1. MA5233: Computational Mathematics
Weizhu Bao
Department of Mathematics
& Center for Computational Science and Engineering
National University of Singapore
URL:

2. Computational Science
Third paradigm for
Discovery in Science
Solving scientific &
engineering problems
Interdisciplinary
Problem-driven
Mathematical models
Numerical methods
Algorithmic aspects—
computer science
Programming
Software
Applications, ……

3. Dynamics of soliton in quantum physics

4. Wave interaction in plasma physics

5. Wave interaction in particle physics

6. Vortex-pair dynamics in superfluidity

7. Vortex-dipole dynamics in superfluidity

8. Vortex lattice dynamics in superfluidity

9. Vortex lattice dynamics in BEC

10. Computational Science
Computational Mathematics – Scientific computing/numerical analysis
Computational Physics
Computational Chemistry
Computational Biology
Computational Fluid Dynamics
Computational Enginnering
Computational Materials Sciences
……...

11. Steps for solving a practical problems
Physical or engineering problems
Mathematical model – physical laws
Analytical methods – existence, regularity, solution, …
Numerical methods – discretization
Programming -- coding
Results -- computing
Compare with experimental results

12. Computational Mathematics
Numerical analysis/Scientific computing
A branch of mathematics interested in constructive methods
Obtain numerically the solution of mathematical problems
Theory or foundation of computational science
Develop new numerical methods
Computational error analysis:
Stability
Convergence
Convergence rate or order of accuracy,….

13. History
Numerical analysis can be traced back to a symposium with the title ``Problems for the Numerical Analysis of the Future, UCLA, July 29-31, 1948.
Volume 15 in Applied Mathematics Series, National Bureau of Standards
Boom of research and applications: Fluid flow, weather prediction, semiconductor, physics, ……

14. Milestone Algorithms 1901: Runge-Kutta methods for ODEs
1903: Cholesky decomposition
1926: Aitken acceleration process
1946: Monte Carlo method
1947: The simplex algorithm
1955: Romberg method
1956: The finite element method

15. Milestone algorithms 1957: The Fortran optimizing compiler
1959: QR algorithm
1960: Multigrid method
1965: Fast Fourier transform (FFT)
1969: Fast matrix manipulations
1976: High Performance computing & packages: LAPACK, LINPACK – Matlab
1982: Wavelets
1982: Fast Multipole method

16. Top 10 Algorithms 1946: Monte Carlo method
1947: Simplex method for linear programming
1950: Krylov subspace iterative methods
1951: Decompositional approach for matrix computation
1957: Fortran optimizing compiler
: QR algorithms
1962: Quicksort
1965: Fast Fourier Transform (FFT)
1977: Integer relation detection algorithm
1982: Fast multipole algorithm

17. Contents Basic numerical methods Numerical linear algebra
Round-off error
Function approximation and interpolation
Numerical integration and differentiation
Numerical linear algebra
Linear system solvers
Eigenvalue probems
Numerical ODE
Nonlinear equations solvers & optimization

18. Contents Numerical PDE Problem driven research:
Finite difference method (FDM)
Finite element method (FEM)
Finite volume method (FVM)
Spectral method
Problem driven research:
Computational Fluid dynamics (CFD)
Computational physics
Computational biology, ……

19. How to do it well Three key factors Ability for a graduate student
Master all kinds of different numerical methods
Know and aware the progress in the applied science
Know and aware the progress in PDE or ODE
Ability for a graduate student
Solve problem correctly
Write your results neatly
Speak your results well and clear – presentation
Find good problems to solve

20. Numerical error
Example 1:
Example 2:
Example 3:
Example 4:

21. Numerical error Truncation error or error of the method
Round-off error: due to finite digits of numbers in computer
Numerical errors for practical problems
Truncation error
Round-off error
Model error & observation error & empirical error etc.

22. Absolute error
Absolute error:
Absolute error bound (not unique!!):

23. Relative error
An example:
Relative error:
Relative error bound:

24. Absolute error bounds for basic operations
Suppose
Error bounds

25. Significant digits An example Definition: n significant digits Method:
Write in the standard form
Count the number of digits after decimal

26. Error spreading: An example
Algorithm 1:
Use 4 significant digits for practical computation
Results

27. Error spreading: An example
Algorithm 2
Result
Truncation error analysis

28. Convergence and its rate
Numerical integration
Exact solution

29. Numerical methods Composite midpoint rule Composite Simpson’s rule
Composite trapezoidal rule
Error estimate

30. Numerical results

31. Numerical errors

32. Observations Before h0 After h1 Between h0 and h1
Truncation error is too large !!
After h1
Round-off error is dominated!!
Between h0 and h1
Clear order of accuracy is observed for the method
We can observe clear convergence rate for proper region of the mesh size!!!

33. Numerical Differentiation
The total error

34. Numerical Differentiation

35. Numerical Differentiation
Total error depends
Truncation error:
Round-off error:
Minimizer of E(h):
Double precision:
Clear region to observe truncation error:

36. How to determine order of accuracy
Numerical approximation or method
How to determine p and C??
By plot log E(h) vs log h

37. How to determine order of accuracy
By quotation