Abstract
We consider the fully adaptive space–time discretization of a class of nonlinear heat equations by Rothe’s method. Space discretization is based on adaptive polynomial collocation which relies on equidistribution of the defect of the numerical solution, and the time propagation is realized by an adaptive backward Euler scheme. From the known scaling laws, we infer theoretically the optimal grids implying error equidistribution, and verify that our adaptive procedure closely approaches these optimal grids.
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Ascher, U., Ruuth, S., Wetton, T.R.: Implicit–explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)
Bebernes, J., Bricher, S.: Final time blowup profiles for semilinear parabolic equations via center manifold theory. SIAM J. Math. Anal. 23(4), 852–869 (1992)
Budd, C.J., Williams, J.: How to adaptively resolve evolutionary singularities in differential equations with symmetry ? J. Eng. Math. 66, 217–236 (2010)
Budd, C.J., Huang, W., Russell, R.D.: Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17, 305–327 (1996)
Huang, W., Russell, R .D.: Adaptive Moving Mesh Methods. Springer, New York (2010)
Kitzhofer, G., Koch, O., Pulverer, G., Simon, C., Weinmüller, E.: The new MATLAB code BVPSUITE for the solution of singular implicit boundary value problems. JNAIAM J. Numer. Anal. Ind. Appl. Math. 5, 113–134 (2010)
Pulverer, G., Söderlind, G., Weinmüller, E.: Automatic grid control in adaptive BVP solvers. Numer. Algorithms 56(1), 61–92 (2011)
Quarteroni, A., Saleri, F.: Approximation of Functions and Data. Springer, Berlin (2006)
Söderlind, G.: Digital filters in adaptive time-stepping. ACM Trans. Math. Softw. 29, 1–26 (2003)
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Budd, C., Koch, O., Taghizadeh, L. et al. Asymptotic properties of the space–time adaptive numerical solution of a nonlinear heat equation. Calcolo 55, 43 (2018). https://doi.org/10.1007/s10092-018-0286-z
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DOI: https://doi.org/10.1007/s10092-018-0286-z