Objectives
To introduce advanced discretization techniques for hyperbolic and
variational/Hamiltonian partial differential equations.
This course surveys a broad selection
of numerical methods for such problems, adresses stability theory of finite
difference methods, dispersion analysis, backward error analysis, discrete
numerical operator theory and discrete differential forms.
Period
Semester 2
Lecturer:
Dr. Jason Frank
(Thursdays) KdV Institute, University of Amsterdam
Plantage Muidergracht 24, Room 2.31
tel: +31 20 5255374
(Rest of the week)
Center for Mathematics and Computer Science (CWI)
Kruislaan 413, Room M259
tel: +31 20 5924096
e-mail:
www: http://www.cwi.nl/~jason
Contents
Preservation of mathematical structure under
discretization will play a central role in the discussion. We consider
hyperbolic wave equations in conservation law form and dispersive wave
equations with variational or Hamiltonian structure, for which conservation laws
can be derived from the equations of motion. Modern numerical discretization
techniques attempt to mimic the conservation laws in the discrete approximation,
to enhance stability and ensure that the simulated solution is qualitatively
correct.
We begin with a discussion of conservation laws and their discrete analogs, and how these yield global conservation under appropriate (numerical) boundary conditions. This theory is directly applied to hyperbolic wave equations, for which application of finite volume methods automatically leads to the discrete conservation laws. In the presence of shock waves and discontinuities, this approach must be supplemented with Riemann solvers and monotonicity-preserving time integrators to ensure the conservation of monotonicity of the solution (and prevent spurious numerical oscillations).
For dispersive wave equations, the relevant conservation laws are often not readily evident from the equations of motion, but instead are derived from some underlying structure. Care must be taken to ensure that analogous manipulations can be carried out on the discrete system. For example, discretizations starting from a Hamiltonian structure or utilizing a discrete variational principle. Modified equation analysis is important for understanding the qualitative behavior of numerical methods.
Adaptivity, implementation of numerical boundary conditions, and conservative time integration are additional aspects which must be dealt with.
Specific topics we will cover are: finite volume discretization of conservation laws, Riemann solvers, monotonicity-preserving discretizations, (pseudo-)spectral methods, symplectic and symmetric time integrators, energy-conserving finite difference methods, discrete variational principles, discrete Poisson brackets, numerical dispersion, backward error analysis, stability and boundary conditions, nonuniform meshes and mimetic discretizations/discrete differential forms.
Prerequisites
Calculus and an understanding of differential equations are necessary.
Helpful are: multivariable calculus, linear algebra, ordinary and partial
differential equations, numerical analysis, familiarity with Matlab. Please
contact the instructor if you are in doubt. Students from physics and related
fields are encouraged to participate.
Registration
Via http://studieweb.student.uva.nl.
or via the
Education Office, phone: 525 7100, e-mail:
ondwns@science.uva.nl
Format
Lectures and instruction.
Class time (2008)
Thursdays, 10.00-13.00, Room I1.02 (except May 29: A5.06)
February 7 - May 29, 2008
Study materials
Lecture notes.
Assessment
Based on homework exercises and more involved projects.