Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations |
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Authors: | Hong‐Lin Liao Zhi‐Zhong Sun |
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Affiliation: | 1. Department of Mathematics, Southeast University, Nanjing 210096, People's Republic of China;2. Department of Applied Mathematics and Physics, Institute of Sciences, PLA University of Science and Technology, Nanjing 211101, People's Republic of China |
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Abstract: | Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth‐order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher‐order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 |
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Keywords: | ADI scheme asymptotic expansion compact ADI scheme discrete energy method parabolic equation Richardson extrapolation |
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