MA5233: Computational Mathematics Weizhu Bao Department of Mathematics & Center for Computational Science and Engineering National University of Singapore Email: bao@math.nus.edu.sg URL: http://www.math.nus.edu.sg/~bao

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, ……

Dynamics of soliton in quantum physics

Wave interaction in plasma physics

Wave interaction in particle physics

Vortex-pair dynamics in superfluidity

Vortex-dipole dynamics in superfluidity

Vortex lattice dynamics in superfluidity

Vortex lattice dynamics in BEC

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

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

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,….

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, ……

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

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

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 1959-61: QR algorithms 1962: Quicksort 1965: Fast Fourier Transform (FFT) 1977: Integer relation detection algorithm 1982: Fast multipole algorithm http://amath.colorado.edu/resources/archive/topten.pdf

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

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, ……

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

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

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.

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

Relative error An example: Relative error: Relative error bound:

Absolute error bounds for basic operations Suppose Error bounds

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

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

Error spreading: An example Algorithm 2 Result Truncation error analysis

Convergence and its rate Numerical integration Exact solution

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

Numerical results

Numerical errors

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!!!

Numerical Differentiation The total error

Numerical Differentiation

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

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

How to determine order of accuracy By quotation