Trees and Numerical Methods for
Ordinary Differential Equations
John Butcher
The University of Auckland
New Zealand
Università degli Studi di Salerno
9 September 2008
Trees and numerical methods for ordinary differential equations – p. 1/41
Contents
What are trees?
What are differential equations?
What are Runge–Kutta methods?
Trees and numerical methods for ordinary differential equations – p. 2/41
Contents
What are trees?
What are differential equations?
What are Runge–Kutta methods?
Order of methods: First approach
Order of methods: Second approach
Trees and numerical methods for ordinary differential equations – p. 2/41
Contents
What are trees?
What are differential equations?
What are Runge–Kutta methods?
Order of methods: First approach
Order of methods: Second approach
Taylor expansion of exact solution
Taylor expansion for numerical approximation
Order conditions
Trees and numerical methods for ordinary differential equations – p. 2/41
Contents
What are trees?
What are differential equations?
What are Runge–Kutta methods?
Order of methods: First approach
Order of methods: Second approach
Taylor expansion of exact solution
Taylor expansion for numerical approximation
Order conditions
Fourth order methods
A method of order 4 or 5
Algebraic interpretation
Trees and numerical methods for ordinary differential equations – p. 2/41
Contents
What are trees?
What are differential equations?
What are Runge–Kutta methods?
Order of methods: First approach
Order of methods: Second approach
Taylor expansion of exact solution
Taylor expansion for numerical approximation
Order conditions
Fourth order methods
A method of order 4 or 5
Algebraic interpretation
Appendix
Trees and numerical methods for ordinary differential equations – p. 2/41
What are trees?
In this talk Trees means Rooted Trees.
I will just give diagrams for the first few (rooted) trees
and hope this is enough of an explanation.
Trees and numerical methods for ordinary differential equations – p. 3/41
What are trees?
In this talk Trees means Rooted Trees.
I will just give diagrams for the first few (rooted) trees
and hope this is enough of an explanation.
Trees and numerical methods for ordinary differential equations – p. 3/41
What are trees?
In this talk Trees means Rooted Trees.
I will just give diagrams for the first few (rooted) trees
and hope this is enough of an explanation.
In each case the root is at the bottom of the diagram.
Trees and numerical methods for ordinary differential equations – p. 3/41
What are trees?
In this talk Trees means Rooted Trees.
I will just give diagrams for the first few (rooted) trees
and hope this is enough of an explanation.
In each case the root is at the bottom of the diagram.
There are applications in genealogies and the
representation of nested arithmetic operations
Trees and numerical methods for ordinary differential equations – p. 3/41
What are trees?
In this talk Trees means Rooted Trees.
I will just give diagrams for the first few (rooted) trees
and hope this is enough of an explanation.
In each case the root is at the bottom of the diagram.
There are applications in genealogies and the
representation of nested arithmetic operations
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Trees and numerical methods for ordinary differential equations – p. 3/41
What are trees?
In this talk Trees means Rooted Trees.
I will just give diagrams for the first few (rooted) trees
and hope this is enough of an explanation.
In each case the root is at the bottom of the diagram.
There are applications in genealogies and the
representation of nested arithmetic operations
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a
Z
rg
Ma
ie
ald la
h
r
n
p
e
El
Fin So
Em
ael
h
c
Mi
ite
r
e
u
J oh
n
An
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y
x
z
+
∗
(x+y)∗z
Trees and numerical methods for ordinary differential equations – p. 3/41
What are differential equations?
In this talk, differential equations are initial value
problems such as
y ′ (x) = f (y(x)),
y(x0 ) = y0 ,
f : RN → RN
(1)
or
y ′ (x) = f (x, y(x)), y(x0 ) = y0 , f : R×RN → RN (2)
Trees and numerical methods for ordinary differential equations – p. 4/41
What are differential equations?
In this talk, differential equations are initial value
problems such as
y ′ (x) = f (y(x)),
y(x0 ) = y0 ,
f : RN → RN
(1)
or
y ′ (x) = f (x, y(x)), y(x0 ) = y0 , f : R×RN → RN (2)
It looks deceptively easy to assume that N = 1 but this
doesn’t capture all the flavour of the problem, even for
problem (2).
Trees and numerical methods for ordinary differential equations – p. 4/41
What are differential equations?
In this talk, differential equations are initial value
problems such as
y ′ (x) = f (y(x)),
y(x0 ) = y0 ,
f : RN → RN
(1)
or
y ′ (x) = f (x, y(x)), y(x0 ) = y0 , f : R×RN → RN (2)
It looks deceptively easy to assume that N = 1 but this
doesn’t capture all the flavour of the problem, even for
problem (2).
If we make no such assumption, then (1) is simpler than
(2) and really just as general, because we can always add
an additional differential equation whose solution is x.
Trees and numerical methods for ordinary differential equations – p. 4/41
What are Runge–Kutta methods?
In terms of the autonomous initial value system
y ′ (x) = f (y(x)), y(x0 ) = y0 , the aim of a Runge–Kutta
method is to calculate approximations:
y1 ≈ y(x1 ),
y2 ≈ y(x2 ),
....
The basic example is the simple Euler method
yn = yn−1 + hf (yn−1 ),
h = xn − xn−1
Trees and numerical methods for ordinary differential equations – p. 5/41
What are Runge–Kutta methods?
In terms of the autonomous initial value system
y ′ (x) = f (y(x)), y(x0 ) = y0 , the aim of a Runge–Kutta
method is to calculate approximations:
y1 ≈ y(x1 ),
y2 ≈ y(x2 ),
....
The basic example is the simple Euler method
yn = yn−1 + hf (yn−1 ),
h = xn − xn−1
This can be made more accurate by using
the
mid-point
quadrature formula:
1
yn = yn−1 + hf yn−1 + 2 hf (yn−1 ) .
Trees and numerical methods for ordinary differential equations – p. 5/41
What are Runge–Kutta methods?
In terms of the autonomous initial value system
y ′ (x) = f (y(x)), y(x0 ) = y0 , the aim of a Runge–Kutta
method is to calculate approximations:
y1 ≈ y(x1 ),
y2 ≈ y(x2 ),
....
The basic example is the simple Euler method
yn = yn−1 + hf (yn−1 ),
h = xn − xn−1
This can be made more accurate by using either the
mid-point or the trapezoidal rule quadrature formula:
1
yn = yn−1 + hf yn−1 + 2 hf (yn−1 ) .
yn = yn−1 + 21 hf (yn−1 ) + 12 hf yn−1 + hf (yn−1 ) .
