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Abstract In this paper, we focus on a discrete physical model describing granular crystals, whose equations of motion can be described by a system of differential difference equations. After revisiting earlier continuum approximations, we propose a regularized continuum model variant to approximate the discrete granular crystal model through a suitable partial differential equation. We then compute, both analytically and numerically, its travelling wave and periodic travelling wave solutions, in addition to its conservation laws. Next, using the periodic solutions, we describe quantitatively various features of the dispersive shock wave (DSW) by applying Whitham modulation theory and the DSW fitting method. Finally, we perform several sets of systematic numerical simulations to compare the corresponding DSW results with the theoretical predictions and illustrate that the continuum model provides a good approximation of the underlying discrete one.
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Mixed finite element methods for nonlinear reaction–diffusion equations with interfaces
We develop mixed finite element methods for nonlinear reaction–diffusion equations with interfaces which have Robin-type interface conditions. We introduce the velocity of chemicals as new variables and reformulate the governing equations. The stability of semidiscrete solutions, existence and the a priori error estimates of fully discrete solutions are proved by fixed point theorem and continuous/discrete Gronwall inequalities. Numerical results illustrating our theoretical analysis are included.
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- Award ID(s):
- 2110781
- PAR ID:
- 10538372
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Computational and Applied Mathematics
- Volume:
- 443
- Issue:
- C
- ISSN:
- 0377-0427
- Page Range / eLocation ID:
- 115756
- Subject(s) / Keyword(s):
- Reaction–diffusion equations Mixed finite element methods Interface conditions Error analysis
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.cam.2024.115756
