.
• Among the A-stable implicit multistep methods,
  the trapezoidal method has the smallest local
  truncation error.
• Any symmetric s-stage and O( h2s ) order implicit
  Runge-Kutta method (implicit Gauss formula) are
  A-stable. Also Radau, Lobatto formulas.