Abstract
In this paper, the importance of both the analysis and computation is emphasized, in relation to a bifurcation problem in a non-equilibrium Richard’s equation from hydrology. The extension of this PDE model for the water saturation S, to take into account additional dynamic memory effects gives rise to an extra third-order mixed space-time derivative term in the PDE of the form τ ∇⋅ [f(S)∇(S t)]. In one space dimension, travelling wave analysis is able to predict the formation of steep non-monotone waves depending on the parameter τ. In two space dimensions, the parameters τ and the frequency ω of a small perturbation term, predict that the waves may become unstable, thereby initiating so-called gravity-driven fingering structures. For the numerical experiments of the time-dependent PDE model, we have used a sophisticated adaptive grid r-refinement technique based on a scaled monitor function.
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Zegeling, P.A. (2021). Adaptive Grids for Non-monotone Waves and Instabilities in a Non-equilibrium PDE Model. In: Garanzha, V.A., Kamenski, L., Si, H. (eds) Numerical Geometry, Grid Generation and Scientific Computing. Lecture Notes in Computational Science and Engineering, vol 143. Springer, Cham. https://doi.org/10.1007/978-3-030-76798-3_11
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