Trees and numerical methods for ordinary differential equations – p. 5/41
These methods from Runge’s 1895 paper are “second
order” because the error in a single step behaves like
O(h3 ). At a specific output point the error is O(h2 ).
Trees and numerical methods for ordinary differential equations – p. 6/41
These methods from Runge’s 1895 paper are “second
order” because the error in a single step behaves like
O(h3 ). At a specific output point the error is O(h2 ).
It is recognised that for accurate calculations, high order
is better than low order and Runge’s work was followed
by contributions by Heun, Kutta, Nyström and others,
pushing the available order up to 3, 4, 5 and higher.
Trees and numerical methods for ordinary differential equations – p. 6/41
These methods from Runge’s 1895 paper are “second
order” because the error in a single step behaves like
O(h3 ). At a specific output point the error is O(h2 ).
It is recognised that for accurate calculations, high order
is better than low order and Runge’s work was followed
by contributions by Heun, Kutta, Nyström and others,
pushing the available order up to 3, 4, 5 and higher.
Our aim now will be to formulate a typical step in an “s
stage method” and then find criteria for this method to
have a specific order.
Trees and numerical methods for ordinary differential equations – p. 6/41
In carrying out a step we evaluate s stage values
Y1 ,
Y2 ,
...,
Ys
and s stage derivatives
F1 , F2 ,
...,
Fs ,
using the formula Fi = f (Yi ).
Trees and numerical methods for ordinary differential equations – p. 7/41
In carrying out a step we evaluate s stage values
Y1 ,
Y2 ,
...,
Ys
and s stage derivatives
F1 , F2 ,
...,
Fs ,
using the formula Fi = f (Yi ).
Each Yi is found as a linear combination of the Fj added
on to y0 :
s
X
Yi = y0 + h
aij Fj
j=1
Trees and numerical methods for ordinary differential equations – p. 7/41
In carrying out a step we evaluate s stage values
Y1 ,
Y2 ,
...,
Ys
and s stage derivatives
F1 , F2 ,
...,
Fs ,
using the formula Fi = f (Yi ).
Each Yi is found as a linear combination of the Fj added
on to y0 :
s
X
Yi = y0 + h
aij Fj ≈ y(x0 + ci h)
j=1
Trees and numerical methods for ordinary differential equations – p. 7/41
In carrying out a step we evaluate s stage values
Y1 ,
Y2 ,
...,
Ys
and s stage derivatives
F1 , F2 ,
...,
Fs ,
using the formula Fi = f (Yi ).
Each Yi is found as a linear combination of the Fj added
on to y0 :
s
X
Yi = y0 + h
aij Fj ≈ y(x0 + ci h)
j=1
and the approximation at x1 = x0 + h is found from
s
X
bi Fi
y1 = y0 + h
i=1
Trees and numerical methods for ordinary differential equations – p. 7/41
In carrying out a step we evaluate s stage values
Y1 ,
Y2 ,
...,
Ys
and s stage derivatives
F1 , F2 ,
...,
Fs ,
using the formula Fi = f (Yi ).
Each Yi is found as a linear combination of the Fj added
on to y0 :
s
X
Yi = y0 + h
aij Fj ≈ y(x0 + ci h)
j=1
and the approximation at x1 = x0 + h is found from
s
X
bi Fi ≈ y(x0 + h).
y1 = y0 + h
i=1
Trees and numerical methods for ordinary differential equations – p. 7/41
We represent the method by a tableau:
c1 a11
c2 a21
.. ..
. .
cs as1
b1
a12
a22
..
.
as2
b2
· · · a1s
· · · a2s
..
.
· · · ass
· · · bs
Trees and numerical methods for ordinary differential equations – p. 8/41
We represent the method by a tableau:
c1 a11 a12 · · · a1s
c2 a21 a22 · · · a2s
.. ..
..
..
. .
.
.
cs as1 as2 · · · ass
b1 b2 · · · bs
or, if the method is explicit, by the simplified tableau
0
c2 a21
.. . . .
.. ..
.
. .
cs as1 as2 · · · as,s−1
b1 b2 · · · bs−1 bs
Trees and numerical methods for ordinary differential equations – p. 8/41
Examples:
1
y1 = y0 + 0hf (y0 ) + 1hf y0 + 2 hf (y0 )
0
1
2
1
2
0 1
Trees and numerical methods for ordinary differential equations – p. 9/41
Examples:
1
y1 = y0 + 0hf (y0 ) + 1hf y0 + 2 hf (y0 )
Y1
0
1
2
1
2
Y2
0 1
Trees and numerical methods for ordinary differential equations – p. 9/41
Examples:
1
y1 = y0 + 0hf (y0 ) + 1hf y0 + 2 hf (y0 )
0
Y1
1
2
Y2
1
2
0 1
y1 = y0 +
1
hf (y0 )
2
+
1
hf
2
y0 + 1hf (y0 )
0
1 1
1
2
1
2
Trees and numerical methods for ordinary differential equations – p. 9/41
Examples:
1
y1 = y0 + 0hf (y0 ) + 1hf y0 + 2 hf (y0 )
0
Y1
1
2
Y2
1
2
0 1
y1 = y0 +
Y1
1
hf (y0 )
2
+
1
hf
2
y0 + 1hf (y0 )
0
1 1
1
2
Y2
1
2
Trees and numerical methods for ordinary differential equations – p. 9/41
Order of methods: First approach
This approach has a proud history andv has been used by
Runge, Heun, Kutta, Nyström and Huta to obtain
methods up to order 6.
Trees and numerical methods for ordinary differential equations – p. 10/41
Order of methods: First approach
This approach has a proud history andv has been used by
Runge, Heun, Kutta, Nyström and Huta to obtain
methods up to order 6.
We start with the differential equation
Trees and numerical methods for ordinary differential equations – p. 10/41
Order of methods: First approach
This approach has a proud history andv has been used by
Runge, Heun, Kutta, Nyström and Huta to obtain
methods up to order 6.
We start with the differential equation
dy
y(x0 ) = y0
dx = f (x, y),
Trees and numerical methods for ordinary differential equations – p. 10/41
Order of methods: First approach
This approach has a proud history andv has been used by
Runge, Heun, Kutta, Nyström and Huta to obtain
methods up to order 6.
We start with the differential equation
dy
y(x0 ) = y0
dx = f (x, y),
We want the first few terms in the formal Taylor series
about x0 : so we calculate the next few derivatives
y ′′ = fx + f fy ,
′′′
2
y = fxx + 2f fxy + f fyy +
2
f fy
Trees and numerical methods for ordinary differential equations – p. 10/41
Order of methods: First approach
This approach has a proud history andv has been used by
Runge, Heun, Kutta, Nyström and Huta to obtain
methods up to order 6.
