ENGR 232
Dynamic Engineering
Systems
Lecture 1.2 - Appendix
Introduction to Numerical
Solution
s
with Differential Equations
Dr. Michael Ryan
Runge-Kutta Methods
• Euler uses information from the current pair (tn, yn) to determine value at the next
pair (tn+1, yn+1)
• Runge-Kutta (2nd and 4th order methods) use a look ahead to get the slopes at
strategically chosen nearby points which are weighted to maximize the accuracy of
the approximation.
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 201/06/2016
Second-order Runge-Kutta
(aka midpoint method)
• Use slope at (tn, yn) to look a half-step ahead to the point
• Compute slope at this midpoint and use this slope to move from (tn, yn)
to (tn+1, yn+1)
• The algorithm follows
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 3
𝒕𝒏 +
𝒉
𝟐
, 𝒚𝒏+
𝒉
𝟐
𝒇(𝒕𝒏, 𝒚𝒏)
01/06/2016
Second-order Runge-Kutta
(aka midpoint method)
Given the IVP 𝑦′ = 𝑓 𝑡, 𝑦 𝑦 𝑡0 = 𝑦0
And a step size of h
To go from 𝑡𝑛,𝑦𝑛 𝑡𝑜 𝑡𝑛+1, 𝑦𝑛+1 𝑢𝑠𝑒:
where
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 4
𝒌𝒏𝟐 = 𝒇 𝒕𝒏 +
𝒉
𝟐
, 𝒚𝒏 +
𝒉
𝟐
𝒌𝒏𝟏
𝒌𝒏𝟏 = 𝒇 𝒕𝒏, 𝒚𝒏
𝒕𝒏+𝟏 = 𝒕𝒏 + 𝒉
𝒚𝒏+𝟏 = 𝒚𝒏 + 𝒉𝒌𝒏𝟐
01/06/2016
Numerical Example
Consider Euler h = 0.5 step from t = 0.0 to 0.5
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 5
t y Euler dydt y analytic error
0 1 -1 1.000 0.000
0.5 0.5 0 0.713 0.213
𝒌𝒏𝟏 = 𝒇 𝒕𝒏, 𝒚𝒏 = 𝐟 𝟎, 𝟏 = (𝐭 − 𝐲) = −𝟏
𝒕𝒏+𝟏 = 𝒕𝒏 + 𝒉 = 𝟎. 𝟓
𝒚𝒏+𝟏 = 𝒚𝒏 + 𝒉𝒌𝒏𝟐 = 𝟏 − 𝟎. 𝟐𝟓 = 𝟎. 𝟕𝟓
𝒌𝒏𝟐 = 𝒇 𝒕𝒏 +
𝒉
𝟐
, 𝒚𝒏 +
𝒉
𝟐
𝒌𝒏𝟏 = 𝒇 𝟎. 𝟐𝟓,𝟎. 𝟕𝟓 = −𝟎.𝟓
Using Runge-Kutta
at t = 0.5, y = 0.75
Compare to Euler;
RK error is -0.037
01/06/2016
Forth-order Runge-Kutta
Given the IVP 𝑦′ = 𝑓 𝑡, 𝑦 𝑦 𝑡0 = 𝑦0
And a step size of h; to go from 𝑡𝑛, 𝑦𝑛 𝑡𝑜 𝑡𝑛+1, 𝑦𝑛+1 :
First computing
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 6
𝒌𝒏𝟏 = 𝒇 𝒕𝒏, 𝒚𝒏
𝒕𝒏+𝟏 = 𝒕𝒏 + 𝒉
𝒚𝒏+𝟏 = 𝒚𝒏 +
𝒉
𝟔
𝒌𝒏𝟏 + 𝟐𝒌𝒏𝟐 + 𝟐𝒌𝒏𝟑 + 𝒌𝒏𝟒
𝒌𝒏𝟐 = 𝒇 𝒕𝒏 +
𝒉
𝟐
, 𝒚𝒏 +
𝒉
𝟐
𝒌𝒏𝟏
𝒌𝒏𝟑 = 𝒇 𝒕𝒏 +
𝒉
𝟐
, 𝒚𝒏 +
𝒉
𝟐
𝒌𝒏𝟐 𝒌𝒏𝟒 = 𝒇 𝒕𝒏 +
𝒉
𝟐
, 𝒚𝒏 + 𝒉𝒌𝒏𝟑
01/06/2016
Error Comparison
Method One step result Actual Error
Exact y(1) = 0.1487…
Euler y(1) = 0 0.1487…
2nd
Runge-Kutta
y(1) = 0.125 0.0237…
4th
Runge-Kutta
y(1) = 0.15 0.0013…
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 7
...
1. ENGR 232
Dynamic Engineering
Systems
Lecture 1.2 - Appendix
Introduction to Numerical
Solution
s
with Differential Equations
Dr. Michael Ryan
Runge-Kutta Methods
• Euler uses information from the current pair (tn, yn) to
determine value at the next
pair (tn+1, yn+1)
2. • Runge-Kutta (2nd and 4th order methods) use a look ahead to
get the slopes at
strategically chosen nearby points which are weighted to
maximize the accuracy of
the approximation.
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 201/06/2016
Second-order Runge-Kutta
(aka midpoint method)
• Use slope at (tn, yn) to look a half-step ahead to the point
• Compute slope at this midpoint and use this slope to move
from (tn, yn)
to (tn+1, yn+1)
• The algorithm follows
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 3
�� +
�
3. �
, ��+
�
�
�(��, ��)
01/06/2016
Second-order Runge-Kutta
(aka midpoint method)
Given the IVP �′ = � �, � � �0 = �0
And a step size of h
To go from ��,�� �� ��+1, ��+1 ���:
where
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 4
4. ��� = � �� +
�
�
, �� +
�
�
���
��� = � ��, ��
��+� = �� + �
��+� = �� + ����
01/06/2016
Numerical Example
Consider Euler h = 0.5 step from t = 0.0 to 0.5
6. Using Runge-Kutta
at t = 0.5, y = 0.75
Compare to Euler;
RK error is -0.037
01/06/2016
Forth-order Runge-Kutta
Given the IVP �′ = � �, � � �0 = �0
And a step size of h; to go from ��, �� �� ��+1, ��+1 :
First computing
ENGR 232 Winter 16 Lecture 1.2 – Dr. M. Ryan 6
��� = � ��, ��
��+� = �� + �
��+� = �� +
�
