Conference ID: 925-756-01590
PIN Code: 605793
The ubiquity of semilinear parabolic equations has been illustrated in their numerous applications ranging from physics, biology, to materials and social sciences. In this talk, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form $u_t=\hL u+f[u]$ with $\hL$ being some linear elliptic operator (or its various approximations) and $f$ being some nonlinear operator, namely a maximum bound principle, or more precisely, a time-invariant uniform bound in the sense that, the time-dependent solution $u$ defined in a spatial domain preserves for all time a spatially uniform pointwise bound satisfied by its initial data. We first study an analytical framework for some sufficient conditions on $\hL$ and $f$ that lead to such a maximum bound principle for the time-continuous dynamic system.
Then, we utilize the exponential time differencing approach and stabilization techniques to develop first and second order accurate temporal discretization schemes that satisfy such maximum bound principle unconditionally in the time-discrete setting. Error estimates of the schemes are derived along with their energy stability. Extensions to vector- and matrix-valued systems are also discussed.
We demonstrate through several examples that the analytical framework developed here offers an effective and unified approach to study the maximum bound principle of the abstract evolution equation that cover a wide variety of well-known model equations and their numerical discretization schemes.
Dr. Zhonghua Qiao is an associate professor at the Department of Applied Mathematics at the Hong Kong Polytechnic University.
He obtained his Ph.D. degree in Computational Mathematics from Hong Kong Baptist University in 2006. Recently, his research is mainly focused on numerical analysis and simulations of phase field problems. Dr. Qiao has received several awards in recognition of his research, including the 2013-2014 Early Career Award from the Research Grants Council of Hong Kong in 2013 and Hong Kong Mathematical Society Award for Young Scholars from the Hong Kong Mathematical Society in 2018.