Abstract
In many applications, the governing PDE to be solved numerically contains a stiff component. When this component is linear, an implicit time stepping method that is unencumbered by stability restrictions is often preferred. On the other hand, if the stiff component is nonlinear, the complexity and cost per step of using an implicit method is heightened, and explicit methods may be preferred for their simplicity and ease of implementation. In this article, we analyze new and existing linearly stabilized schemes for the purpose of integrating stiff nonlinear PDEs in time. These schemes compute the nonlinear term explicitly and, at the cost of solving a linear system with a matrix that is fixed throughout, are unconditionally stable, thus combining the advantages of explicit and implicit methods. Applications are presented to illustrate the use of these methods.
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Notes
We will refer to these simply as IMEX methods.
To address this deficiency, [1] recommended mCNAB, a scheme closely related to CNAB but with stronger damping of high frequencies. As it turns out, the two are equivalent within this linear stabilization framework.
For simplicity, going forward we will refer to linearly stabilized IMEX methods without prefacing by “linearly stabilized”. For example, we will refer to the “the linearly stabilized CNAB method” as CNAB and the “the linearly stabilized SBDF3 method” as SBDF3, etc.
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We are grateful to the referees for their constructive input.
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The authors gratefully acknowledge the financial support of NSERC Canada (RGPIN 2016-04361).
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Chow, K., Ruuth, S.J. Linearly Stabilized Schemes for the Time Integration of Stiff Nonlinear PDEs. J Sci Comput 87, 95 (2021). https://doi.org/10.1007/s10915-021-01477-0
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DOI: https://doi.org/10.1007/s10915-021-01477-0