Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes

Du, Qiang; Ju, Lili; Li, Xiao; Qiao, Zhonghua

doi:10.1137/19M1243750

Title: Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes

Journal Article · Fri May 01 00:00:00 EDT 2020 · SIAM Review

DOI:https://doi.org/10.1137/19M1243750· OSTI ID:1631279

Du, Qiang ^[1]; Ju, Lili ^[2]; Li, Xiao ^[3]; Qiao, Zhonghua ^[3]

Columbia Univ., New York, NY (United States)
Univ. of South Carolina, Columbia, SC (United States)
Hong Kong Polytechnic Univ. (Hong Kong)

The ubiquity of semilinear parabolic equations has been highlighted in their numerous applications ranging from physics, biology, to materials and social sciences. Here, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $$u_t=\mathcal{L}u+f[u]$$ with $$\mathcal{L}$$ being a linear dissipative operator and $$f$$ being a nonlinear operator in space, namely a time-invariant maximum bound principle, in the sense that the time-dependent 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 some sufficient conditions on $$\mathcal{L}$$ 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 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 study 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.

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Research Organization:: Univ. of South Carolina, Columbia, SC (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Biological and Environmental Research (BER). Biological Systems Science Division; National Science Foundation (NSF); National Natural Science Foundation of China (NSFC); Hong Kong Research Council

Grant/Contract Number:: SC0020270; SC0016540

OSTI ID:: 1631279

Journal Information:: SIAM Review, Vol. 36, Issue 2; ISSN 0036-1445

Publisher:: Society for Industrial and Applied MathematicsCopyright Statement

Country of Publication:: United States

Language:: English

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