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

We present an adaptive moving mesh method for the numerical solution of thin liquid film spreading flows with surface tension. We follow the r-adaptive moving mesh technique which 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. Numerical experiments are performed on two one dimensional thin film flow equations to test the accuracy and efficiency of the method. This technique accurately resolves the multiple one-dimensional structures observed in these test problems. Moreover, it reduces the computational effort in comparison to the numerical solution using the finite difference scheme on a fixed uniform mesh

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

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

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

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

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

Methods of Ensemble Data Assimilation on Adaptive Moving Meshes

Numerical models solved on adaptive moving meshes have become increasingly prevalent in recent years. In particular, neXtSIM is a 2D model of sea-ice that is numerically solved on a Lagrangian mesh that does not conserve the number of mesh points. In this dissertation, we present two novel approaches to the formulation of ensemble data assimilation for models with this underlying computational structure. Specifically, we map ensemble members onto a common reference mesh, where the Ensemble Kalman Filter (EnKF) can be performed. Numerical experiments are carried out using 1D prototypical models: Burgers and Kuramoto-Sivashinsky equations, with both Eulerian and Lagrangian synthetic observations assimilated. One of the approaches is very effective, while the other is significantly less so. We also present a novel approach in the formulation of the Local Ensemble Transform Kalman Filter (LETKF) on a conservative moving mesh model. This is also achieved by mapping the ensemble members onto a common reference mesh, but it done in a significantly different manner than from the previous two approaches. Specifically, the common mesh is formed by taking an equidistributing mesh for the previous output of the algorithm. The preliminary results of this method from an application to Burgers equation are encouraging.Doctor of Philosoph