An adaptive moving mesh method for thin film flow equations with surface tension

Alharbi, Abdulghani; Naire, Shailesh

doi:10.1016/j.cam.2017.01.019

Cited by 25 publications

(24 citation statements)

References 46 publications

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“…Adapting the mesh can prove computationally efficient in that an adaptive mesh generally requires fewer points than a fixed mesh to attain the same level of accuracy (Huang and Russell, 2010). Some important application areas where adaptive meshes have been used are: groundwater equations (Huang et al, 2002), and thin 15 film equations (Alharbi and Naire, 2017) as well as large geophysical systems (Pain et al, 2005;Davies et al, 2011).…”

Section: Adaptive Mesh Modelsmentioning

confidence: 99%

Data assimilation using adaptive, non-conservative, moving mesh models

Aydoğdu¹,

Carrassi²,

Guider³

et al. 2019

Preprint

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Numerical models solved on adaptive moving meshes have become increasingly prevalent in recent years. Motivating problems include the study of fluids in a Lagrangian frame and the presence of highly localized structures such as shock waves or interfaces. In the former case, Lagrangian solvers move the nodes of the mesh with the dynamical flow; in the latter, mesh resolution is increased in the proximity of the localized structure. Mesh adaptation can include remeshing, a procedure that adds or 5 removes mesh nodes according to specific rules reflecting constraints in the numerical solver. In this case, the number of mesh nodes will change during the integration and, as a result, the dimension of the model's state vector will not be conserved. This work presents a novel approach to the formulation of ensemble data assimilation for models with this underlying computational structure. The challenge lies in the fact that remeshing entails a different state space dimension across members of the ensemble, thus impeding the usual computation of consistent ensemble-based statistics. Our methodology adds one forward and one 10 backward mapping step before and after the EnKF analysis respectively. This mapping takes all the ensemble members onto a fixed, uniform, reference mesh where the EnKF analysis can be performed. We consider a high-(HR) and a low-resolution (LR) fixed uniform reference mesh, whose resolutions are determined by the remeshing tolerances. This way the reference meshes embed the model numerical constraints and also are upper and lower uniform meshes bounding the resolutions of the individual ensemble meshes. Numerical experiments are carried out using 1D prototypical models: Burgers and Kuramoto-15 Sivashinsky equations, and both Eulerian and Lagrangian synthetic observations. While the HR strategy generally outperforms that of LR, their skill difference can be reduced substantially by an optimal tuning of the data assimilation parameters. The LR case is appealing in high-dimensions because of its lower computational burden. Lagrangian observations are shown to be very effective in that fewer of them are able to keep the analysis error at a level comparable to the more numerous observers for the Eulerian case. This study is motivated by the development of suitable EnKF strategies for 2D models of the sea-ice that are 20 numerically solved on a Lagrangian mesh with remeshing.and Lagrangian (where observation locations move with the flow) observations. We test the strategy for two well-known PDEs:the viscous Burgers and Kuramoto-Sivashinsky equation, whose associated computational models we refer to as BGM and KSM, respectively. The Burgers' equation, which can be viewed as modeling a one-dimensional fluid, is a canonical example for which a localized structure, in this case a shock wave, develops and an adaptive moving mesh will get denser near the shock front. The Kuramoto-Sivashinsky equation exhibits chaotic behavior and this provides a natural test-bed for DA in a dynamical Nonlin. Processes Geophys. ...

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Section: Adaptive Mesh Modelsmentioning

confidence: 99%

Data assimilation using adaptive, non-conservative, moving mesh models

Aydoğdu¹,

Carrassi²,

Guider³

et al. 2019

Preprint

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“…In our earlier paper [32], we had listed four of the commonly used one dimensional MMPDEs, the so-called MMPDEs 4, 5 and 6 and modified MMPDE5, and two mesh density functions based on arc-length and curvature. In this work, we extend MMPDEs 4 and 5 to two dimensions.…”

Section: Moving Mesh Partial Differential Equations (Mmpdes) and Meshmentioning

confidence: 99%

“…Here, α and β are adaptivity parameters (or weight parameters) of the mesh density function [33,37,38]. These are usually taken to be constant but could be dependent on the spatial variables if there are multiple regions over which the solution characteristics vary rapidly (see example in Alharbi & Naire [32]). The above monitor functions can also be extended to include multiple solution components (see example in Alharbi & Naire [32]).…”

Section: Moving Mesh Partial Differential Equations (Mmpdes) and Meshmentioning

confidence: 99%

“…These are usually taken to be constant but could be dependent on the spatial variables if there are multiple regions over which the solution characteristics vary rapidly (see example in Alharbi & Naire [32]). The above monitor functions can also be extended to include multiple solution components (see example in Alharbi & Naire [32]). In addition, it is common practice in the context of moving mesh methods to smooth the monitor function in order to obtain a smoother mesh and also to make the MMPDEs easier to integrate.…”

Section: Moving Mesh Partial Differential Equations (Mmpdes) and Meshmentioning

confidence: 99%

“…The above adaptive mesh schemes were shown to capture and resolve the moving contact line and the associated fingering instability accurately and computationally efficiently compared to a fixed uniform grid scheme. In a recent paper, Alharbi & Naire [32] successfully implemented the r-adaptive moving mesh technique [33,34] for a one-dimensional thin film equation with surface tension. This technique utilises a mesh density function and moving mesh partial differential equations (MMPDEs) to adapt and move the mesh coupled to the PDE(s) describing the thin film flow problem.…”

Section: Introductionmentioning

confidence: 99%

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An adaptive moving mesh method for two-dimensional thin film flow equations with surface tension

Alharbi

Naire

2019

Journal of Computational and Applied Mathematics

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In this paper, we extend our previous work [A. Alharbi and S. Naire, An adaptive moving mesh method for thin film flow equations with surface tension, J. Computational and Applied Mathematics, 319 (2017), pp. 365-384.] on a one-dimensional r-adaptive moving mesh technique based on a mesh density function and moving mesh partial differential equations (MMPDEs) to two dimensions. As a test problem, we consider the gravitydriven thin film flow down an inclined and pre-wetted plane including surface tension and a moving contact line. This technique accurately captures and resolves the moving contact line and associated fingering instability. Moreover, the computational effort is hugely reduced in comparison to a fixed uniform mesh. AbstractIn this paper, we extend our previous work [A. Alharbi and S. Naire, An adaptive moving mesh method for thin film flow equations with surface tension, J. Computational and Applied Mathematics, 319 (2017), pp. 365-384.] on a one-dimensional r-adaptive moving mesh technique based on a mesh density function and moving mesh partial differential equations (MMPDEs) to two dimensions. As a test problem, we consider the gravity-driven thin film flow down an inclined and pre-wetted plane including surface tension and a moving contact line. This technique accurately captures and resolves the moving contact line and associated fingering instability. Moreover, the computational effort is hugely reduced in comparison to a fixed uniform mesh.

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Analytical and numerical investigations of the modified Camassa–Holm equation

Abdelrahman

Alharbi

2021

Pramana - J Phys

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An adaptive moving mesh method for thin film flow equations with surface tension

Cited by 25 publications

References 46 publications

Data assimilation using adaptive, non-conservative, moving mesh models

Data assimilation using adaptive, non-conservative, moving mesh models

An adaptive moving mesh method for two-dimensional thin film flow equations with surface tension

Analytical and numerical investigations of the modified Camassa–Holm equation

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