×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adjerid, S.; Aiffa, M.; Flaherty, J.E., High-order finite element methods for singularly-perturbed elliptic and parabolic problems, SIAM J. appl. math., 55, 520-543, (1995) · Zbl 0827.65097
[2] Ascher, U.; Christiansen, J.; Russell, R.D., Collocation software for boundary value odes, ACM trans. math. software, 7, 209-222, (1981) · Zbl 0455.65067
[3] Adjerid, S.; Flaherty, J.E.; Moore, P.K.; Wang, Y., High-order adaptive methods for parabolic systems, Physica D, 60, 94-111, (1992) · Zbl 0790.65088
[4] Adjerid, S.; Flaherty, J.E.; Wang, Y., A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems, Numer. math., 65, 1-21, (1993) · Zbl 0791.65070
[5] Ascher, U.; Mattheij, R.M.M.; Russell, R.D., Numerical solution of boundary value problems for ordinary differential equations, (1995), SIAM Philadelphia, PA
[6] De Boor, C., Package for calculating with B-splines, SIAM J. numer. anal., 14, 441-472, (1977) · Zbl 0364.65008
[7] De Boor, C., A practical guide to splines, (1978), Springer New York · Zbl 0406.41003
[8] Berzins, M.; Capon, P.J.; Jimack, P.K., On spatial adaptivity and interpolation when using the method of lines, Appl. numer. math., 26, 117-133, (1998) · Zbl 0890.65102
[9] Brenan, K.E.; Campbell, S.L.; Petzold, L.R., Numerical solution of initial-value problems in differential – algebraic equations, (1996), SIAM Philadelphia, PA · Zbl 0844.65058
[10] Blom, J.G.; Sanz-Serna, J.M.; Verwer, J.G., On simple moving grid methods for one-dimensional evolutionary partial differential equations, J. comput. phys., 74, 191-213, (1988) · Zbl 0645.65060
[11] Blom, J.G.; Zegeling, P.A., Algorithm 731: A moving-grid interface for systems of one-dimensional time-dependent partial differential equations, ACM trans. math. software, 20, 194-214, (1994) · Zbl 0889.65099
[12] Carroll, J., A composite integration scheme for the numerical solution of systems of parabolic PDEs in one space dimension, J. comput. appl. math., 46, 327-343, (1993) · Zbl 0779.65058
[13] Diaz, J.C.; Fairweather, G.; Keast, P., FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM trans. math. software, 9, 358-375, (1983) · Zbl 0516.65013
[14] Huang, W.; Russell, R.D., A moving collocation method for solving time dependent partial differential equations, Appl. numer. math., 20, 101-116, (1996) · Zbl 0859.65112
[15] Keast, P.; Muir, P.H., Algorithm 688. EPDCOL: A more efficient PDECOL code, ACM trans. math. software, 17, 153-166, (1991) · Zbl 0900.65270
[16] Moore, P.K., A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension, SIAM J. numer. anal., 31, 149-169, (1994) · Zbl 0798.65089
[17] Moore, P.K., Comparison of adaptive methods for one dimensional parabolic systems, Appl. numer. math., 16, 471-488, (1995) · Zbl 0822.65069
[18] Moore, P.K., Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension, Numer. math., 90, 149-177, (2001) · Zbl 0997.65103
[19] Miller, K.; Miller, R.N., Moving finite elements. I, SIAM J. numer. anal., 18, 1019-1032, (1981) · Zbl 0518.65082
[20] Madsen, N.K.; Sincovec, R.F., Algorithm 540. PDECOL, general collocation software for partial differential equations, ACM trans. math. software, 5, 326-351, (1979) · Zbl 0426.35005
[21] L.R. Petzold, A description of DASSL: A differential/algebraic system solver, SAND82-8637, Sandia Labs, Livermore, CA, 1982
[22] Wang, R., High order adaptive method of lines for 1-D parabolic equations, (1999), M.Sc. Thesis, Dalhousie University, Halifax
[23] Wang, R., High order adaptive collocation software for 1-D parabolic pdes, (2002), Ph.D. Thesis, Dalhousie University, Halifax
[24] Wang, R.; Keast, P.; Muir, P.H., A comparison of adaptive software for 1-D parabolic pdes, J. comput. appl. math., (2004), in press · Zbl 1070.65564
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.