This document discusses numerical methods for differentiation and integration. It begins by reviewing Richardson's extrapolation for numerical integration. It then discusses numerical differentiation, including examples of calculating the first and second derivatives using forward, backward, and central difference methods. It explains that numerical differentiation is useful when only discrete data is available or when exact formulas are too complex. The accuracy and order of different numerical differentiation methods is compared based on the number of grid points used in the approximation. Queries about the course content can be addressed to the instructor.

5. Numerical methods and
computation Physics
Course code: PHC023P1M
Credits: 2-0-2
Instructions: Dr. Sanat Kumar Tiwari
TA: Ab Rauoof Wani

6. Numerical integration - Error correction - Rechardson’s extrapolation
Numerical di
ff
erentiation: basics
What we learnt in previous lecture?

7. Rechardson’s extrapolation
How the expression takes form for step size = ?
h2 h1/2
I(h) = the approximation of the trapezoidal rule with step size h = (b - a)/n,
For two step sizes and :
h1 h2

8. Numerical differentiation: Example
• Fourier’s law of heat conduction quanti
fi
es the observation that
heat
fl
ows from regions of high to low temperature
• Gradient/Derivative tells the direction of
fl
ow of the heat.
q(x) = − κ
dT
dx

9. Numerical differentiation: Example
• Fourier’s law of heat conduction quanti
fi
es the observation that
heat
fl
ows from regions of high to low temperature
• Gradient/Derivative tells the direction of
fl
ow of the heat.
q(x) = − κ
dT
dx

10. Engineering equations with derivatives

11. Numerical differentiation

12. Numerical differentiation
• A simple example for the
fi
rst derivative:
• We are aware, it comes from the Taylor’s expansion, let’s
calculate the error
• Based on formula, let’s re-write the
fi
rst derivative:

13. Numerical differentiation: Why we need it?
• Even for known functional forms that we need to di
ff
erentiate, the
data is usually sampled without knowing the function itself.
• There are some cases where we have is a discrete data set only.
We may still be interested in studying changes in the data
through derivatives.
• At times, exact formulas are available but are complicated and
di
ff
erentiation requires a lot of function evaluations. It is simpler
to approximate the derivative instead.
• When approximating solutions to ordinary (or partial) di
ff
erential
equations, we typically represent the solution as a discrete
approximation that is de
fi
ned on a grid.

14. Taylor’s series

15. Numerical differentiation: First derivative
Forward di
ff
erence
Backward di
ff
erence
Central di
ff
erence

16. First derivative: Central difference: Derivation
Aim is to get
fi
rst derivative, so subtract second eqn from 1st

17. First derivative: the accuracy vs grid points
i i + 1 i + 2 i + 3
i − 1
i − 2
i − 3

18. i + 3

19. First derivative: the accuracy vs grid points
i i + 1 i + 2 i + 3
i − 1
i − 2
i − 3
• A th order derivative needs minimum data points on
stencil
• The order of accuracy of nth derivative can be improved taking
into accounts grid points more than
n n + 1
n + 1

20. Second derivative: the accuracy vs grid points
i i + 1 i + 2 i + 3
i − 1
i − 2
i − 3
• A th order derivative needs minimum data points on
stencil
• The order of accuracy of nth derivative can be improved taking
into accounts grid points more than
n n + 1
n + 1

21. Second derivative: the forward difference
i i + 1 i + 2 i + 3
i − 1
i − 2
i − 3
Forward di
ff
erence

22. Second derivative: Forward difference
Forward di
ff
erence

23. Second derivative: Central difference
i i + 1 i + 2 i + 3
i − 1
i − 2
i − 3
Central di
ff
erence

24. Second derivative: Central difference
Central di
ff
erence

25. First derivative: FD: accuracy
O(h2
)
Forward di
ff
erence
i i + 1 i + 2 i + 3 i + 4 i + 6
i + 5

26. First derivative: FD: accuracy
O(h2
)
Forward di
ff
erence

31. Queries/concerns
sanat.tiwari@iitjammu.ac.in
Course LMS page
https://lms.iitjammu.ac.in/course/view.php?id=233