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Liu,
Yingjie
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Yingjie
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ItemHierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws(Georgia Institute of Technology, 20090731) Liu, Yingjie ; Shu, ChiWang ; Xu, ZhiliangThe hierarchical reconstruction (HR) [Liu, Shu, Tadmor and Zhang, SINUM ’07] can effectively reduce spurious oscillations without local characteristic decomposition for numerical capturing of discontinuous solutions. However, there are still small re maining overshoots/undershoots in the vicinity of discontinuities. HR with partial neighboring cells [Xu, Liu and Shu, JCP ’09] essentially overcomes this drawback for the third order case, and in the mean time further improves the resolution of the numer ical solution. Extending the technique to higher order cases we observe the returning of overshoots/undershoots. In this paper, we introduce a new technique to work with HR on partial neighboring cells, which lowers the order of the remainder while maintaining the theoretical order of accuracy, essentially eliminates overshoots/undershoots for the fourth and fifth order cases (in one dimensional numerical examples) and reduces the numerical cost.

ItemSimulation of Bubbles in Foam With The Volume Control Method(Georgia Institute of Technology, 2007) Kim, Byungmoon ; Liu, Yingjie ; Llamas, Ignacio ; Jiao, Xiangmin ; Rossignac, JarekLiquid and gas interactions often produce bubbles that stay for a long time without bursting on the surface, making a dry foam structure. Such long lasting bubbles simulated by the level set method can suffer from a small but steady volume error that accumulates to a visible amount of volume change. We propose to address this problem by using the volume control method. We track the volume change of each connected region, and apply a carefully computed divergence that compensates undesired volume changes. To compute the divergence, we construct a mathematical model of the volume change, choose control strategies that regulate the modeled volume error, and establish methods to compute the control gains that provide robust and fast reduction of the volume error, and (if desired) the control of how the volume changes over time.

ItemNonOscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes(Georgia Institute of Technology, 20061030) Liu, Yingjie ; Shu, ChiWang ; Tadmor, Eitan ; Zhang, MengpingThis is the continuation of the paper "central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction" by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to nite volume schemes on nonstaggered grids. This takes a new nite volume approach for approximating nonsmooth solutions. A critical step for high order nite volume schemes is to reconstruct a nonoscillatory high degree polynomial approximation in each cell out of nearby cell averages. In the paper this procedure is accomplished in two steps: first to reconstruct a high degree polynomial in each cell by using e.g., a central reconstruction, which is easy to do despite the fact that the reconstructed polynomial could be oscillatory; then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution. All numerical computations for systems of conservation laws are performed without characteristic decomposition. In particular, we demonstrate that this new approach can generate essentially nonoscillatory solutions even for 5th order schemes without characteristic decomposition.

ItemCentral Discontinuous Galerkin Methods on Overlapping Cells with a NonOscillatory Hierarchical Reconstruction(Georgia Institute of Technology, 20060828) Liu, Yingjie ; Shu, ChiWang ; Tadmor, Eitan ; Zhang, MengpingThe central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990)] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys, 160 (2000)] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu; J. Comput. Phys, 209 (2005)]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunovtype nite volume schemes. Overlapping cells lend themselves to the development of a centraltype discontinuous Galerkin (DG) method, following the series of work by Cockburn and Shu [J. Comput. Phys. 141 (1998)] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities, beyond those of Godunovtype schemes. In particular, the central DG is coupled with a novel reconstruction procedure which postprocesses the central DG solution to remove spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine and Flaherty [Appl. Numer. Math. 14 (1994)], but is otherwise di fferent in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes and retains the overall preprocessed order of accuracy while eff ectively removes spurious oscillations around shocks.

ItemSimulation of Bubbles and Liquid Films(Georgia Institute of Technology, 2006) Kim, Byungmoon ; Liu, Yingjie ; Llamas, Ignacio ; Rossignac, JarekLiquid and gas interactions often contain bubbles surrounded by thin liquid films. Simulation of these liquid films is challenging since they quickly become thinner than the grid resolution, which leads to premature bursting or merging of the bubbles. We prevent this thinning process by applying a disjoining force to the film, obtaining bubbles that last much longer without bursting or merging. The surface tension on the liquid film is the next diffuculty. Since the level set is not differentiable at the center of the thin liquid film, the curvature computed from the level set gradient is noisy, and the thin liquid film ruptures quickly. To prevent this, we compute the surface tension from the local isosurface, obtaining longlasting liquid films. However, since bubbles stay longer without bursting or merging, the volume loss of each bubble is noticeable. To solve this problem, we modify the pressure projection to produce a velocity field whose divergence is controlled by the proportional and integral feedback. This allows us to preserve the volume or, if desired, to inflate or deflate the bubbles. In addition to premature bursting and volume change, another difficulty is the complicated liquid surface, which increases memory and computational costs. To reduce storage requirement, we collocate the velocity and pressure to simplify the octree mesh. To reduce the computational complexity of the pressure projection, we use a multigrid method.

