By Alain Vande Wouwer, A Vande Wouwer, Ph. Saucez, W.E. Schiesser
The overall approach to traces (MOL) approach presents a versatile structure for the answer of the entire significant sessions of partial differential equations (PDEs) and is very like minded to evolutionary, nonlinear wave PDEs. regardless of its software, despite the fact that, there are really few texts that discover it at a extra complex point and replicate the method's present kingdom of development.Written via unique researchers within the box, Adaptive approach to strains displays the variety of suggestions and purposes concerning the MOL. so much of its chapters concentrate on a specific software but in addition offer a dialogue of underlying philosophy and process. specific recognition is paid to the idea that of either temporal and spatial adaptivity in fixing time-dependent PDEs. Many very important rules and techniques are brought, together with relocating grids and grid refinement, static and dynamic gridding, the equidistribution precept and the concept that of a computer screen functionality, the minimization of a sensible, and the relocating finite point procedure. functions addressed comprise shallow water circulation, combustion and flame propagation, shipping in porous media, fuel dynamics, chemical engineering strategies, solitary waves, and magnetohydrodynamics.As the 1st complex textual content to symbolize the trendy period of the tactic of strains, this monograph deals a good chance to find new suggestions, examine new options, and discover a variety of purposes.
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Extra info for Adaptive method of lines
Numerical Results This example motivates the use of an adaptive grid technique with a variable number of nodes. Indeed, more nodes are required for reproducing the two separate solitons traveling in opposite direction than for capturing the interaction of these two entities. 5, and Nadapt = 5. Tolerances atol = rtol = 10−6 are imposed for the time integration with RADAU5. The number of nodes varies between N = 89 and 109 over the time interval (0, 45). 6 show snapshots at t = 0, 20, 45, respectively, and compare the adaptive grid solution (dots) with a reference solution computed with 2001 fixed nodes (solid line) as well as with a fixed grid solution with N = 140 nodes (dashed line), which requires the same computational expense as the adaptive grid solution.
In the latter case, time integration can be performed without interruption. However, it is interesting to consider the situation where it is halted after every time integration step, in order to evaluate the computational costs associated with the solver restarts (apparently, about 50% more computation time is required when restarting the solver periodically, resulting however in a slightly better overall accuracy). 1 are average values over the time span of interest. Clearly, the adaptive grid algorithm performs very satisfactorily, both in terms of accuracy and computational costs.
Schiesser, Numerical experiments with the (gradient-weighted) finite element method, submitted. G. G. M. A. E. J. S. D. , Adaptive Methods for Partial Differential Equations, SIAM, Philadelphia (1989), 160–175. J. J. Baines, On the structure of the moving finite-element equations, IMA J. Numer. , 5, (1985), 161–182. J. Wathen, Mesh-independent spectra in the moving finite element equations, SIAM J. , 23, (1986), 797–814. B. White, On the numerical solution of initial/boundary-value problems in one-space dimension, SIAM J.
Adaptive method of lines by Alain Vande Wouwer, A Vande Wouwer, Ph. Saucez, W.E. Schiesser