Abstract
This paper considered cross-diffusion equations. With those equations the concentration development in a certain region during an interesting time-interval can be described.
Cross-diffusion means the diffusion of some species which influence each other. The population dynamics of different species is a famous example of cross-diffusion.
The implicit time-integration of such parabolic equations leads to nonlinear equation systems which requires a huge computational amount.
To avoid this amount we discuss a linear scheme proposed by Murakawa [2] and investigate his properties.
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Notes
- 1.
The function [q] gives the largest integer back, which is less than q.
References
Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theor. Biol. 79, 83–99 (1979)
Murakawa, H.: A linear scheme to approximate nonlinear cross-diffusion systems. ESAIM: M”AN 45, 1141–1161 (2011)
Murakawa, H.: Error estimates for discrete-time approximation of nonlinear cross-diffusion systems. SIAM J. Numer. Anal. 52(2), 955–974 (2014)
Chen, L., Jüngel, A.: Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal. 36, 301–322 (2006)
Baumbach, J.: Numerische Untersuchungen eines linearen Zeitintegrationsschemas für nichtlineare Kreuz-Diffusionsprobleme. Master-thesis, TU Berlin (2018)
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Bärwolff, G., Baumbach, J. (2019). Numerical Solution of Nonlinear Cross Diffusion Problems. In: Misra, S., et al. Computational Science and Its Applications – ICCSA 2019. ICCSA 2019. Lecture Notes in Computer Science(), vol 11619. Springer, Cham. https://doi.org/10.1007/978-3-030-24289-3_2
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DOI: https://doi.org/10.1007/978-3-030-24289-3_2
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