Download Numerical solution of the ODEs

Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Numerical solution of the ODEs
Using difference equations
Dalibor Karásek
Czech Technical University in Prague
Faculty of Nuclear Sciences and Physical Engineering
Student Colloquium and School on Mathematical Physics,
Stará Lesná 23.08–29.08. 2010
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Outline
1
Symmetries of equations
2
Difference equations
3
Differential equations
4
Difference schema
5
Example
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Outline
1
Symmetries of equations
2
Difference equations
3
Differential equations
4
Difference schema
5
Example
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Infinitesimal symmetries of algebraic equation
Algebraic equation F (x1 , . . . , xn ) = 0.
Rn ).
Infinitesimal symmetries are X ∈ X(
X (F ) F =0 = 0.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Infinitesimal symmetries for ODE
We can use same pattern for the ODE of nth order
F (x , y , y 0 , . . . , y (n) ) = 0.
Problem: x , y , . . . , y (n) obviously aren’t independent.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Infinitesimal symmetries for ODE
We can use same pattern for the ODE of nth order
F (x , y , y 0 , . . . , y (n) ) = 0.
Problem: x , y , . . . , y (n) obviously aren’t independent.
R2) → X(Rn ).
pr X (F ) F =0 = 0 for X ∈ X(R2 ).
Take map pr : X(
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Prolongation for ODE
X = ξ(x , y )∂x + η(x , y )∂y .
Prolongation has form
pr X = ξ(x , y )∂x + η i (x , y , y 0 , . . . , y (i) )∂y (i) ,
η 0 = η,
η i = Dx η i−1 − y (i) · Dx ξ,
Dx = ∂x + y (i+1) ∂y (i) .
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Construction of equation with particular symmetry
R
Suppose Xi ∈ X( k ) are prescribed infinitesimal symmetries. How
does general equation with these symmetries look like?
1 Find invariants I(x , . . . , x ) of fields X using method of
1
n
i
characteristics.
b
∀i ∈ n
Dalibor Karásek
pr Xi (I) = 0.
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Construction of equation with particular symmetry
R
Suppose Xi ∈ X( k ) are prescribed infinitesimal symmetries. How
does general equation with these symmetries look like?
1 Find invariants I(x , . . . , x ) of fields X using method of
1
n
i
characteristics.
b
∀i ∈ n
2
pr Xi (I) = 0.
Every symmetrical equation can be written in the form
F (I1 , . . . , Ik ) = 0.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Construction of equation with particular symmetry
R
Suppose Xi ∈ X( k ) are prescribed infinitesimal symmetries. How
does general equation with these symmetries look like?
1 Find invariants I(x , . . . , x ) of fields X using method of
1
n
i
characteristics.
b
∀i ∈ n
2
pr Xi (I) = 0.
Every symmetrical equation can be written in the form
F (I1 , . . . , Ik ) = 0.
3
(pr Xi )(F (I1 , . . . , Ik )) =
k
X
j=1
Dalibor Karásek
∂j F · (pr Xi )(Ij ) =
k
X
j=1
Numerical solution of the ODEs
∂j F · 0 = 0.
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Outline
1
Symmetries of equations
2
Difference equations
3
Differential equations
4
Difference schema
5
Example
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Prolongation for difference equation
Difference equation of nth order.
Stencil x0 , . . . , xn , y0 , . . . , yn .
Equation F (x0 , . . . , xn , y0 , . . . , yn ) = 0.
Infinitesimal symmetry X = ξ(x , y )∂x + η(x , y )∂y .
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Prolongation for difference equation
Difference equation of nth order.
Stencil x0 , . . . , xn , y0 , . . . , yn .
Equation F (x0 , . . . , xn , y0 , . . . , yn ) = 0.
Infinitesimal symmetry X = ξ(x , y )∂x + η(x , y )∂y .
Prolongation
pr X =
n
X
ξ(xi , yi )∂xi + η(xi , yi )∂yi .
i=0
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Continuous limit of difference equation
Way of approximating of differential equations. Let hi := xi − xi−1
and F (x0 , . . . , xn , y0 , . . . , yn ) a difference equation.
Think of yi as y (xi ) and formally Taylor-expand it around x0 .
Substitute into the difference equation and expand it again.
Continuous limit of the difference equation is the term ∝ h0 .
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Example — continuous limit
Consider the difference equation
y1 − y0
− sin(x1 ) = 0
x1 − x0
.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Example — continuous limit
Consider the difference equation
y1 − y0
− sin(x1 ) = 0
x1 − x0
.
1
y0 + y 0 h1 + O(h2 ) − y0
− sin(x0 + h1 ) = 0.
h1
2
y 0 h1 + O(h2 )
− sin(x0 ) − cos(x0 )h1 + O(h2 ) = 0.
h1
y 0 − sin(x ) + O(h) = 0.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Outline
1
Symmetries of equations
2
Difference equations
3
Differential equations
4
Difference schema
5
Example
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Solving of ODE using symmetries
F (x , y , y 0 , . . . , y (n) ) = 0.
Can be solved by quadratures when infinitesimal symmetries
form solvable Lie algebra of dimension at least n.