We start with the differential equation
dy
y(x0 ) = y0
dx = f (x, y),
We want the first few terms in the formal Taylor series
about x0 : so we calculate the next few derivatives
y ′′ = fx + f fy ,
′′′
2
y = fxx + 2f fxy + f fyy +
2
f fy
Once we have the Taylor series for y(x0 + h) we find
also the series for a numerical approximation and we get
a number of “order conditions”.
Trees and numerical methods for ordinary differential equations – p. 10/41
Order of methods: First approach
This approach has a proud history andv has been used by
Runge, Heun, Kutta, Nyström and Huta to obtain
methods up to order 6.
We start with the differential equation
dy
y(x0 ) = y0
dx = f (x, y),
We want the first few terms in the formal Taylor series
about x0 : so we calculate the next few derivatives
y ′′ = fx + f fy ,
′′′
2
y = fxx + 2f fxy + f fyy +
2
f fy
Once we have the Taylor series for y(x0 + h) we find
also the series for a numerical approximation and we get
a number of “order conditions”.
For order p call the number of conditions Cp .
Trees and numerical methods for ordinary differential equations – p. 10/41
The number of parameters in an explicit Runge–Kutta
method with s stages is s(s + 1)/2. It might be expected
that s should be chosen so that there are more parameters
than conditions, as in the table
p
1
2
3
4
Cp
1
2
4
8
s
1
2
3
4
s(s+1)
2
5
16
6
21
6
31
8
36
1
3
6
10
Trees and numerical methods for ordinary differential equations – p. 11/41
The number of parameters in an explicit Runge–Kutta
method with s stages is s(s + 1)/2. It might be expected
that s should be chosen so that there are more parameters
than conditions, as in the table
p
1
2
3
4
Cp
1
2
4
8
s
1
2
3
4
s(s+1)
2
5
6
16
31
6
7
8
21
28
36
1
3
6
10
Surprisingly, order 6 can be achieved with s = 7
Trees and numerical methods for ordinary differential equations – p. 11/41
Order of methods: Second approach
We make two changes; these are
1. Remove x from f (x, y) — in other words make the
problem autonomous
Trees and numerical methods for ordinary differential equations – p. 12/41
Order of methods: Second approach
We make two changes; these are
1. Remove x from f (x, y) — in other words make the
problem autonomous
2. Regard y as a vector-valued function
Trees and numerical methods for ordinary differential equations – p. 12/41
Order of methods: Second approach
We make two changes; these are
1. Remove x from f (x, y) — in other words make the
problem autonomous
2. Regard y as a vector-valued function
The first change seems to make everything simpler
because all terms involving fx , fxy etc just disappear.
Trees and numerical methods for ordinary differential equations – p. 12/41
Order of methods: Second approach
We make two changes; these are
1. Remove x from f (x, y) — in other words make the
problem autonomous
2. Regard y as a vector-valued function
The first change seems to make everything simpler
because all terms involving fx , fxy etc just disappear.
The second change seems to make everything more
complicated, because terms like f 2 fy fyy , which are
identical in one dimension, have to be distinguished from
each other in N dimensions.
Trees and numerical methods for ordinary differential equations – p. 12/41
How are we going to write these complicated terms
involving compositions of what will be linear and
multilinear operators?
Trees and numerical methods for ordinary differential equations – p. 13/41
How are we going to write these complicated terms
involving compositions of what will be linear and
multilinear operators?
It is possible to work in terms of tensors but I prefer to
use Frechet derivatives.
Trees and numerical methods for ordinary differential equations – p. 13/41
How are we going to write these complicated terms
involving compositions of what will be linear and
multilinear operators?
It is possible to work in terms of tensors but I prefer to
use Frechet derivatives.
I will now write the first, second and third derivatives in
this way with the one-dimensional formulae written
beside them for comparison.
y ′ = f,
y′ = f
y ′′ = fx + f fy ,
y ′′ = f ′ f
y ′′′ = fxx + 2f fxy + f 2 fyy + f fy2 , y ′′′ = f ′′ (f, f) + f ′ f ′ f
where f = f (y(x)), f ′ = f ′ (y(x)) etc.
Trees and numerical methods for ordinary differential equations – p. 13/41
If we count the number of order conditions in this second
formulation — call this number Dp for the moment —
and include Cp for comparison, we get this table
Trees and numerical methods for ordinary differential equations – p. 14/41
If we count the number of order conditions in this second
formulation — call this number Dp for the moment —
and include Cp for comparison, we get this table
p
Cp D p
1
2
3
4
5
6
7
8
1
2
4
8
16
31
1
2
4
8
17
37
85
200
Trees and numerical methods for ordinary differential equations – p. 14/41
If we count the number of order conditions in this second
formulation — call this number Dp for the moment —
and include Cp for comparison, we get this table
p
Cp D p
1
1
1
2
2
2
3
4
4
4
8
8
5
16 17
6
31 37
7
85
8
200
I don’t know if C7 , C8 , . . . are known
Trees and numerical methods for ordinary differential equations – p. 14/41
If we count the number of order conditions in this second
formulation — call this number Dp for the moment —
and include Cp for comparison, we get this table
p
Cp D p
1
1
1
2
2
2
3
4
4
4
8
8
5
16 17
6
31 37
7
85
8
200
I don’t know if C7 , C8 , . . . are known but I don’t think
we should really care what their values are.
Trees and numerical methods for ordinary differential equations – p. 14/41
In the second approach, we again list the formulae for the
derivatives; but this time go as far as order 4. Note the
tree-like structures:
Trees and numerical methods for ordinary differential equations – p. 15/41
In the second approach, we again list the formulae for the
derivatives; but this time go as far as order 4. Note the
tree-like structures:
y ′ = f,
f
Trees and numerical methods for ordinary differential equations – p. 15/41
In the second approach, we again list the formulae for the
derivatives; but this time go as far as order 4. Note the
tree-like structures:
y ′ = f,
f
y ′′ = f ′ f,
f′
f
Trees and numerical methods for ordinary differential equations – p. 15/41
In the second approach, we again list the formulae for the
derivatives; but this time go as far as order 4. Note the
tree-like structures:
y ′ = f,
f
y ′′ = f ′ f,
y ′′′ = f ′′ (f, f)
+ f ′ f ′ f,
f′
f
f f′′
f
f′
f′
f
Trees and numerical methods for ordinary differential equations – p. 15/41
In the second approach, we again list the formulae for the
derivatives; but this time go as far as order 4. Note the
tree-like structures:
y ′ = f,
f
f′
f
f f′′
f
y ′′ = f ′ f,
y ′′′ = f ′′ (f, f)
f′
f′
f
+ f ′ f ′ f,
y
(4)
f f f
f ′′′
′′′
= f (f, f, f)
+ 3f ′′ (f, f ′ f)
f
f
+ f ′ f ′′ (f, f)
′ ′ ′
+f f f f
f′
f′′
f
f′′
f′
f
f′
f′
f′
f
Trees and numerical methods for ordinary differential equations – p. 15/41
The plan now is to formulate Runge–Kutta methods in a
systematic way and to explore the order conditions.
