This section is woefully incomplete without references. I will have to come back to it and put those in as well as hooks into the previous section where a bunch of new methods are devised. The section is largely narrative and a bit presumptive. The devil would indeed be in the details.
When Van Leer’s fourth installment on his series was published in 1977, the concept of limiters to provide non-oscillatory results was still novel. Boris and Book had introduced a similarly motivated technique of flux-corrected transport in a series of papers. Loosely speaking, the idea was to provide the non-oscillatory character of first-order methods such as upwind (or donor cell) while providing higher accuracy (resolution) associated with second-order or higher methods. The geometric character of the approach was particularly appealing and surely encouraged the subsequent adoption of the methodology by the remap community starting in the early 1980’s.
The adoption of these ideas was by no means limited to the remap community, as non-oscillatory methods rapidly became the standard for solving hyperbolic conservation laws. Extending and improving these non-oscillatory methods became an acute research focus, and many new methods arose in the 1980’s and 1990’s although the pace of research has abated in the 2000’s. A key aspect of the work on new non-oscillatory method is the property of early limiters to reduce the accuracy of approximation to first-order at discontinuities, and extrema in the solution. Along with the diminishing accuracy, there is large amount of requisite numerical dissipation with the first-order methods. To combat the unappealing aspects of first-order results, the non-oscillatory research sought to create approaches that could maintain accuracy at extrema. At discontinuities the solution will be first-order for any practical method based on the shock capturing philosophy.
One manner of viewing this effort is seeking nonlinear stability for the approximations. This compliments the linear stability so essential in basic numerical method development. Nonlinear stability is associated with the detailed nature of the solution, and not simply the combination of accuracy and classical stability.
One of the most focused efforts to overcome the limitations of the first generation of limiters are essentially non-oscillatory (ENO) methods. These methods are designed to maintain accuracy at discontinuities and accuracy by adaptively selecting the stencil from the smoothest part of the local solution. ENO methods were extended successfully to a more practical algorithm in the weighted ENO methods, which define weights that blend the stencils together continuously allowing the method to reach higher order accuracy with the same stencil when the solution is sufficiently smooth. These methods among the most actively developed current methods.
Take for example third-order ENO methods based on deciding which of three stencils will approximate the flux to update . Five cell-center values participate in the scheme, . The standard approach then derives parabolic approximations based on stencil using cells, , , and . By some well-defined means the smoothest stencil among the three is chosen to update . In the semi-discrete case only the edge value for the update is chosen, but follows similar principles. WENO methods predominantly adopt this approach. There the parabolic stencils are simply evaluated at the edge, for the case descried above , , and . Weights are defined by using the derivatives of the corresponding parabolas to measure the smoothness of each. The weights are chosen so that in smooth regions the weights give a fifth-order approximation to the edge value, . The point is that the same principles could be used for other schemes, indeed they already have been in the case of Hermitian WENO methods.
Given a set of self-consistent stencils for methods utilizing cell-averages, edge values, moments, and edge-derivatives one can easily envision an essentially non-oscillatory approach to providing new schemes with nonlinear stability. The key is to not use stencils across discontinuous structures in the solution. The question to answer through future research is the ability of the stencil selection to provide stability with compact stencils provided by the alternative methods.
Schemes III and VI are based on moments and are equivalent to discontinuous Galerkin (DG) methods. DG methods are quite actively being developed due to their excellent theoretical properties as exemplified by the results Van Leer showed. The development of nonlinear stability for discontinuous Galerkin has not been entirely successful because limiters often deeply undermine the properties of the linear schemes so severely.
Cell based schemes have been the focus of virtually all of the work on non-oscillatory methods starting with Van Leer (and collaterally with FCT). A key example is the associated with PPM where monotonicity-preserving techniques were employed on the higher order approximations. Extrema-preserving approaches have been devised by several groups including the author and similar work by Sekora in collaboration with Colella. As noted by the author in his work the broad efforts with ENO methods including the wildly successful WENO could easily be incorporated into the PPM framework because of its notable flexibility. A third approach are the edge-based limiters developed by Suresh and Huynh that have been somewhat successfully incorporated to reduce some of the poor behavior of very high-order WENO methods. These methods would be thought of as combining the montonicity-preserving limiters with aspects of the essentially non-oscillatory approaches based on the nature of the solution. Results indicate that these methods produce better practical accuracy.
For the new methods Suresh and Huynh’s approach would appear to be particularly useful particularly with the semi-discrete formalism. The application of their approach would be rather straightforward.