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A high-order global spatially adaptive collocation method for 1-D parabolic PDEs. (English) Zbl 1049.65110
Summary: We describe a high-order solver, adaptive in space and time, for the efficient numerical solution of one-dimensional parabolic partial differential equations (PDEs). Collocation at Gaussian points is employed for the spatial discretization, using a B-spline basis. A modification of the well-known differential-algebraic equation solver, DASSL is used for the time integration. An a posteriori spatial error estimate is calculated at each successful time step. A new mesh selection strategy based on an equidistribution principle is presented for controlling the spatial error which is balanced with respect to the temporal error. This new mesh adaptation algorithm is shown to be robust, and particularly efficient for problems having solutions with rapid variation.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65M15 Error bounds 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
Full Text: DOI
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