Convex ENO High Order Multi-dimensional Schemes without Field by Field Decomposition or Staggered Grids

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) Component-wise 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 component-wise 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 SGIO2workstation, over methods based on field-by-field decomposition limiting. The new decision process leads to new, “convex” ENO schemes which, we believe, are of interest in a more general setting. Our numerical results show the value of these new methods.