Central Compact-Reconstruction WENO Methods

Abstract

High-order compact upwind schemes produce block-tridiagonal systems due to performing the reconstruction in the characteristic variables, which is necessary to avoid spurious oscillations. Consequently they are less efficient than their non-compact counterparts except on high-frequency features. Upwind schemes lead to many practical drawbacks as well, so it is desirable to have a compact scheme that is more computationally efficient at all wavenumbers that does not require a characteristic decomposition. This goal cannot be achieved by upwind schemes so we turn to the central schemes, which by design require neither a Riemann solver nor a determination of upwind directions by characteristic decomposition. In practice, however, central schemes of fifth or higher order apparently need the characteristic decomposition to fully avoid spurious oscillations. The literature provides no explanation for this fact that is entirely convincing; however, a comparison of two central WENO schemes suggests one. Pursuing that possibility leads to the first main contribution of this work, which is the development of a fifth-order, central compact-reconstruction WENO (CCRWENO) method. That method retains the key advantages of central and compact schemes but does not entirely avoid characteristic variables as was desired. The second major contribution is to establish that the role of characteristic variables is to to make flux Jacobians within a stencil more diagonally dominant, having ruled out some plausible alternative explanations. The CCRWENO method cannot inherently improve the diagonal dominance without compromising its key advantages, so some strategies are explored for modifying the CCRWENO solution to prevent the spurious oscillations. Directions for future investigation and improvement are proposed.