Trees and numerical methods for ordinary differential equations – p. 16/41
The plan now is to formulate Runge–Kutta methods in a
systematic way and to explore the order conditions.
This will all be done using what I am calling the second
approach.
Trees and numerical methods for ordinary differential equations – p. 16/41
The plan now is to formulate Runge–Kutta methods in a
systematic way and to explore the order conditions.
This will all be done using what I am calling the second
approach.
Later, right at the end, I plan to come back to the first
approach and show why I don’t think it is completely
satisfactory.
Trees and numerical methods for ordinary differential equations – p. 16/41
The plan now is to formulate Runge–Kutta methods in a
systematic way and to explore the order conditions.
This will all be done using what I am calling the second
approach.
Later, right at the end, I plan to come back to the first
approach and show why I don’t think it is completely
satisfactory.
Before we go on, note that the numbers Dp , is just the
number of rooted trees with no more than p vertices.
Trees and numerical methods for ordinary differential equations – p. 16/41
Taylor expansion of exact solution
Recalculate formulae for the second, third, . . . ,
derivatives.
Trees and numerical methods for ordinary differential equations – p. 17/41
Taylor expansion of exact solution
Recalculate formulae for the second, third, . . . ,
derivatives.
y ′ (x) = f (y(x))
Trees and numerical methods for ordinary differential equations – p. 17/41
Taylor expansion of exact solution
Recalculate formulae for the second, third, . . . ,
derivatives.
y ′ (x) = f (y(x))
y ′′ (x) = f ′ (y(x))y ′ (x)
= f ′ (y(x))f (y(x)))
Trees and numerical methods for ordinary differential equations – p. 17/41
Taylor expansion of exact solution
Recalculate formulae for the second, third, . . . ,
derivatives.
y ′ (x) = f (y(x))
y ′′ (x) = f ′ (y(x))y ′ (x)
= f ′ (y(x))f (y(x)))
′′′
′′
′
′
′
′
y (x) = f (y(x))(f (y(x)), y (x)) + f (y(x))f (y(x))y (x)
=f ′′ (y(x))(f (y(x)), f (y(x)))+f ′ (y(x))f ′ (y(x))f (y(x))
Trees and numerical methods for ordinary differential equations – p. 17/41
Taylor expansion of exact solution
Recalculate formulae for the second, third, . . . ,
derivatives.
y ′ (x) = f (y(x))
y ′′ (x) = f ′ (y(x))y ′ (x)
= f ′ (y(x))f (y(x)))
′′′
′′
′
′
′
′
y (x) = f (y(x))(f (y(x)), y (x)) + f (y(x))f (y(x))y (x)
=f ′′ (y(x))(f (y(x)), f (y(x)))+f ′ (y(x))f ′ (y(x))f (y(x))
As we have found out, this becomes increasingly
complicated as we evaluate higher derivatives.
Trees and numerical methods for ordinary differential equations – p. 17/41
Taylor expansion of exact solution
Recalculate formulae for the second, third, . . . ,
derivatives.
y ′ (x) = f (y(x))
y ′′ (x) = f ′ (y(x))y ′ (x)
= f ′ (y(x))f (y(x)))
′′′
′′
′
′
′
′
y (x) = f (y(x))(f (y(x)), y (x)) + f (y(x))f (y(x))y (x)
=f ′′ (y(x))(f (y(x)), f (y(x)))+f ′ (y(x))f ′ (y(x))f (y(x))
As we have found out, this becomes increasingly
complicated as we evaluate higher derivatives.
Hence we return to the known systematic pattern:
Trees and numerical methods for ordinary differential equations – p. 17/41
Taylor expansion of exact solution
Recalculate formulae for the second, third, . . . ,
derivatives.
y ′ (x) = f (y(x))
y ′′ (x) = f ′ (y(x))y ′ (x)
= f ′ (y(x))f (y(x)))
′′′
′′
′
′
′
′
y (x) = f (y(x))(f (y(x)), y (x)) + f (y(x))f (y(x))y (x)
=f ′′ (y(x))(f (y(x)), f (y(x)))+f ′ (y(x))f ′ (y(x))f (y(x))
As we have found out, this becomes increasingly
complicated as we evaluate higher derivatives.
Hence we return to the known systematic pattern:
Recall the notation f = f (y(x)), f′ = f ′ (y(x)),
f′′ = f ′′ (y(x)), . . . .
Trees and numerical methods for ordinary differential equations – p. 17/41
′
y (x) = f
f
′′
′
f
f′
′′′
′′
y (x) = f f
y (x) = f (f, f)
+ f′ f′ f
f
f
f′′
f
f′
f′
Trees and numerical methods for ordinary differential equations – p. 18/41
′
y (x) = f
f
′′
′
f
f′
′′′
′′
y (x) = f f
y (x) = f (f, f)
+ f′ f′ f
f
f
f′′
f
f′
f′
The various terms have a structure related to rooted-trees.
Trees and numerical methods for ordinary differential equations – p. 18/41
′
y (x) = f
f
′′
′
f
f′
′′′
′′
y (x) = f f
y (x) = f (f, f)
+ f′ f′ f
f
f
f′′
f
f′
f′
The various terms have a structure related to rooted-trees.
Hence, we introduce the set of all rooted trees and some
functions on this set.
