490 research outputs found
An Adaptive Characteristic-wise Reconstruction WENOZ scheme for Gas Dynamic Euler Equations
Due to its excellent shock-capturing capability and high resolution, the WENO
scheme family has been widely used in varieties of compressive flow simulation.
However, for problems containing strong shocks and contact discontinuities,
such as the Lax shock tube problem, the WENO scheme still produces numerical
oscillations. To avoid such numerical oscillations, the characteristic-wise
construction method should be applied. Compared to component-wise
reconstruction, characteristic-wise reconstruction leads to much more
computational cost and thus is not suite for large scale simulation such as
direct numeric simulation of turbulence. In this paper, an adaptive
characteristic-wise reconstruction WENO scheme, i.e. the AdaWENO scheme, is
proposed to improve the computational efficiency of the characteristic-wise
reconstruction method. The new scheme performs characteristic-wise
reconstruction near discontinuities while switching to component-wise
reconstruction for smooth regions. Meanwhile, a new calculation strategy for
the WENO smoothness indicators is implemented to reduce over-all computational
cost. Several one dimensional and two dimensional numerical tests are performed
to validate and evaluate the AdaWENO scheme. Numerical results show that
AdaWENO maintains essentially non-oscillatory flow field near discontinuities
as the characteristic-wise reconstruction method. Besieds, compared to the
component-wise reconstruction, AdaWENO is about 40\% faster which indicates its
excellent efficiency
High-order conservative finite difference GLM-MHD schemes for cell-centered MHD
We present and compare third- as well as fifth-order accurate finite
difference schemes for the numerical solution of the compressible ideal MHD
equations in multiple spatial dimensions. The selected methods lean on four
different reconstruction techniques based on recently improved versions of the
weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving
(MP) schemes as well as slope-limited polynomial reconstruction. The proposed
numerical methods are highly accurate in smooth regions of the flow, avoid loss
of accuracy in proximity of smooth extrema and provide sharp non-oscillatory
transitions at discontinuities. We suggest a numerical formulation based on a
cell-centered approach where all of the primary flow variables are discretized
at the zone center. The divergence-free condition is enforced by augmenting the
MHD equations with a generalized Lagrange multiplier yielding a mixed
hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175
(2002) 645-673). The resulting family of schemes is robust, cost-effective and
straightforward to implement. Compared to previous existing approaches, it
completely avoids the CPU intensive workload associated with an elliptic
divergence cleaning step and the additional complexities required by staggered
mesh algorithms. Extensive numerical testing demonstrate the robustness and
reliability of the proposed framework for computations involving both smooth
and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics
(Aug 7 2009
Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes
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.The research of Y. Liu was supported in part by NSF grant DMS-0511815. The research of C.-W. Shu was supported in part by the Chinese Academy of Sciences while this author was visiting the University of Science
and Technology of China (grant 2004-1-8) and the Institute of Computational Mathematics and Scienti c/Engineering Computing. Additional support was provided by ARO grant W911NF-04-1-0291 and NSF grant DMS-0510345. The research of E. Tadmor was supported in part by NSF grant 04-07704 and ONR grant N00014-91-J-1076. The research of M. Zhang was supported in part by the Chinese Academy of Sciences grant 2004-1-8
- …