Convex ENO High Order Multi-dimensional Schemes without Field by Field Decomposition or Staggered Grids
✍ Scribed by Xu-Dong Liu; Stanley Osher
- Book ID
- 102970232
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 753 KB
- Volume
- 142
- Category
- Article
- ISSN
- 0021-9991
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✦ Synopsis
Second order accurate (first order at extrema) cell averaged based approximations extending the Lax-Friedrichs central scheme, using component-wise rather than field-by-field limiting, have been found to give surprisingly good results for a wide class of problems involving shocks (see H. Nessyahu and E. Tadmor, J. Comput. Phys. 87, 408, 1990). The advantages of component-wise limiting compared to its counterpart, field-by-field limiting, are apparent: (1) No complete set of eigenvectors is needed and hence weakly hyperbolic systems can be solved. (2) Componentwise limiting is faster than field-by-field limiting. (3) The programming is much simpler, especially for complicated coupled systems of many equations. However, these methods are based on cell-averages in a staggered grid and are thus a bit complicated to extend to multiple dimensions. Moreover the staggering causes slight difficulties at the boundaries. In this work we modify and extend this componentwise central differencing based procedure in two directions: (1) Point values, rather than cell averages are used, thus removing the need for staggered grids, and also making the extension to multi-dimensions quite simple. We use TVD Runge-Kutta time discretizations to update the solution. (2) A new type of decision process, which follows the general ENO philosophy is introduced and used. This procedure enables us to extend our method to a third order component-wise central ENO scheme, which apparently works well and is quite simple to implement in multi-dimensions. Additionally, our numerical viscosity is governed by the local magnitude of the maximum eigenvalue of the Jacobian, thus reducing the smearing in the numerical results. We found a speed up of a factor of 2 in each space dimension, on a SGI O 2 workstation, over methods based on field-by-field decomposition limiting. The new decision process leads to new, "convex" ENO schemes which, we believe, are of