Adaptive grid methods for Q-tensor theory of liquid crystals : a one-dimensional feasibility study

This paper illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory of liquid crystals. We present the results of an initial study using a simple one-dimensional test problem which illustrates the feasibility of applying adaptive grid techniques in such situations. We describe how the grids are computed using an equidistribution principle, and investigate the comparative accuracy of adaptive and uniform grid strategies, both theoretically and via numerical examples

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to Geosciences. Other author's papers can be downloaded at http://www.denys-dutykh.com

Symplectic integrators with adaptive time steps

In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall into two categories. In the first, the time step is considered a function of time alone, \Delta=\Delta(t). In this case, backwards error analysis shows that while the algorithms remain symplectic, parametric instabilities arise because of resonance between oscillations of \Delta(t) and the orbital motion. In the second category the time step is a function of phase space variables \Delta=\Delta(q,p). In this case, the system of equations to be solved is analyzed by introducing a new time variable \tau with dt=\Delta(q,p) d\tau. The transformed equations are no longer in Hamiltonian form, and thus are not guaranteed to be stable even when integrated using a method which is symplectic for constant \Delta. We analyze two methods for integrating the transformed equations which do, however, preserve the structure of the original equations. The first is an extended phase space method, which has been successfully used in previous studies of adaptive time step symplectic integrators. The second, novel, method is based on a non-canonical mixed-variable generating function. Numerical trials for both of these methods show good results, without parametric instabilities or spurious growth or damping. It is then shown how to adapt the time step to an error estimate found by backward error analysis, in order to optimize the time-stepping scheme. Numerical results are obtained using this formulation and compared with other time-stepping schemes for the extended phase space symplectic method.Comment: 23 pages, 9 figures, submitted to Plasma Phys. Control. Fusio

BEM simulation for glass parisons

+124hlm.;24c

Unfitted finite element methods for axisymmetric two-phase flow

We propose and analyze unfitted finite element approximations for the two-phase incompressible Navier--Stokes flow in an axisymmetric setting. The discretized schemes are based on an Eulerian weak formulation for the Navier--Stokes equation in the 2d-meridian halfplane, together with a parametric formulation for the generating curve of the evolving interface. We use the lowest order Taylor--Hood and piecewise linear elements for discretizing the Navier--Stokes formulation in the bulk and the moving interface, respectively. We discuss a variety of schemes, amongst which is a linear scheme that enjoys an equidistribution property on the discrete interface and good volume conservation. An alternative scheme can be shown to be unconditionally stable and to conserve the volume of the two phases exactly. Numerical results are presented to show the robustness and accuracy of the introduced methods for simulating both rising bubble and oscillating droplet experiments