Trees and numerical methods for ordinary differential equations – p. 18/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
Trees and numerical methods for ordinary differential equations – p. 19/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
In this table, t will denote a typical tree
Trees and numerical methods for ordinary differential equations – p. 19/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
In this table, t will denote a typical tree
r(t)
order of t = number of vertices
Trees and numerical methods for ordinary differential equations – p. 19/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
In this table, t will denote a typical tree
r(t)
order of t = number of vertices
σ(t)
symmetry of t = order of automorphism group
Trees and numerical methods for ordinary differential equations – p. 19/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
In this table, t will denote a typical tree
r(t)
order of t = number of vertices
σ(t)
symmetry of t = order of automorphism group
γ(t)
density of t
Trees and numerical methods for ordinary differential equations – p. 19/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
In this table, t will denote a typical tree
r(t)
order of t = number of vertices
σ(t)
symmetry of t = order of automorphism group
γ(t)
density of t
α(t)
number of ways of labelling with an ordered set
Trees and numerical methods for ordinary differential equations – p. 19/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
In this table, t will denote a typical tree
r(t)
order of t = number of vertices
σ(t)
symmetry of t = order of automorphism group
γ(t)
density of t
α(t)
number of ways of labelling with an ordered set
β(t)
number of ways of labelling with an unordered set
Trees and numerical methods for ordinary differential equations – p. 19/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
In this table, t will denote a typical tree
r(t)
order of t = number of vertices
σ(t)
symmetry of t = order of automorphism group
γ(t)
density of t
α(t)
number of ways of labelling with an ordered set
β(t)
number of ways of labelling with an unordered set
F (t)(y0 ) elementary differential
Trees and numerical methods for ordinary differential equations – p. 19/41
Let T denote the set of rooted trees:
n
T =
,
,
,
,
,
,
,
,
...
o
We identify the following functions on T .
In this table, t will denote a typical tree
r(t)
order of t = number of vertices
σ(t)
symmetry of t = order of automorphism group
γ(t)
density of t
α(t)
number of ways of labelling with an ordered set
β(t)
number of ways of labelling with an unordered set
F (t)(y0 ) elementary differential
We will give examples of these functions based on t =
Trees and numerical methods for ordinary differential equations – p. 19/41
t =
Trees and numerical methods for ordinary differential equations – p. 20/41
t =
1 23 4
r(t) = 7
5
6
7
Trees and numerical methods for ordinary differential equations – p. 20/41
t =
1 23 4
r(t) = 7
5
6
7
σ(t) = 8
Trees and numerical methods for ordinary differential equations – p. 20/41
t =
1 23 4
r(t) = 7
5
6
7
σ(t) = 8
1 11 1
γ(t) = 63
3
3
7
Trees and numerical methods for ordinary differential equations – p. 20/41
t =
1 23 4
5
r(t) = 7
6
7
σ(t) = 8
1 11 1
3
γ(t) = 63
α(t) =
r(t)!
σ(t)γ(t)
3
7
= 10
Trees and numerical methods for ordinary differential equations – p. 20/41
t =
1 23 4
5
r(t) = 7
6
7
σ(t) = 8
1 11 1
3
γ(t) = 63
3
7
α(t) =
r(t)!
σ(t)γ(t)
= 10
β(t) =
r(t)!
σ(t)
= 630
Trees and numerical methods for ordinary differential equations – p. 20/41
t =
1 23 4
5
r(t) = 7
6
7
σ(t) = 8
1 11 1
3
γ(t) = 63
3
7
α(t) =
r(t)!
σ(t)γ(t)
= 10
β(t) =
r(t)!
σ(t)
= 630
f f f f
′′
′′
′′
F (t) = f f (f, f), f (f, f)
f′′
f′′
f′′
Trees and numerical methods for ordinary differential equations – p. 20/41
These functions are easy to compute up to order 4 trees:
t
r(t) 1 2
3
3
4
4
4
4
σ(t) 1 1
2
1
6
1
2
1
γ(t) 1 2
3
6
4
8
12
24
α(t) 1 1
1
1
1
3
1
1
β(t) 1 2
3
6
4
24
12
24
′
′′
′ ′
′′′
′′
′
′ ′′
′ ′ ′
F (t) f f f f (f, f) f f f f (f, f, f) f (f, f f) f f (f, f) f f f f
Trees and numerical methods for ordinary differential equations – p. 21/41
The formal Taylor expansion of the solution at x0 + h is
y(x0 + h) = y0 +
X α(t)hr(t)
t∈T
r(t)!
F (t)(y0 )
Trees and numerical methods for ordinary differential equations – p. 22/41
The formal Taylor expansion of the solution at x0 + h is
y(x0 + h) = y0 +
X α(t)hr(t)
t∈T
r(t)!
F (t)(y0 )
Using the known formula for α(t), we can write this as
y(x0 + h) = y0 +
X
t∈T
hr(t)
F (t)(y0 )
σ(t)γ(t)
Trees and numerical methods for ordinary differential equations – p. 22/41
The formal Taylor expansion of the solution at x0 + h is
y(x0 + h) = y0 +
X α(t)hr(t)
t∈T
r(t)!
F (t)(y0 )
Using the known formula for α(t), we can write this as
y(x0 + h) = y0 +
X
t∈T
hr(t)
F (t)(y0 )
σ(t)γ(t)
Our aim will now be to find a corresponding formula for
the result computed by one step of a Runge-Kutta
method.
Trees and numerical methods for ordinary differential equations – p. 22/41
The formal Taylor expansion of the solution at x0 + h is
y(x0 + h) = y0 +
X α(t)hr(t)
t∈T
r(t)!
F (t)(y0 )
Using the known formula for α(t), we can write this as
y(x0 + h) = y0 +
X
t∈T
hr(t)
F (t)(y0 )
σ(t)γ(t)
Our aim will now be to find a corresponding formula for
the result computed by one step of a Runge-Kutta
method.
By comparing these formulae term by term, we will
obtain conditions for a specific order of accuracy.
Trees and numerical methods for ordinary differential equations – p. 22/41
Taylor expansion for numerical approximation
We need to evaluate various expressions which depend
on the tableau for a particular method.
Trees and numerical methods for ordinary differential equations – p. 23/41
Taylor expansion for numerical approximation
We need to evaluate various expressions which depend
on the tableau for a particular method.
These are known as “elementary weights”.
Trees and numerical methods for ordinary differential equations – p. 23/41
Taylor expansion for numerical approximation
We need to evaluate various expressions which depend
on the tableau for a particular method.
These are known as “elementary weights”.
We use the example tree we have already considered to
illustrate the construction of the elementary weight Φ(t).
l
t=
m n
j
o
k
i
Trees and numerical methods for ordinary differential equations – p. 23/41
Taylor expansion for numerical approximation
We need to evaluate various expressions which depend
on the tableau for a particular method.
These are known as “elementary weights”.
