Application of adaptive methods based on finite difference discretizations to systems of PDAEs

Brito, Paulo; Portugal, António

http://hdl.handle.net/10198/1373

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Authors

Brito, Paulo

Portugal, António

Abstract(s)

In this paper two adaptive algorithms are presented for the solution of systems of evolutive one-dimensional Partial Differential/ AIgebraic Equations (PDAEs). A spatial discretization based on finite difference approximations on arbitrarily spaced grids: transforms the original problem in a set of Ordinary Differential Equations (ODEs), solved via an implicit integrator package (DASSL). The temporal integration is coupled with a spatial adapting strategy. The identification of the spatial subdomains: where the introduction of grid adaptivity is needed, is done through the comparison of the solutions computed with two fixed grids of different sizes. The subproblems generated are solved by two adaptive strategies: the Grid Refinement Method (GRM), that refines the subgrids detected in the previous step, and the Moving Mesh Method (MMM), that includes an additional differential equation for the nodal mobility in each original subproblem. In this paper, these algorithms were successfully applied to the solution of two problems: an isothermal tubular reactor model and a fiame propagation system described by two PDEs referring to fuel mass density and temperature dynamics. The performance of each algorithm is compared to the results obtained by Duarte [1], based on the application of a formulation of the Moving Finite Elements Method, with cubic Hermite polynomials approximations. The MMM algorithm revealed its robustness in dealing with the chosen models. The GRM algorithm originated poorer results, mainly due to errors associated with the boundary conditions procedure.

Keywords

Adaptive methods Grid refinement Moving nodes Partial differential equations Finite difference approximations

URI

http://hdl.handle.net/10198/1373

Citation

Brito, Paulo; Portugal, António (1999). Application of adaptive methods based on finite difference discretizations to systems of PDAEs. Meeting on Numerical Methods for Differential Equations. Coimbra. p.410-419