Prof Choi-Hong Lai – University of Greenwich
On transformation methods and the induced parallel properties for the temporal domain
Many engineering and applied science problems are described by time dependent nonlinear partial differential equations. Numerical methods of handling transient problems are usually based on temporal integration methods such as Euler’s method, Runge-Kutta methods, multi-step methods, amongst many others. In relation to the nature of a given problem which may or may not require fine solution details at intermediate time steps, one usually has to choose a fine or a coarse time stepping. In the case of fine details are required the traditional method is to use temporal integration methods with fine time steps. These temporal integration methods are very difficult to parallelise because of their intrinsic sequential properties. In the case where fine details are not required it is still not possible to use a very large time step in an implicit scheme. There are restrictions imposed on the temporal step size usually due to stability criteria of an explicit scheme or the truncation errors of an implicit scheme in approximating the temporal derivatives. Computing time of such numerical methods inevitably becomes significant. There are also many problems which require solution details not at each time step of the time-marching scheme, but only at a few crucial steps and the steady state. Therefore effort in finding fine details of the solutions using many intermediate time steps is considered being wasted. Such effort becomes significant in the case of nonlinear problems where a linearisation process, which amounts to an inner iterative loop within the time-marching scheme, is required. It would be a significant save in computing time when the linearisation process and the time-marching scheme can both be done in parallel. The main objective of the present work is to remove the time stepping and to combine it with parallel/distributed computers.
To investigate the parallelisation of the temporal domain, this talk begins with a concise overview of classical temporal integration methods, including time-stepping restrictions of an explicit scheme, truncation errors in an implicit scheme, and other advantages and disadvantages of using a time marching scheme, and a brief discussion is given of several attempts by various researchers in parallelising temporal integration methods. Second, the use of transformation methods and their relations to possibly induce parallel properties to certain intrinsic sequential problems are examined. These transformation methods include the Boltzmann transformations, general stretch transformations, Fourier transformation, and Laplace transformation. Several examples related to these transformations are discussed, including diffusion-convection and image processing problems. Finally, discussions and conclusions are presented.
G.J. Moridis and D.L. Reddell. The Laplace transform finite difference method for simulation of flow through porous media. Water Resources Research, 27, 1873 – 1884, 1991.
C.-H. Lai, A. K. Parrott, S. Rout. A distributed algorithm for European options with nonlinear volatility. Computers and Mathematics with Applications, 49, 885 – 894, 2005.
C.-H. Lai. On transformation methods and the induced parallel properties for the temporal domain. In Substructuring Techniques and Domain Decomposition Methods, ed F. Magoules, 45 – 70, Saxe-Coburg Publications, 2010.
Prof. Lai completed a BSc in Mathematics with Engineering at Queen Mary, University of London, with a 1st class honours in 1981 and a PhD in the area of aerodynamics and numerical partial differential equations at the same institution in 1985. He stayed at Queen Mary as a research fellow, working in the area of parallel finite element techniques, until 1989 before joining University of Greenwich in August 1989 as a lecturer. His main research interests include numerical partial differential equations, distributed and parallel algorithms, inverse problems, image processing, computational biology, and computational aeroacoustics. He is currently a professor of numerical mathematics at the Department of Mathematical Sciences, University of Greenwich. He also holds three visiting professorships are Buckingham University, Jiangnan University, China, and Fuzhou University, China.
School of Computing, Robert Gordon University, St Andrew Street, Aberdeen, Lecture Room C48, 14:15 – 15:15.