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An adaptive multiresolution method for parabolic PDEs with time‐step control
Author(s) -
Domingues M. O.,
Roussel O.,
Schneider K.
Publication year - 2008
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2501
Subject(s) - discretization , grid , partial differential equation , mathematics , multiresolution analysis , algorithm , mathematical optimization , reduction (mathematics) , representation (politics) , spacetime , runge–kutta methods , computer science , numerical analysis , mathematical analysis , geometry , discrete wavelet transform , physics , wavelet transform , quantum mechanics , artificial intelligence , politics , political science , wavelet , law
We present an efficient adaptive numerical scheme for parabolic partial differential equations based on a finite volume (FV) discretization with explicit time discretization using embedded Runge–Kutta (RK) schemes. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. Compact RK methods of second and third order are then used to choose automatically the new time step while controlling the approximation error in time. Non‐admissible choices of the time step are avoided by limiting its variation. The implementation of the multiresolution representation uses a dynamic tree data structure, which allows memory compression and CPU time reduction. This new numerical scheme is validated using different classical test problems in one, two and three space dimensions. The gain in memory and CPU time with respect to the FV scheme on a regular grid is reported, which demonstrates the efficiency of the new method. Copyright © 2008 John Wiley & Sons, Ltd.