»ã±¨±êÌâ (Title)£ºAdaptive grids for detecting non-monotone waves and instabilities in a non-equilibrium PDE model from porous media £¨Ì½²â¶à¿×½éÖÊ·ÇƽºâPDEÄ£ÐÍÖзǵ¥µ÷²¨Óë²»²»±äÐÔµÄ×ÔÊÊÓ¦Íø¸ñ£©

»ã±¨ÈË (Speaker)£ºZegeling, P.A. ½ÌÊÚ£¨Utrecht University£©

»ã±¨¹¦·ò (Time)£º2025Äê11ÔÂ17ÈÕ£¨ÖÜÒ»£©10:00-12:00

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿GJ303

Ô¼ÇëÈË(Inviter)£ºÀƷ¡¢²ÌÃô

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»ã±¨ÌáÒª£ºSpace-time evolution described by nonlinear PDE models involves patterns and qualitative changes induced by parameters. In this talk I will emphasize the importance of both the analysis and computation 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 was suggested by Hassanizadeh and Gray in the 90's. This gives rise to an extra {\it third-order mixed} space-time derivative term in the PDE of the form $\tau ~ \nabla \cdot [T(S) \nabla (S_t)]$.

In one space dimension traveling wave analysis is able to predict the formation of steep non-monotone waves depending on $\tau$. In 2D, the parameters $\tau$ and the frequency $\omega$ included in a small perturbation term, predict that the waves may become {\it unstable}, thereby initiating so-called gravity-driven fingering structures. This phenomenon can be analysed with a linear stability analysis and its effects are supported by the numerical experiments of the 2D time-dependent PDE model. For this purpose, we have used a sophisticated adaptive grid r-refinement technique based on a recently developed monitor function. The numerical experiments in one and two space dimension show the effectiveness of the adaptive grid solver.