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Title: Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes
Authors: Du, Q
Ju, L
Li, X 
Qiao, Z 
Issue Date: 2021
Source: SIAM review, 2021, v. 63, no. 2, p. 317-359
Abstract: The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form ut = L u + f[u], with L a linear dissipative operator and f a nonlinear operator in space, namely, a time-invariant maximum bound principle, in the sense that the timedependent solution u preserves for all time a uniform pointwise bound in absolute value imposed by its initial and boundary conditions. We first study an analytical framework for sufficient conditions on L and f that lead to such a maximum bound principle for the time-continuous dynamic system of infinite or finite dimensions. Then we utilize a suitable exponential time-differencing approach with a properly chosen generator of the contraction semigroup to develop first- and second-order accurate temporal discretization schemes that satisfy the maximum bound principle unconditionally in the time-discrete setting. Error estimates of the proposed schemes are derived along with their energy stability. Extensions to vector- and matrix-valued systems are also discussed. We demonstrate that the abstract framework and analysis techniques developed here offer an effective and unified approach to studying the maximum bound principle of the abstract evolution equation that covers a wide variety of well-known models and their numerical discretization schemes. Some numerical experiments are also carried out to verify the theoretical results.
Keywords: Energy stability
Error estimate
Exponential time differencing
Maximum bound principle
Numerical approximation
Semilinear parabolic equation
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM review 
ISSN: 0036-1445
EISSN: 1095-7200
DOI: 10.1137/19M1243750
Rights: First Published in SIAM Review in Volume 63, Issue 2, published by the Society for Industrial and Applied Mathematics (SIAM)
© 2021, Society for Industrial and Applied Mathematics
Unauthorized reproduction of this article is prohibited.
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