We use the example tree we have already considered to
illustrate the construction of the elementary weight Φ(t).
l
t=
Φ(t) =
s
X
m n
j
o
k
i
bi aij ajl ajmaik aknako
i,j,k,l,m,n,o=1
Trees and numerical methods for ordinary differential equations – p. 23/41
Taylor expansion for numerical approximation
We need to evaluate various expressions which depend
on the tableau for a particular method.
These are known as “elementary weights”.
We use the example tree we have already considered to
illustrate the construction of the elementary weight Φ(t).
l
t=
Φ(t) =
m n
j
s
X
o
k
i
bi aij ajl ajmaik aknako
i,j,k,l,m,n,o=1
Simplify by summing over l, m, n, o:
Φ(t) =
s
X
2
2
biaij cj aik ck
i,j,k=1
Trees and numerical methods for ordinary differential equations – p. 23/41
Now add Φ(t) to the table of functions:
t
r(t)
α(t)
β(t)
Φ(t)
1
1
1
P
bi
2
1
2
P
bi ci
3
1
3
P 2
bi ci
3
1
6
P
bi aij cj
t
r(t)
4
4
4
4
1
3
1
1
α(t)
β(t)
4
24
12
24
P
P
P 3 P
2
Φ(t)
bi aij cj
bi aij ajk ck
bi ci
bi ci aij cj
Trees and numerical methods for ordinary differential equations – p. 24/41
The formal Taylor expansion of the computed solution at
x0 + h is
y1 = y0 +
X β(t)hr(t)
t∈T
r(t)!
Φ(t)F (t)(y0 )
Trees and numerical methods for ordinary differential equations – p. 25/41
The formal Taylor expansion of the computed solution at
x0 + h is
y1 = y0 +
X β(t)hr(t)
t∈T
r(t)!
Φ(t)F (t)(y0 )
Using the known formula for β(t), we can write this as
y1 = y0 +
X hr(t)
t∈T
σ(t)
Φ(t)F (t)(y0 )
Trees and numerical methods for ordinary differential equations – p. 25/41
Order conditions
To match the Taylor series
y(x0 + h) = y0 +
X
t∈T
y1 = y0 +
hr(t)
F (t)(y0 )
σ(t)γ(t)
X hr(t)
t∈T
σ(t)
Φ(t)F (t)(y0 )
up to hp terms we need to ensure that
1
Φ(t) =
,
γ(t)
Trees and numerical methods for ordinary differential equations – p. 26/41
Order conditions
To match the Taylor series
y(x0 + h) = y0 +
X
t∈T
y1 = y0 +
hr(t)
F (t)(y0 )
σ(t)γ(t)
X hr(t)
t∈T
σ(t)
Φ(t)F (t)(y0 )
up to hp terms we need to ensure that
for all trees such that
1
Φ(t) =
,
γ(t)
r(t) ≤ p.
Trees and numerical methods for ordinary differential equations – p. 26/41
Order conditions
To match the Taylor series
y(x0 + h) = y0 +
X
t∈T
y1 = y0 +
hr(t)
F (t)(y0 )
σ(t)γ(t)
X hr(t)
t∈T
σ(t)
Φ(t)F (t)(y0 )
up to hp terms we need to ensure that
for all trees such that
1
Φ(t) =
,
γ(t)
r(t) ≤ p.
These are the “order conditions”.
Trees and numerical methods for ordinary differential equations – p. 26/41
Fourth order methods
The definitive work on fourth order methods is Kutta’s
1901 paper.
Trees and numerical methods for ordinary differential equations – p. 27/41
Fourth order methods
The definitive work on fourth order methods is Kutta’s
1901 paper.
Kutta found several families of methods; today I will
review only methods based on Simpson’s rule:
0
1
2
1
2
1
2
1
2
− a32
1 1 − a42 − a43
1
6
a32
a42 a43
2
3 − b3 b3
1
6
Trees and numerical methods for ordinary differential equations – p. 27/41
Fourth order methods
The definitive work on fourth order methods is Kutta’s
1901 paper.
Kutta found several families of methods; today I will
review only methods based on Simpson’s rule:
0
1
2
1
2
1
2
1
2
− a32
1 1 − a42 − a43
a32
a42 a43
1
2
1
−
b
b
3
3 6
6
3
Because the underlying quadrature formula has order 4,
some order conditions are automatically satisfied:
Trees and numerical methods for ordinary differential equations – p. 27/41
It is necessary in fourth order methods with 4 stages that
P
−1
a4j = b4 (bj (1 − cj ) − i<4 bi aij )
Trees and numerical methods for ordinary differential equations – p. 28/41
It is necessary in fourth order methods with 4 stages that
P
−1
a4j = b4 (bj (1 − cj ) − i<4 bi aij )
and it also implies that three further trees become
automatically satisfied:
0
1
2
1
2
1
2
1
2
− a32
a32
1 6b3 a32 −1 2−3b3 −6b3 a32 3b3
1
2
b3
6
3 − b3
1
6
Trees and numerical methods for ordinary differential equations – p. 28/41
It is necessary in fourth order methods with 4 stages that
P
−1
a4j = b4 (bj (1 − cj ) − i<4 bi aij )
and it also implies that three further trees become
automatically satisfied:
0
1
2
1
2
1
2
1
2
− a32
a32
1 6b3 a32 −1 2−3b3 −6b3 a32 3b3
1
2
b3
6
3 − b3
1
6
Trees and numerical methods for ordinary differential equations – p. 28/41
The remaining order condition is
X
bi aij cj =
which reduces to
1
8
b3 (1 − a32 ) =
1
6
Trees and numerical methods for ordinary differential equations – p. 29/41
The remaining order condition is
X
bi aij cj =
which reduces to
1
8
b3 (1 − a32 ) =
1
6
Substitute a32 = 1 − 1/6b3 and the method becomes
0
1
2
1
2
1
2
− 12 1 − 6b13
1 6b3 − 2 3 − 9b3 3b3
1
2
b3
6
3 − b3
1
6b3
1
6
Trees and numerical methods for ordinary differential equations – p. 29/41
What about the value of b3 ?
Trees and numerical methods for ordinary differential equations – p. 30/41
What about the value of b3 ?
√
In 1951, S. Gill proposed a new value b3 = 31 + 62 ,
because this was more efficient in terms of memory than
the classical choice proposed by Kutta 50 years earlier.
Trees and numerical methods for ordinary differential equations – p. 30/41
What about the value of b3 ?
√
In 1951, S. Gill proposed a new value b3 = 31 + 62 ,
because this was more efficient in terms of memory than
the classical choice proposed by Kutta 50 years earlier.
But everyone seems to like the classical method of Kutta
with b3 = 31 .
