Dr Simon J.A. Malham

• Room No. : CM T.21
• Address :    Mathematics Department   
                      Heriot-Watt University 
                      Edinburgh EH14 4AS 
                      Scotland
• Telephone No. : +44 131 451 3254
• Fax No. : +44 131 451 3249
• E-Mail Address : simonm[at]ma.hw.ac.uk

Research

• The Navier-Stokes equations and turbulent flows. Do solutions of the Navier-Stokes equations develop a singularity in finite time? Since Leray suggested these singular solutions may signify the onset of turbulence in a real fluid flow, it is also an interesting engineering issue. This remains an important open problem and is currently one of the millenium prize problems set by the Clay Institute and worth a million dollars (see the criteria devised by Charles Fefferman).
  
  LMS-EPSRC Short Course: Theoretical Fluid Dynamics
  Heriot-Watt University: August 29th - September 2nd 2011
  
  
• Reaction-diffusion phenomena:
• Travelling waves;
• Autocatalysis;
• Combustion.
• Computational spectral theory:
• Shooting and matching, Evans function;
• Geometric/Lie group integration;
• Grassmannian and Riccati numerical methods;
• Multi-dimensional stability.
• Stochastic differential equations:
• Strong approximation;
• Lie group integration;
• Hopf algebra structure;
• Implicit methods;
• Positivity preservation;
• Linear-quadratic optimal control.

If you would like to know more or are interested in doing a PhD, email me: simonm [at] ma.hw.ac.uk.

Publications

• Bartuccelli, M.V., Gibbon, J.D. and Malham, S.J.A. 1993, Lattice methods and the pressure field for solutions of the Navier-Stokes equations (pdf), Nonlinearity 6(1), pp. 79-92.
• Bartuccelli, M.V., Doering, C.R., Gibbon, J.D. and Malham, S.J.A. 1993, Length scales in solutions of the Navier-Stokes equations (pdf), Nonlinearity 6(4), pp. 549-568.
• Malham, S. and Xin, J. 1998, Global solutions to a reactive Boussinesq system with front data on an infinite domain (pdf), Comm. Math. Phys. 193(2), pp. 287-316.
• Balmforth, N.J., Craster, R.V. and Malham, S.J.A. 1999, Unsteady fronts in an autocatalytic system (pdf), Proc. R. Soc. Lond. A 455, pp. 1401-1433.
• Malham, S.J.A. and Oliver, M. 2000, Accelerating fronts in autocatalysis (pdf) , Proc. R. Soc. Lond. A 456, pp. 1609-1624.
• Ford, R., Malham, S.J.A. and Oliver, M. 2002, A new model for shallow water in the low Rossby-number limit (pdf) , JFM 450, pp. 287-296.
• Aparicio, N.D., Malham, S.J.A. and Oliver, M. 2005, Numerical evaluation of the Evans function by Magnus integration (pdf), BIT 45, pp. 219-258.
• Lord, G., Malham, S.J.A. and Wiese, A. 2008, Efficient strong integrators for linear stochastic systems (arxiv link), SIAM J. on Numerical Analysis 46(6) (2008), pp. 2892-2919.
• Malham, S. and Niesen, J., Evaluating the Evans function: order reduction in numerical methods (arxiv link), Math. Comp. 77 (2008), pp. 159-179.
• Malham, S.J.A. and Wiese, A., Stochastic Lie group integrators (arxiv link), SIAM J. Sci. Comput. 30 (2) (2008), pp. 597-617.
• Ledoux, V., Malham, S.J.A. and Thummler, V., Grassmannian spectral shooting (arxiv link), Math. Comp., published electronically Jan 2010.
• Malham, S.J.A. and Wiese, A. 2011, Positive volatility simulation in the Heston model (arxiv link), submitted.
• Ledoux, V., Malham, S.J.A., Niesen, J. and Thummler, V., Computing stability of multi-dimensional travelling waves (arxiv link), SIAM J. on Applied Dynamical Systems 8(1) (2009), pp. 480-507.
• Malham, S.J.A. and Wiese, A., Stochastic expansions and Hopf algebras (arxiv link), Proc. R. Soc. A 465 (2009), pp. 3729-3749. doi:10.1098/rspa.2009.0203
• Malham, S.J.A. and Wiese, A. 2011, Levy area logistic expansion and simulation (arxiv link), submitted.
• Ebrahimi-Fard, K., Lundervold, A., Malham, S.J.A., Munthe-Kaas, H. and Wiese, A. 2011, Algebraic structure of stochastic expansions and efficient simulation (arxiv link), submitted.

Unpublished reviews/notes/thesis

Thesis

Regularity assumptions and length scales for the Navier-Stokes equations (July 1993)

Reviews

Review notes: Introductory Schubert calculus (with V. Ledoux, v1: Jan 2010, v2: Sept 2010)

An introduction to SDE simulation (with A. Wiese, Feb 2010)

Review notes: Introductory Fredholm theory and computation (with I. Karambal, V. Ledoux and J. Niesen, Nov 2010)

Lecture notes

Differential equations and linear algebra (March 2008) (whole course)

An introduction to asymptotic analysis (Nov 2005)

An introduction to Lagrangian and Hamiltonian mechanics (Jan 2010) (whole course)

Introductory fluid mechanics (August 2011)

Miscellaneous

• Recent talks
• Surfing