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On stability issues for IMEX schemes applied to 1D scalar hyperbolic equations with stiff reaction terms


Authors: R. Donat, I. Higueras and A. Martínez-Gavara
Journal: Math. Comp. 80 (2011), 2097-2126
MSC (2010): Primary 35L65, 65M20, 65L06, 65L20, 65M12
DOI: https://doi.org/10.1090/S0025-5718-2011-02463-4
Published electronically: February 4, 2011
MathSciNet review: 2813350
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Abstract: The application of a Method of Lines to a hyperbolic PDE with source terms gives rise to a system of ODEs containing terms that may have very different stiffness properties. In this case, Implicit-Explicit Runge-Kutta (IMEX-RK) schemes are particularly useful as high order time integrators because they allow an explicit handling of the convective terms, which can be discretized using the highly developed shock capturing technology, together with an implicit treatment of the source terms, necessary for stability reasons.

Motivated by the structure of the source term in a model problem introduced by LeVeque and Yee in [J. Comput. Phys. 86 (1990)], in this paper we study the preservation of certain invariant regions as a weak stability constraint. For the class of source terms considered in this paper, the unit interval is an invariant region for the model balance law. In the first part of the paper, we consider first order time discretizations, which are the basic building blocks of higher order IMEX-RK schemes, and study the conditions that guarantee that $ [0,1]$ is also an invariant region for the numerical scheme. In the second part of the paper, we study the conditions that ensure the preservation of this property for higher order IMEX schemes.


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Additional Information

R. Donat
Affiliation: Departament de Matemàtica Aplicada, Universitat de València, 46100 Burjassot, Spain
Email: donat@uv.es

I. Higueras
Affiliation: Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain
Email: higueras@unavarra.es

A. Martínez-Gavara
Affiliation: Departamento de Matemática Aplicada I, Universidad de Sevilla, 41012 Sevilla, Spain
Email: gavara@us.es

DOI: https://doi.org/10.1090/S0025-5718-2011-02463-4
Received by editor(s): July 31, 2009
Received by editor(s) in revised form: April 26, 2010, and July 6, 2010
Published electronically: February 4, 2011
Additional Notes: The authors acknowledge support from projects MTM2008-00974, MTM2008-00785, MTM2006-01275 and MTM2009-07719.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.