ItemAdvections with Significantly Reduced Dissipation and Diffusion(Georgia Institute of Technology, 2006) Kim, Byungmoon ; Liu, Yingjie ; Llamas, Ignacio ; Rossignac, JarekBack and Forth Error Compensation and Correction (BFECC) can be applied to reduce dissipation and diffusion in advection steps, such as velocity, smoke density, and image advections. It can be implemented trivially as a small modification of the firstorder upwind or semiLagrangian integration of advection equations. It provides secondorder accuracy in both space and time and reduces volume loss significantly. We demonstrate its benefits on the simulation of smoke, bubbles, and interaction between water, a solid, and air.

ItemCentral Schemes and Central Discontinuous Galerkin Methods on Overlapping Cells(Georgia Institute of Technology, 2005) Liu, YingjieThe central scheme of Nessyahu and Tadmor (J. Comput. Phys, 87(1990)) has the bene t of not having to deal with the solution within the Riemann fan for solving hyperbolic conservation laws and related equations. But the staggered averaging causes large dissipation when the time step size is small comparing to the mesh size. The recent work of Kurganov and Tadmor (J. Comput. Phys, 160(2000)) overcomes the problem by use of a variable control volume and obtains a semidiscrete nonstaggered central scheme. Motivated by this work, we introduce overlapping cell averages of the solution at the same discrete time level, and develop a simple alternative technique to control the O(1/∆t) dependence of the dissipation. Semidiscrete form of the central scheme can also be obtained. This technique is essentially independent of the reconstruction and the shape of the mesh, thus could also be useful for Voronoi mesh. The overlapping cell representation of the solution also opens new possibilities for reconstructions. Generally more compact reconstruction can be achieved. We demonstrate through numerical examples that combining two classes of the overlapping cells in the reconstruction can achieve higher resolution. Overlapping cells create self similarity in the grid and enable the development of central type discontinuous Galerkin methods for convection di ffusion equations and elliptic equations with convection, following the series works of Cockburn and Shu (Math. Comp. 52(1989)).

ItemFlowFixer: Using BFECC for Fluid Simulation(Georgia Institute of Technology, 2005) Kim, Byungmoon ; Liu, Yingjie ; Llamas, Ignacio ; Rossignac, JarekBack and Forth Error Compensation and Correction (BFECC) was recently developed for interface computation by using the level set method. We show that it can be applied to reduce dissipation and diffusion encountered in various advection steps in fluid simulation such as velocity, smoke density and image advections. BFECC can be implemented easily on top of the first order upwinding or semiLagrangian integration of advection equations, while providing second order accuracy both in space and time. When applied to level set evolution, BFECC reduces volume loss significantly. We combine these techniques with variable density projection and show that they yield a realistic animations of twophase flows. We demonstrate the benefits of this approach on the image advection and on the simulation of smoke, of bubbles in water, and of a highly dynamic interaction between water, a solid, and air.

ItemCentral Schemes on Overlapping Cells(Georgia Institute of Technology, 2004) Liu, YingjieNessyahu and Tadmor's central scheme (J. Comput. Phys, 87(1990)) has the benefi t of not using Riemann solvers or characteristic decomposition for solving hyperbolic conservation laws and related convection diffusion equations. But the staggered averaging causes large dissipation when the time step size is small comparing to the mesh size. The recent work of Kurganov and Tadmor (J. Comput. Phys, 160(2000)) overcomes the problem by use of a variable control volume and obtains a semidiscrete nonstaggered central scheme. Motivated by this work, we introduce overlapping cell averages of the solution at the same discrete time level, and develop a simple alternative technique to control the O(1/∆t) dependence of the dissipation. Semidiscrete form of the central scheme can also be obtained to which the TVD RungeKutta time discretization of Shu and Osher (J. Comput. Phys, 77(1988)) can be applied. This technique is essentially independent of the reconstruction and the shape of the mesh, thus could also be useful for unstructured mesh. The overlapping cell representation of the solution also opens new possibilities for reconstructions. Generally more compact reconstruction can be achieved. We demonstrate through numerical examples that combining two classes of the overlapping cells in the reconstruction can achieve higher resolution.

ItemBack and Forth Error Compensation and Correction Methods for SemiLagranging Schemes with Application to Interface Computation Using Level Set Method(Georgia Institute of Technology, 2004) Dupont, Todd F. ; Liu, YingjieSemiLagranging schemes have been explored by several authors recently for transport problems in particular for moving interfaces using level set method. We incorporate the backward error compensation method developed in [2] into the semiLagranging schemes with almost the same simplicity and three times the complexity of a first order semiLagranging scheme but improve the order of accuracy. When applying this simple semiLagranging scheme to the level set method in interface computation, we observe good improvement comparable to results computed with other more complicated methods.