Numerical Solution of
Differential Equations
3rd year JMC group project ● Summer Term 2004
Supervisor: Prof. Jeff Cash
Saeed Amen
Paul Bilokon
Adam Brinley Codd
Minal Fofaria
Tejas Shah
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Agenda
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Adam: Differential equations and numerical
methods
Paul: Basic methods (FE, BE, TR). Problem:
“circle”
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation
Minal: Stiff equations
Tejas: Derivation of 6th order method
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Agenda





Adam: Differential equations and numerical
methods
Paul: Basic methods (FE, BE, TR). Problem:
“circle”
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation
Minal: Stiff equations
Tejas: Derivation of 6th order method
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Introduction to Differential
Equations
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Equations involving a function y of x, and its
derivatives
Model real world systems
General Equation:
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Introduction to Differential
Equations
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Simple example
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Solution obtained by integrating both sides
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Initial values can determine c
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Introduction to Differential
Equations
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Special case when equations do not involve x
E.g.
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Initial values y(0) = 1 and y'(0) = 0, solution is
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Introduction to Differential
Equations
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Kepler’s Equations of Planetary Motion
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Difficult to solve analytically – we use
numerical methods instead
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Introduction to Differential
Equations
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Forward Euler and Backward Euler
Trapezium Rule
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General method
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–
–
–
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Use initial value of y at x=0
Calculate next value (x=h for small h) using gradient
Call this y1 and repeat
Need formula for yn+1 in terms of yn
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Agenda





Adam: Differential equations and numerical
methods
Paul: Basic methods (FE, BE, TR). Problem:
“circle”
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation
Minal: Stiff equations
Tejas: Derivation of 6th order method
© Imperial College London
Agenda





Adam: Differential equations and numerical
methods P
Paul: Basic methods (FE, BE, TR). Problem:
“circle”
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation
Minal: Stiff equations
Tejas: Derivation of 6th order method
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Euler’s Methods
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Simplest way of solving
ODEs numerically
Does not always
produce reasonable
solutions
Forward Euler (explicit)
and Backward Euler
(implicit)
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Forward Euler
y
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True solution
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(x2, y2)
y2
Slope f(x1, y1)
Slope f(x0, y0)
Approximated
solution
y1
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y(x + h) = y(x) + h y’(x)
Explicit
Eloc = ½h2y(2)()  h2
First order
Asymmetric
(x1, y1)
y0
(x0, y0)
h
x0
h
x1
x2
x
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Backward Euler
y
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(x2, y2)
y2
True solution
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Slope f(x2, y2)
Slope f(x1, y1)
Approximated
solution
(x1, y1)
y1
y0
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y(x + h) = y(x) + h y’(x +
h)
Implicit
Eloc = - ½h2y(2)()  h2
First order
Asymmetric
(x0, y0)
h
x0
h
x1
x2
x
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Trapezium Rule
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y(x + h) = y(x) + ½ h [y’(x) + y’(x + h)]
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Average of FE and BE
Implicit
Eloc = - (h3/12) f(2)()  h3
Second order
Symmetric
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Example Problem: “Circle”
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Consider y’’ = -y, with
initial conditions
–
–
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y(0) = 1
y’(0) = 0
The analytical solution is
y = cos x, that can be
used for comparison with
numerical solutions
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Plots for the True Solution
Time series plot (xy)
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Phase plane plot (zy)
Time Series
y
x
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Phase Plane Plots: FE & BE
Forward Euler
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Backward Euler
Phase Plane Plots: TR
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FE & BE fail: non-periodic
TR: OK, periodic soln
Why?
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Symmetricity
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TR is symmetric, whereas FE & BE are not
y(x + h) = y(x) + ½ h [y’(x) + y’(x + h)]
h  -h
y(x - h) = y(x) - ½ h [y’(x) + y’(x - h)]
X := x - h
y(X + h) = y(X) + ½ h [y’(X) + y’(X +
h)]
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Symmetricity
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TR is symmetric, whereas FE & BE are not
y(x + h) = y(x) + h y’(x)
h  -h
y(x - h) = y(x) - h y’(x)
X := x - h
y(X + h) = y(X) + h y’(X + h)
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Time Step
h = 10 -1
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h = 10 -2
Agenda





Adam: Differential equations and numerical
methods P
Paul: Basic methods (FE, BE, TR). Problem:
“circle”
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation
Minal: Stiff equations
Tejas: Derivation of 6th order method
© Imperial College London
Agenda