Often we can’t get explicit form y = f (x ).
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Solving of ODE using symmetries
F (x , y , y 0 , . . . , y (n) ) = 0.
Can be solved by quadratures when infinitesimal symmetries
form solvable Lie algebra of dimension at least n.
Often we can’t get explicit form y = f (x ).
It’s usually nasty implicit form containing inverses of wild
functions.
Therefore we are forced to use numerical methods.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Common numerical methods
Euler More or less theoretical importance
Runge-Kutta Accurate, but very CPU-demanding.
rough difference methods Usually using Lagrange interpolation.
Quick, but less accurate.
And not only that neither of them uses symmetries of ODE, they
break them completely.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Outline
1
Symmetries of equations
2
Difference equations
3
Differential equations
4
Difference schema
5
Example
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Issues of usually used solution method
Why are ordinary methods (like Runge-Kutta) ineffective?
1
They violate symmetries.
2
Use brute force (correction of correction).
3
Prioritize uniform lattice.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Maintaining of symmetries
Approximate differential equation with difference equation.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Maintaining of symmetries
Approximate differential equation with difference equation.
Both should have same symmetry algebra.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Maintaining of symmetries
Approximate differential equation with difference equation.
Both should have same symmetry algebra.
Difference equation should transform to given ODE in
continuous limit.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Lattice
Lattice is set of point {. . . , x−1 , x0 , x1 , ...}.
Instead of uniform lattice we can use lattice respecting
symmetries.
It can be fixed by a difference equation, e.g. h1 = h2 .
It is no longer lattice in the sense of an additive subgroup of
.
R
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Synergy of the ideas
Let an ODE be given.
1
Find it’s symmetries.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Synergy of the ideas
Let an ODE be given.
1
Find it’s symmetries.
2
Find difference equations which have same symmetries.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Synergy of the ideas
Let an ODE be given.
1
Find it’s symmetries.
2
Find difference equations which have same symmetries.
Construct two difference equation.
3
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Synergy of the ideas
Let an ODE be given.
1
Find it’s symmetries.
2
Find difference equations which have same symmetries.
Construct two difference equation.
3
One should go to the ODE in continuous limit. (It
approximates the ODE.)
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Synergy of the ideas
Let an ODE be given.
1
Find it’s symmetries.
2
Find difference equations which have same symmetries.
Construct two difference equation.
3
One should go to the ODE in continuous limit. (It
approximates the ODE.)
One should go to the identity (0 = 0). (It fixes the lattice).
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Synergy of the ideas
Let an ODE be given.
1
Find it’s symmetries.
2
Find difference equations which have same symmetries.
Construct two difference equation.
3
One should go to the ODE in continuous limit. (It
approximates the ODE.)
One should go to the identity (0 = 0). (It fixes the lattice).
4
Set initial condition x0 , . . . , xn−1 , y0 , . . . , yn−1 and solve these
equations with respect to xn , yn .
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Synergy of the ideas
Let an ODE be given.
1
Find it’s symmetries.
2
Find difference equations which have same symmetries.
Construct two difference equation.
3
One should go to the ODE in continuous limit. (It
approximates the ODE.)
One should go to the identity (0 = 0). (It fixes the lattice).
4
Set initial condition x0 , . . . , xn−1 , y0 , . . . , yn−1 and solve these
equations with respect to xn , yn .
5
Iterate last step with x1 , . . . , xn , y1 , . . . , yn as the new initial
conditions.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Outline
1
Symmetries of equations
2
Difference equations
3
Differential equations
4
Difference schema
5
Example
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
ODE
ODE
y 00 = (y 0 )3
Symmetries
∂x , ∂y , 2x ∂x + y ∂y
Difference invariants
(2)
P
h1 (P0 ) , h2 P1 P0 , 0 3 , ...
(P0 )
2
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Difference schema
We construct following difference schema:
(2)
P0
= 1,
(P0 )3
(1)
h1 (P0 )2 = h2 P1 P0 .
One of it’s solution is:
y2 = 2y1 − y0 ,
1
h1
x2 = x1 − 2 −
+
2
P0
Dalibor Karásek
s
1
h12 3h1
+
+ 2.
4
P04
P0
Numerical solution of the ODEs
(2)
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Comparison
Comparison of Runge-Kutta — (time 0.2s) and difference — (time
less than 0.04s).
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Conclusions and problems
ODE solving methods which respects symmetries can be
constructed.
They are CPU-cheap and sufficiently precise.
However:
They have to be pre-solved by hands (at least as far as I am
concerned).
We don’t know how to choose best difference schemas for the
given ODE.
Mathematical prove that they are better is missing.
Dalibor Karásek
Numerical solution of the ODEs
Symmetries of equations
Difference equations
Differential equations
Difference schema
Example
Thank you for your attention.
A. Bourlioux, C. Cyr-Gagnon, and P. Winternitz
“Difference schemes with point symmetries and their
numerical tests”
Journal of Physics A: Mathematical and General, vol. 39,
no. 22, p. 6877, 2006
Dalibor Karásek
Numerical solution of the ODEs