Trees and numerical methods for ordinary differential equations – p. 30/41
What about the value of b3 ?
√
In 1951, S. Gill proposed a new value b3 = 31 + 62 ,
because this was more efficient in terms of memory than
the classical choice proposed by Kutta 50 years earlier.
But everyone seems to like the classical method of Kutta
with b3 = 31 .
It is simple, easy to remember and works well:
Trees and numerical methods for ordinary differential equations – p. 30/41
What about the value of b3 ?
√
In 1951, S. Gill proposed a new value b3 = 31 + 62 ,
because this was more efficient in terms of memory than
the classical choice proposed by Kutta 50 years earlier.
But everyone seems to like the classical method of Kutta
with b3 = 31 .
It is simple, easy to remember and works well:
0
1
2
1
2
1
2
1
2
0
1 0
0
1
1
6
1
3
1
3
1
6
Trees and numerical methods for ordinary differential equations – p. 30/41
A method of order 4 or 5
When we were counting order conditions, we saw that in
the “First approach”, there were 16 conditions for order 5
but in the “Second approach” there were 17 conditions.
Trees and numerical methods for ordinary differential equations – p. 31/41
A method of order 4 or 5
When we were counting order conditions, we saw that in
the “First approach”, there were 16 conditions for order 5
but in the “Second approach” there were 17 conditions.
The reason is quite simple: the two elementary
differentials
f ′′ (f, f ′ f ′ f) and
correspond to identical terms
f ′ f ′′ (f, f ′ f)
(f fyy + fxy )fy (f fy + fx )
in the “First approach” formulation.
Trees and numerical methods for ordinary differential equations – p. 31/41
A method of order 4 or 5
When we were counting order conditions, we saw that in
the “First approach”, there were 16 conditions for order 5
but in the “Second approach” there were 17 conditions.
The reason is quite simple: the two elementary
differentials
f ′′ (f, f ′ f ′ f) and
correspond to identical terms
f ′ f ′′ (f, f ′ f)
(f fyy + fxy )fy (f fy + fx )
in the “First approach” formulation.
The consequence is a confusion of two order conditions
P
P
1
1
bi ci aij ajk ck + bi aij cj ajk ck = 30
+ 40
Trees and numerical methods for ordinary differential equations – p. 31/41
In a short 1995 paper, I showed how to construct a
method which satisfies all the equations for order 5 in the
first approach but not for the second approach.
I tried this method out for two closely related problems
dy
y−x
=
,
y(0) = 1
(1)
dx y + x
d y
1
y−x
y(1)
1
,
=
(2)
=p
x(1)
0
dt x
x2 + y 2 y + x
For (1), order 5 was observed numerically but for (2)
only order 4 seems to be achieved.
Trees and numerical methods for ordinary differential equations – p. 32/41
Algebraic interpretation
We will introduce an algebraic system which represents
individual Runge-Kutta methods and also compositions
of methods.
Trees and numerical methods for ordinary differential equations – p. 33/41
Algebraic interpretation
We will introduce an algebraic system which represents
individual Runge-Kutta methods and also compositions
of methods.
This centres on the meaning of order for Runge-Kutta
methods and leads to a possible generalisation to
“effective order”.
Trees and numerical methods for ordinary differential equations – p. 33/41
Algebraic interpretation
We will introduce an algebraic system which represents
individual Runge-Kutta methods and also compositions
of methods.
This centres on the meaning of order for Runge-Kutta
methods and leads to a possible generalisation to
“effective order”.
Denote by G the group consisting of mappings of
(rooted) trees to real numbers where the group operation
is defined according to the algebraic theory of
Runge-Kutta methods or to the (equivalent) theory of
B-series.
Trees and numerical methods for ordinary differential equations – p. 33/41
Algebraic interpretation
We will introduce an algebraic system which represents
individual Runge-Kutta methods and also compositions
of methods.
This centres on the meaning of order for Runge-Kutta
methods and leads to a possible generalisation to
“effective order”.
Denote by G the group consisting of mappings of
(rooted) trees to real numbers where the group operation
is defined according to the algebraic theory of
Runge-Kutta methods or to the (equivalent) theory of
B-series.
We will illustrate this operation in a table
Trees and numerical methods for ordinary differential equations – p. 33/41
Algebraic interpretation
We will introduce an algebraic system which represents
individual Runge-Kutta methods and also compositions
of methods.
This centres on the meaning of order for Runge-Kutta
methods and leads to a possible generalisation to
“effective order”.
Denote by G the group consisting of mappings of
(rooted) trees to real numbers where the group operation
is defined according to the algebraic theory of
Runge-Kutta methods or to the (equivalent) theory of
B-series.
We will illustrate this operation in a table, where we also
introduce the special member E ∈ G.
Trees and numerical methods for ordinary differential equations – p. 33/41
i ti
1
2
3
4
5
6
7
8
Trees and numerical methods for ordinary differential equations – p. 34/41
r(ti )
1
2
3
i ti
1
2
3
3
4
4
5
4
6
4
7
4
8
Trees and numerical methods for ordinary differential equations – p. 34/41
r(ti )
1
2
3
i ti α(ti ) β(ti )
1
α1
β1
2
α2
β2
α3
β3
3
3
4
4
5
α4
α5
β4
β5
4
6
α6
β6
4
7
α7
β7
4
8
α8
β8
Trees and numerical methods for ordinary differential equations – p. 34/41
r(ti )
1
2
3
i ti α(ti ) β(ti )
1
α1
β1
2
α2
β2
α3
β3
3
(αβ)(ti )
α1 + β1
α2 + α1 β1 + β2
α3 + α12 β1 + 2α1 β2 + β3
3
4
4
5
α4
α5
4
6
α6
4
7
α7
β4
α4 + α2 β1 + α1 β2 + β4
β5 α5 + α13 β1 + 3α12 β2 + 3α1 β3 + β5
2
α
+
α
α
β
+
(α
6
1
2
1
1 + α2 )β2
β6
+ α1 (β3 + β4 ) + β6
β7 α7 + α3 β1 + α12 β2 + 2α1 β4 + β7
4
8
α8
β8
α8 + α4 β1 + α2 β2 + α1 β4 + β8
Trees and numerical methods for ordinary differential equations – p. 34/41
r(ti )
1
2
3
i ti α(ti ) β(ti )
1
α1
β1
2
α2
β2
α3
β3
3
3
4
4
5
α4
α5
β4
β5
4
6
α6
β6
4
7
α7
β7
4
8
α8
β8
(αβ)(ti )
α1 + β1
α2 + α1 β1 + β2
α3 + α12 β1 + 2α1 β2 + β3
E(ti )
1
α8 + α4 β1 + α2 β2 + α1 β4 + β8
1
24
1
2
1
3
1
α4 + α2 β1 + α1 β2 + β4
6
α5 + α13 β1 + 3α12 β2 + 3α1 β3 + β5 14
1
α6 + α1 α2 β1 + (α12 + α2 )β2
8
+ α1 (β3 + β4 ) + β6
1
α7 + α3 β1 + α12 β2 + 2α1 β4 + β7 12
Trees and numerical methods for ordinary differential equations – p. 34/41
Gp will denote the normal subgroup defined by t 7→ 0 for
r(t) ≤ p.