Adam: Differential equations and numerical
methods P
Paul: Basic methods (FE, BE, TR). Problem:
“circle” P
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation
Minal: Stiff equations
Tejas: Derivation of 6th order method
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After Euler and TR
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Can create higher order methods, which have far
smaller global errors
Methods are more complex and require more
computation on each step
But for same step size more accurate
Introduce concept of half step
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Two Fourth Order Methods
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yn+1 – yn = yn + h/2(y’n) + (h2/12)(4y’’n+½+ 2y’’n)
y’n+1 – y’n = h/6(y’’n+1 + 4y’’n+½+ y’’n) (*)
Then to calculate the half-step we use either A or B
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–
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A) yn+ ½ = ½(yn+1 + yn) – h2/48(y’’n+1 + 4y’’ n+½ + y’’n)
B) yn+ ½ = yn + ½y’n – h2/192(-2y’’n+1 + 12y’’ n+½ + 14y’’n)
Which one is more accurate? (next slide)
Solve iteratively and then apply solution to find
derivative (*)
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Creating Method A
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Start with
–
–
–
–
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yn+1 – yn = h/6[y’n+1 + 4y’n+½ + y’n] (1)
diff. y’n+1 – y’n = h/6[y’’n+1 + 4y’’n+½ + y’’n] (2)
yn+½= ½(yn+1 + y’n) – h/8(y’n+1 – y’n) (3)
diff. y’n+½= ½(y’n+1 + y’’n) – h/8(y’’n+1 – y’’n) (4)
subs (4) into (1) (to eliminate y’n+½ ) and
eliminate y’n+1 using (2),
then use (2) in (3) to get half-step
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Comparing Fourth Order Methods
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Comparing errors when
solving circle problem
Both A and B produce a
much smaller order of
error than Euler’s
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Finding Earth’s Orbit Around Sun
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Kepler’s Equation
Can use it to find the orbit of planets
Use to find orbit of earth around the sun
Work in two dimensions z and y
z’’ = -(GM z) / (y2 + z2)3/2
y’’ = -(GM y) / (y2 + z2)3/2
Constants and initial conditions in report
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Results Plot
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Solution uses small step
size h = 0.01
Becomes difficult to tell
difference between
methods visually
Using h = 0.1 difference
is more marked
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Agenda





Adam: Differential equations and numerical
methods P
Paul: Basic methods (FE, BE, TR). Problem:
“circle” P
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation
Minal: Stiff equations
Tejas: Derivation of 6th order method
© Imperial College London
Agenda





Adam: Differential equations and numerical
methods P
Paul: Basic methods (FE, BE, TR). Problem:
“circle” P
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation P
Minal: Stiff equations
Tejas: Derivation of 6th order method
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Stiff Equations
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Certain systems of ODEs are classified as stiff
A system of ODEs is stiff if there are two or
more very different scales of the independent
variable on which the dependent variables are
changing
Some of the methods used to find numerical
solutions fail to obtain the required solution
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Example

Consider the equation:
2
d y
dx

2
 (   1)
dy
 y  0
dx
With initial conditions:
y (0)  1
y ' (0)  1

There are two solutions to this problem
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Example continued
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Analytical solution: y=e-x
Unwanted solution: y=e-λx
We will now show how forward Euler is not
stable when solving this problem under certain
circumstances
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Example continued
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Let λ = 103 and let (the step size) h = 0.1
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Increasing the range of the x-axis
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When h = 0.001
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Agenda





Adam: Differential equations and numerical
methods P
Paul: Basic methods (FE, BE, TR). Problem:
“circle” P
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation P
Minal: Stiff equations
Tejas: Derivation of 6th order method
© Imperial College London
Agenda





Adam: Differential equations and numerical
methods P
Paul: Basic methods (FE, BE, TR). Problem:
“circle” P
Saeed: Advanced methods (4th order).
Problem: Kepler’s equation P
Minal: Stiff equations P
Tejas: Derivation of 6th order method
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A Sixth Order Numerical Method
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Has an error term with smallest h degree as h7
Idea is to find values for α, A, B, C, D, E, F, C,
D, E, F such that:
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Derivation
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Compare Taylor’s Expansion of LHS and RHS
So for:
–
First expand the LHS
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Derivation
• Now expand individual terms on RHS of our
expression:
• This gives:
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Derivation
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Equate both sides and solve for constants:
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Similarly we can find C, D, E, F and C, D, E, F
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Applying 6th Order Method
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How to obtain results using derived method.
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Produce a set of simultaneous equations and solve.
Find y’n+1 from old values and those just obtained.
Set x, yn , y’n etc. to new values.
Repeat above procedure with updated variables.
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Circle Problem Example
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Circle is produced for phase plane plot
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Kepler’s Equations
• Solution obtained from
z-y plot reinforces fourth
order method results.
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The End
But perhaps you want more?
 Read our report
 Visit our website:
http://www.doc.ic.ac.uk/~pb401/DE