Adaptive moving mesh methods for two-dimensional resistive magneto-hydrodynamic PDE models

Tan, Zhijun

doi:10.1016/j.compfluid.2006.04.005

Cited by 8 publications

(6 citation statements)

References 46 publications

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“…Provided that these conditions are carefully considered, r-adaptive methods can be used with considerable success for many time-evolving systems. Examples of these include computational fluid dynamics (Yanenko, Kroshko, Liseikin, Fomin, Shapeev and Shitov 1976), groundwater flow Zhan 2004, Huang, Zheng andZhan 2002), blow-up problems (Budd, Huang and Russell 1996, Budd and Williams 2006, Ceniceros and Hou 2001, Ren and Wang 2000, chemotaxis systems (Budd, Carretero-Gonzalez and Russell 2005), reaction-diffusion systems (Zegeling and Kok 2004), the nonlinear Schrödinger equation (Sulem and Sulem 1999, Ceniceros 2002, Budd, Chen and Russell 1999a, phase change problems (Mackenzie and Robertson 2002, Mackenzie and Mekwi 2007a, Tan, Lim and Khoo 2007, shear layer calculations (Tang 2005), gas dynamics Petzold 1997, Li, Petzold andRen 1998), hyperbolic conservation laws with high Mach number (Li and Petzold 1997, Tang 2005, Stockie, Mackenzie and Russell 2000, Tang and Tang 2003, problems with high vorticity (Ceniceros and Hou 2001), magneto-hydrodynamics (Tan 2007) and meteorological problems (Budd and Piggott 2005). More details of such applications are given in Section 5.…”

Section: Adaptivity On a Moving Meshmentioning

confidence: 99%

Adaptivity with moving grids

2009

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In this article we survey r-adaptive (or moving grid) methods for solving time-dependent partial differential equations (PDEs). Although these methods have received much less attention than their h- and p-adaptive counterparts, particularly within the finite element community, we review the substantial progress that has been made in developing more robust and reliable algorithms and in understanding the basic principles behind these methods, and we give some numerical examples illustrative of the wide classes of problems for which these methods are suitable alternatives to the traditional ones.More specifically, we first examine the basic geometric properties of moving meshes in both one and higher spatial dimensions, and discuss the discretization process for PDEs on such moving meshes (both structured and unstructured). In particular, we consider the issues of mesh regularity, equidistribution, alignment, and associated variational methods. An overview is given of the general interpolation error analysis for a function or a truncation error on such an adaptive mesh. Guided by these principles, we show how to design effective moving mesh strategies. We then examine in more detail how these strategies can be implemented in practice. The first class of methods which we consider are based upon controlling mesh density and hence are called position-based methods. These make use of a so-called moving mesh PDE (MMPDE) approach and variational methods, as well as optimal transport methods. This is followed by an analysis of methods which have a more Lagrange-like interpretation, and due to this focus are called velocity-based methods. These include the moving finite element method (MFE), the geometric conservation law (GCL) methods, and the deformation map method. Finally, we present a number of specific types of examples for which the use of a moving mesh method is particularly effective in applications. These include scale-invariant problems, blow-up problems, problems with moving fronts and problems in meteorology. We conclude that, whilst r-adaptive methods are still in their relatively early stages of development, with many outstanding questions remaining, they have enormous potential and indeed can produce an optimal form of adaptivity for many problems.

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Section: Adaptivity On a Moving Meshmentioning

confidence: 99%

Adaptivity with moving grids

2009

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“…Our experience shows that the accuracy of this scheme is important to the results of the model, and that a low‐order interpolation can make the solution excessively diffusive. The mesh moving scheme is similar to Tan [2007], although the form of the monitor function differs.…”

Section: Numericsmentioning

confidence: 99%

Grounding line movement and ice shelf buttressing in marine ice sheets

2009

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[1] Understanding the dynamics of marine ice sheets is integral to studying the evolution of the Antarctic ice sheet in both the short and long terms. An important component of the dynamics, grounding line migration, has proved difficult to represent in numerical models, and most successful attempts have made use of techniques that are only readily applicable to flow line models. However, to capture the stress transmission involved in another important component, the buttressing of a marine ice sheet by its ice shelf, the transverse direction must also be resolved. We introduce a model that solves the time-dependent shelfy stream equations and makes use of mesh adaption techniques to overcome the difficulties typically associated with the numerics of grounding line migration. We compare the model output with a recent benchmark for flow line models and show that our model yields an accurate solution while using far less resources than would be required without mesh adaption. We also show that the mesh adapting techniques extend to two horizontal dimensions. Experiments are carried out to determine how both ice shelf buttressing and ice rises affect the marine instability predicted for an ice sheet on a foredeepened bed. We find that buttressing is not always sufficient to stabilize such a sheet but collapse of the grounded portion is still greatly delayed. We also find that the effect of an ice rise is similar to that of narrowing the ice shelf.

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“…At each iteration, the mesh generation equations are linearized by freezing the coefficient on current mesh and then are solved with RBFs collocation method. The iterative algorithm is based on the approach proposed in [26], to do the grid restructuring for two-dimensional time dependent PDEs.…”

Section: Adaptive Mesh Methodsmentioning

confidence: 99%