Trees and numerical methods for ordinary differential equations – p. 35/41
Gp will denote the normal subgroup defined by t 7→ 0 for
r(t) ≤ p.
If α ∈ G then this maps canonically to αGp ∈ G/Gp.
Trees and numerical methods for ordinary differential equations – p. 35/41
Gp will denote the normal subgroup defined by t 7→ 0 for
r(t) ≤ p.
If α ∈ G then this maps canonically to αGp ∈ G/Gp.
If α is defined from the elementary weights for a
Runge-Kutta method then order p can be written as
αGp = EGp.
Trees and numerical methods for ordinary differential equations – p. 35/41
Gp will denote the normal subgroup defined by t 7→ 0 for
r(t) ≤ p.
If α ∈ G then this maps canonically to αGp ∈ G/Gp.
If α is defined from the elementary weights for a
Runge-Kutta method then order p can be written as
αGp = EGp.
Effective order p is defined by the existence of β such
that
βαGp = EβGp.
Trees and numerical methods for ordinary differential equations – p. 35/41
The computational interpretation of this idea is that we
carry out a starting step corresponding to β
Trees and numerical methods for ordinary differential equations – p. 36/41
The computational interpretation of this idea is that we
carry out a starting step corresponding to β and a
finishing step corresponding to β −1
Trees and numerical methods for ordinary differential equations – p. 36/41
The computational interpretation of this idea is that we
carry out a starting step corresponding to β and a
finishing step corresponding to β −1 , with many steps in
between corresponding to α.
Trees and numerical methods for ordinary differential equations – p. 36/41
The computational interpretation of this idea is that we
carry out a starting step corresponding to β and a
finishing step corresponding to β −1 , with many steps in
between corresponding to α.
This is equivalent to many steps all corresponding to
βαβ −1 .
Trees and numerical methods for ordinary differential equations – p. 36/41
The computational interpretation of this idea is that we
carry out a starting step corresponding to β and a
finishing step corresponding to β −1 , with many steps in
between corresponding to α.
This is equivalent to many steps all corresponding to
βαβ −1 .
Thus, the benefits of high order can be enjoyed by high
effective order.
Trees and numerical methods for ordinary differential equations – p. 36/41
We analyse the conditions for effective order 4.
Without loss of generality assume β(t1 ) = 0.
i
(βα)(ti )
(Eβ)(ti )
1
α1
1
1
2
β2 + α2
2 + β2
1
3
β3 + α3
3 + 2β2 + β3
1
4
β4 + β2 α1 + α4
6 + β2 + β4
1
5
β5 + α5
4 + 3β2 + 3β3 + β5
1
3
6
β6 + β2 α2 + α6
+
8
2 β2 + β3 + β4 + β6
1
7
β7 + β3 α1 + α7
12 + β2 + 2β4 + β7
1
1
+
8 β8 + β4 α1 + β2 α2 + α8
24
2 β2 + β4 + β8
Trees and numerical methods for ordinary differential equations – p. 37/41
Of these 8 conditions, only 5 are conditions on α.
Trees and numerical methods for ordinary differential equations – p. 38/41
Of these 8 conditions, only 5 are conditions on α.
Once α is known, there remain 3 conditions on β.
Trees and numerical methods for ordinary differential equations – p. 38/41
Of these 8 conditions, only 5 are conditions on α.
Once α is known, there remain 3 conditions on β.
The 5 order conditions, written in terms of the
Runge-Kutta tableau, are
X
bi = 1
X
bi ci = 21
X
bi aij cj = 61
X
1
bi aij ajk ck = 24
X
X
bi c2i (1 − ci ) +
bi aij cj (2ci − cj ) = 41
Trees and numerical methods for ordinary differential equations – p. 38/41
Appendix
M
ES
S
S
SM
O(hp+1 )
E
Trees and numerical methods for ordinary differential equations – p. 39/41
hF
A U
Y
=
[n]
y [n−1]
B V
y
η
A U
ηD
=
B V
ξ
Eξ
E(t) = 1/γ(t)
D(τ ) = 1
otherwise D(t) = 0
Trees and numerical methods for ordinary differential equations – p. 40/41
Application to TSRK
(n)
Yi
= ui yn−2 + (1 − ui )yn−1
s
s
X
X
(n)
(n−1)
Bij f (Yj )
)+h
+h
Aij f (Yj
j=1
j=1
yn = θyn−2 + (1 − θ)yn−1
s
s
X
X
(n)
(n−1)
bj f (Yj )
)+h
aj f (Yj
+h
j=1
j=1
Trees and numerical methods for ordinary differential equations – p. 41/41
Application to TSRK
(n)
Yi
= ui yn−2 + (1 − ui )yn−1
s
s
X
X
(n)
(n−1)
Bij f (Yj )
)+h
+h
Aij f (Yj
j=1
j=1
yn = θyn−2 + (1 − θ)yn−1
s
s
X
X
(n)
(n−1)
bj f (Yj )
)+h
aj f (Yj
+h
j=1
j=1
η = uE −1 + (1 − u) + AE −1 ηD + BηD
E = θE −1 + (1 − θ) + aT E −1 ηD + bT ηD
Trees and numerical methods for ordinary differential equations – p. 41/41
Application to TSRK
(n)
Yi
= ui yn−2 + (1 − ui )yn−1
s
s
X
X
(n)
(n−1)
Bij f (Yj )
)+h
+h
Aij f (Yj
j=1
j=1
yn = θyn−2 + (1 − θ)yn−1
s
s
X
X
(n)
(n−1)
bj f (Yj )
)+h
aj f (Yj
+h
j=1
j=1
η = uE −1 + (1 − u) + AE −1 ηD + BηD
E = θE −1 + (1 − θ) + aT E −1 ηD + bT ηD
I welcome further discussion about this appendix.
Trees and numerical methods for ordinary differential equations – p. 41/41