Simulation of thin film flows with a moving mesh mixed finite element method

We present an efficient mixed finite element method to solve the fourth-order thin film flow equations using moving mesh refinement. The moving mesh strategy is based on harmonic mappings developed by Li et al. [J. Comput. Phys., 170 (2001), pp. 562-588, and 177 (2002), pp. 365-393]. To achieve a high quality mesh, we adopt an adaptive monitor function and smooth it based on a diffusive mechanism. A variety of numerical tests are performed to demonstrate the accuracy and efficiency of the method. The moving mesh refinement accurately resolves the overshoot and downshoot structures and reduces the computational cost in comparison to numerical simulations using a fixed mesh.Comment: 18 pages, 10 figure

On resistive MHD models with adaptive moving meshes

In this paper we describe an adaptive moving mesh technique and its application to convection-diffusion models from magnetohydrodynamics (MHD). The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which are derived from the minimization of a meshenergy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDE model and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, an implicit-explicit method is applied for the time integration in combination with the iterative method Bi-CGSTAB. The adaptive mesh can be viewed as a 2D variant of the equidistribution principle, and it has the ability to track individual features of the physical solutions in the developing plasma flows. The results of a series of numerical experiments are presented which cover several aspects typifying resistive magnetofluid-dynamics

A Dynamically-Moving Adaptive Grid Method Based on a Smoothed Equidistribution Principle along Coordinate Lines

In this paper a time-dependent moving-grid method is described to numerically solve time-dependent partial differential equations (PDEs) in two space dimensions involving fine scale structures such as steep moving fronts, emerging steep layers, pulses and shocks. The method is based on an equidistribution principle along coordinate lines in the two spatial directions. Smoothing in the spatial direction is employed to control grid clustering and expansion. Additional smoothing in the temporal direction ensures a smooth progression of the grid points in time by preventing the points from responding too quickly to current values of the weight functions. Numerical results are given for a (parabolic) reaction-diffusion model and a (hyperbolic) rotating-pulse model

Tensor-product adaptive grids based on coordinate transformations

In this paper we discuss a two-dimensional adaptive grid method that is based on a tensor-product approach. Adaptive grids are a commonly used tool for increasing the accuracy and reducing computational costs when solving both Partial Differential Equations (PDEs) and Ordinary Differential Equations (ODEs). A traditional and widely used form of adaptivity is the concept of equidistribution, which is well-defined and well-understood in one space dimension. The extension of the equidistribution principle to two or three space dimensions, however, is far from trivial and has been the subject of investigation of many researchers during the last decade. Besides the non-singularity of the transformation that defines the non-uniform adaptive grid, the smoothness of the grid (or transformation) plays an important role as well. We will analyze these properties and illustrate their importance with numerical experiments for a set of time-dependent PDE models with steep moving pulses, fronts, and boundary layers

Pattern formation in the one-dimensional Gray-Scott model

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