On multi-dimensional unstructured mesh adaption

Abstract

Anisotropic unstructured mesh adaption is developed for a truly multi-dimensional upwind fluctuation splitting scheme, as applied to scalar advection-diffusion. The adaption is performed locally using edge swapping, point insertion/deletion, and nodal displacements. Comparisons are made versus the current state of the art for aggressive anisotropic unstructured adaption, which is based on a posteriori error estimates. Demonstration of both schemes to model problems, with features representative of compressible gas dynamics, show the present method to be superior to the a posteriori adaption for linear advection. The performance of the two methods is more similar when applied to nonlinear advection, with a difference in the treatment of shocks. The a posteriori adaption can excessively cluster points to a shock, while the present multi-dimensional scheme tends to merely align with a shock, using fewer nodes. As a consequence of this alignment tendency, an implementation of eigenvalue limiting for the suppression of expansion shocks is developed for the multi-dimensional distribution scheme. The differences in the treatment of shocks by the adaption schemes, along with the inherently low levels of artificial dissipation in the fluctuation splitting solver, suggest the present method is a strong candidate for applications to compressible gas dynamics.

Adaptive grid generation by minimizing residuals

“…This could be done much more economically ( Figure 2). Similar points are made in Reference [11] which also contains a useful literature survey.…”

“…Elements that try to align their edges with characteristic directions experience di culty when these directions intersect, as they do at shocks. We were unable to carry the adaptive grids to a converged state, nor apparently were the authors of Reference [11]. But the possibility of success is hinted at by the fact [10] that degenerate elements exist that recognize non-linear shockwaves.…”

A numerical scheme for ionizing shock waves

“…The FS schemes differ from the FV schemes in the sense that the flow variables are not mesh averages but are kept at the vertices of the mesh and that the numerical fluxes are not used. See [15] for a comparison between the FS and FV schemes.…”

A visual fluctuation splitting scheme for magnetohydrodynamics with a new sonic fix and Euler limit

“…Since it is this quasi-linear form of the system of equations used by FS schemes, the fluxes which are used in finite volume schemes are not estimated. See [32] for a comparison between the fluctuation splitting and finite volume schemes. Although it is not necessary to work in conservative variables in the derivation of the eigensystem of wave models in the FS schemes (see [25], for example), the flow variables (which are updated) are conservative variables in order to obtain correct weak solutions.…”