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Comparison of adaptive methods for one-dimensional parabolic systems. (English) Zbl 0822.65069
This paper compares three adaptive spatial mesh refinement procedures for solving one-dimensional parabolic partial differential equations based on a finite element Galerkin approach with a piecewise polynomial hierarchical spatial basis.
The first approach uses the standard package EPDCOL while the other two techniques are based on a MOL (method of lines)-time integration approach (DASSL) and an integrated spatial and temporal refinement approach in which the time integrators are singly implicit collocation Runge-Kutta methods. In the singly implicit case regridding is allowed at each time step and the stage values are used to predict future spatial discretizations.
Some numerical tests show that the singly implicit Runge-Kutta approach is the most robust and compares favourably with EPDCOL computationally but that, in general, DASSL is much more computationally efficient than either.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
Software:
DASSL; EPDCOL; PDECOL; RODAS
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References:
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