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Hierarchical Reconstruction with up to Second Degree Remainder for Solving Nonlinear Conservation Laws

(Georgia Institute of Technology, 2009-07-31) Liu, Yingjie ; Shu, Chi-Wang ; Xu, Zhiliang

The 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.
Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes

(Georgia Institute of Technology, 2006-10-30) Liu, Yingjie ; Shu, Chi-Wang ; Tadmor, Eitan ; Zhang, Mengping

This is the continuation of the paper "central discontinuous Galerkin methods on overlapping cells with a non-oscillatory 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 non-staggered grids. This takes a new nite volume approach for approximating non-smooth 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 non-oscillatory solutions even for 5th order schemes without characteristic decomposition.
Central Discontinuous Galerkin Methods on Overlapping Cells with a Non-Oscillatory Hierarchical Reconstruction

(Georgia Institute of Technology, 2006-08-28) Liu, Yingjie ; Shu, Chi-Wang ; Tadmor, Eitan ; Zhang, Mengping

The 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 semi-discrete non-staggered 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 Godunov-type nite volume schemes. Overlapping cells lend themselves to the development of a central-type 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 Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which post-processes 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 multi-layer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes and retains the overall pre-processed order of accuracy while eff ectively removes spurious oscillations around shocks.
Central Schemes and Central Discontinuous Galerkin Methods on Overlapping Cells

(Georgia Institute of Technology, 2005) Liu, Yingjie

The 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 semi-discrete non-staggered 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. Semi-discrete 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)).
Central Schemes on Overlapping Cells

(Georgia Institute of Technology, 2004) Liu, Yingjie

Nessyahu 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 semi-discrete non-staggered 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. Semi-discrete form of the central scheme can also be obtained to which the TVD Runge-Kutta 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.
Back and Forth Error Compensation and Correction Methods for Semi-Lagranging Schemes with Application to Interface Computation Using Level Set Method

(Georgia Institute of Technology, 2004) Dupont, Todd F. ; Liu, Yingjie

Semi-Lagranging 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 semi-Lagranging schemes with almost the same simplicity and three times the complexity of a first order semi-Lagranging scheme but improve the order of accuracy. When applying this simple semi-Lagranging scheme to the level set method in interface computation, we observe good improvement comparable to results computed with other more complicated methods.
An Explanation to the Phenomenon of Coupled First Order Shift of Conservative Quantities during the Interaction of Captured Discontinuities

(Georgia Institute of Technology, 2003) Dupont, Todd F. ; Liu, Yingjie

High resolution capturing schemes generally speaking can take advantage of the piece-wise smooth property of the weak solutions of conservation laws and achieve high order accuracy wherever the solution is smooth. Thus In 1D, the positions of smeared discontinuities may supposedly be recovered before they interact with each other using the subcell resolution methods (Harten [9]). An interesting phenomenon is that after their interaction, the recovered positions of smeared discontinuities degenerate to first order. We study this phenomena in an ideal model and give an explanation.
A Backward Point Shifted Levelset Method for Highly Accurate Interface Computation

(Georgia Institute of Technology, 2002) Liu, Yingjie

We propose a technique that could significantly improve the accuracy of the levelset method and has the potential for fully conservative treatment. Level set method uses the levelset function, usually an approximate signed distance function Φ to indirectly represent the interface by the zero set of Φ. When Φ is advanced to the next time level by an advection equation, it is no longer a signed distance function any more, therefore the uneven numerical dissipation associated with the discretization of the advection equation could distort the interface particularly in places where the radius of curvature of the interface changes dramatically or two segments of the interface are getting close. Also an auxiliary equation is usually solved at each time level to restore Φ into a signed distance function, which could further shift the interface position. We address the second problem by using an analytic point shifted algorithm to locally perturb the mesh without geometric reconstruction of the interface so that the zero set of Φ is located at grid nodes and therefore solving the auxiliary equation will not move the interface. Our strategy for solving the first problem is that when applying the advection equation for Φ, it does not initiate from a signed distance function, but ends up with one, which can be achieved by solving the advection equation backward in time. These two techniques are combined with an iterative procedure.
Back and Forth Error Compensation and Correction Methods for Removing Errors Induced by Uneven Gradients of the Level Set Function

(Georgia Institute of Technology, 2002) Dupont, Todd F. ; Liu, Yingjie

We propose a method that signi cantly improves the accuracy of the level set method and could be of fundamental importance for numerical solutions of di fferential equations in general. Level set method uses the level set function, usually an approximate signed distance function Φ to represent the interface as the zero set of Φ. When Φ is advanced to the next time level by an advection equation, its new zero level set will represent the new interface position. But the non zero curvature of the interface will result in uneven gradients of the level set function which induces extra numerical error. Instead of attempting to reduce this error directly, we update the level set function Φ forward in time and then backward to get another copy of the level set function, say Φ[1]. Φ[1] and Φ should have be equal if there were no numerical error. Therefore Φ - Φ[1] provides us the information of error induced by uneven gradients and this information can be used to compensate Φ before updating Φ forward again in time.