APMA2811F - Topics for final presentations

Each student has to choose one of the special methods below and prepare a presentation of this method. As a minimum, the presentation must contain:
  • An introduction to the methods and its application areas
  • Overview of important theoretical and practical properties
  • Comparative discussion of advantages and disadvantages in relation to alternatives
  • Examples to demonstrate performance
The presentation should last about 20 min.

Note: The references listed below are not complete and it is expected that you identify and use other papers and books in addition to these.

List of topics for final presentation:

ADER methods
  • M. Dumpser and C.-D. Munz, Arbitrary high-order discontinuous Galerkin schemes, [Preprint]
  • V.A. Titarev and E.F. Toro, ADER Schemes for three-dimensional non-linear hyperbolic systems. J. Comput. Phys, 204, 715-736, 2005.
  • M. Kaser and A. Iske, ADER schemes on adaptive triangular meshes for scalar conservation laws, J. Comput. Phys 205(2), 486-508, 2005.
  • E.F. Toro and V.A. Titarev, Derivative Riemann solvers for systems of conservation laws and ADER methods, J. Comput. Phys. 212, 150-165, 2006.
  • M. Dumpser, M. Kaser, and E.F. Toro, An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping and p-adaptivity. Geophysical Journal International 171(2), 695-717, 2007.
Presented by:


Differential algebraic equations (DAEs) and methods
  • E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Equations, Springer Series in Computational Mathematics 14, Springer Verlag, 2nd edition, 1996.
  • Uri M. Ascher and Linda R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM Publishing, Philadelphia, 1998.
  • S. Schulz, Four lectures on differential-algebraic equations. [Lecture notes]
  • Series of lectures on DAEs [LINK]
Presented by: S. Tan on December 2, 2009


Multi-rate methods
  • C.W. Gear, D.R. Wells. Multirate linear multistep methods BIT Numerical Methods, 24, 484-502, 1984.
  • M. Gunther, A. Kvaerno and P. Rentrop, Multirate partitioned Runge-Kutta methods, BIT Numerical Methods, 41, 504-514, 2001
  • V. Savcenco, W. Hundsdorfer and J.G. Verwer, A multirate time stepping strategy for stiff ODEs, BIT 47, 137-155, 2007.
Presented by: C. Li on December 2, 2009


Parareal methods
  • J.-L. Lions, Y. Maday, and G. Turinici, A “parareal” in time discretization of PDE’s, C. R. Acad. Sci. Paris S ́er. I Math., 332, 661–668, 2001.
  • G.A. Staff and E.M. Roenquist, Stability of the Parareal Algorithm, Proceedings of DD15. [LINK]
  • M.J. Gander and S. Vandewalle, Analysis of the Parareal Time-Parallel Time-Integration Method, SIAM Journal on Scientific Computing 29(2), 556-578, 2007.
  • G. Bal, On the Convergence and the Stability of the Parareal Algorithm to solve Partial Differential Equations. [Preprint]
Presented by: S. Field on December 2, 2009

Runge-Kutta-Chebyshev methods
  • J.G. Verwer, W.H. Hundsdorfer and B.P. Sommeijer, Convergence properties of the Runge-Kutta-Chebyshev method, Numer. Math. 57, 157-178, 1990.
  • B.P. Sommeijer, L.F. Shampine and J.G. Verwer, RKC: An explicit solver for parabolic PDEs, J. Comp. Appl. Math. 88, 315-326, 1997.
  • J.G. Verwer, B.P. Sommeijer and W. Hundsdorfer, RKC time-stepping for Advection-Diffusion-Reaction Problems, J. Comput. Phys. 201, 61-79, 2004.
  • J.G. Verwer and B.P. Sommeijer, An Implicit-Explicit Runge-Kutta-Chebyshev Scheme for Diffusion-Reaction Equations, SIAM J. Scientific Computing 25, 1824-1835, 2004.
Presented by: Z. Xueyu on December 2, 2009


Runge-Kutta methods optimized for advection problems
  • F.Q. Hu, M.Y. Hussaini, J.L. Manthey, Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics, 124, 177-191, 1996.
  • D. Stanescu and W. G. Habashi, 2N-storage low dissipation and dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics, 143(2), 674-681, 1998.
  • Julien Berland, Christophe Bogey, Christophe Bailly, Low-dissipation and low-dispersion fourth-order Runge-Kutta algorithm, Computers & Fluids, Vol. 35(10), 1459-1463, 2006.
  • M. Bernardini and S. Pirozzoli, A general strategy for the optimization of Runge-Kutta schemes for wave propagation phenomena, Journal of Computational Physics, Volume 228(11), 4182-4199, 2009.
Presented by: T. Seshu on December 9, 2009


Runge-Kutta-Rosenbrock methods
  • H.H. Rosenbrock, Some general implicit processes for the numerical solution of differential equations, Comput. J. , 5, 329–330, 1963.
  • J.G. Verwer, An analysis of Rosenbrock methods for nonlinear stiff initial value problems, SIAM J. Numer. Anal. 19, 155-170, 1982.
  • J.G. Verwer, Instructive experiments with some Runge-Kutta-Rosenbrock methods, Comp. and Math. with Appls. 7, 217-229, 1982.
  • J.G. Verwer, S. Scholz, J.G. Blom and M. Louter-Nool, A class of Runge-Kutta-Rosenbrock methods for stiff differential equations, ZAMM 63, 13-20, 1983.
  • E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Equations, Springer Series in Computational Mathematics 14, Springer Verlag, 2nd edition, 1996.
Presented by: Y. Yu on December 9, 2009


Spectral deferred correction methods
  • A. Dutt, L. Greengard, V. Rokhlin, Spectral Deferred Correction Methods for Ordinary Differential Equations, BIT Numerical Mathematics, 40(2), 241-266, 2000.
  • M.L. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci. 1(3), 471-500, 2003.
  • A. Bourlioux, A.T. Layton, and M.L. Minion, High-Order Multi-Implicit Spectral Deferred Correction Methods for Problems of Reactive Flows, Journal of Computational Physics 189(2), 651-675, 2003.
  • A.T. Laytona, On the efficiency of spectral deferred correction methods for time-dependent partial differential equations, Applied Numerical Mathematics 59(7), 1629-1643, 2009.
Presented by:


TVD/SSP methods
  • S. Gottlieb and C.-W. Shu. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation, 67, 73-85, 1998.
  • S. Gottlieb, C.-W. Shu, and E. Tadmor. Strong stability preserving high-order time discretization methods. SIAM Review, 43, 89-112, 2001.
  • S. J. Ruuth and R. J. Spiteri. Two barriers on strong-stability-preserving time discretization methods. Journal of Scientific Computation, 17, 211-220, 2002.
  • S. Gottlieb. On high order strong stability preserving Runge-Kutta and multi step time discretizations. Journal of Scientific Computing, 25, 105-127, 2005.
  • Extensive biblioghraphy at [LINK] http://www.cfm.brown.edu/people/sg/SSPpage/sspsite/sspbib.html
Presented by: X. Zhong on December 9, 2009


Variational integrators
  • M. West, Variational Integrators. 2003 Thesis. [LINK]
  • A. Lew, J.E. Marsden, M. Ortoz, M. West, An Overview of Variational Integrators. [Preprint]
Presented by: