Quadratic-Reconstruction Finite Volume Scheme for Compressible Flows on Unstructured Adaptive Grids

A high-order accurate unstructured finite volume Newton–Krylov algorithm for inviscid compressible flows

“…Ollivier-Gooch [24,25] experimented with solution-dependent weightings as a way to compute oscillation-free least-squares reconstructions for schemes of up to fourth-order accuracy. Delanaye and Essers [10] and Geuzaine et al [13] proposed a quadratic reconstruction finite volume scheme, including a new approach to monotonicity enforcement. They computed the inviscid flux directly from their quadratic reconstruction; however, viscous terms were obtained through a linear interpolation and were therefore only second order.…”

Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations

“…This limiter is only semi-differentiable (it is not fully differentiable when the control volume with the maximum value changes, for instance), but this is enough for excellent convergence behavior. To address accuracy issues, we apply the limiter selectively: our experience as well as other research [16,18,17] shows that applying the limiter to all derivatives yields a more diffusive solution. Therefore, we employ a differentiable switch r to drop the non-linear terms in the reconstruction when the limiter / differs significantly from one: In smooth regions, the full high-order reconstruction is applied by choosing r = 1.…”

“…The best understood part of this problem is reconstruction of the control volume averages to produce a high-order accurate approximation to the true solution. Numerous researchers, beginning with Barth and Frederickson [8], have explored this topic, both from the k-exact least-squares perspective [6,16,36,17,37] and from the (weighted) essentially non-oscillatory point of view [1,2,20,24]; we choose to follow the k-exact approach. While accurate reconstruction -by whatever means -is of course the foundation of a high-order finite-volume method, accurate flux integration, accurate boundary treatment and robust treatment of discontinuities are equally important, and receive little or no treatment in the literature.…”

“…Finally, treatment of shocks and other solution discontinuities requires particular attention for high-order schemes, both to eliminate overshoots at discontinuities and to avoid ruining accuracy in smooth regions of the flow while retaining good convergence properties. Here, we have experimented with two approaches based on Venkatakrishnan's limiter [44]: a smooth version of the approach of Delanaye and his co-workers [16,18,17], which reduces to a limited linear reconstruction near discontinuities, and a variant of our own that limits but does not eliminate all derivatives [27,28]; in this paper, we will use the former exclusively. We will describe our flow solver in more detail in Section 2.…”

Globalized matrix-explicit Newton-GMRES for the high-order accurate solution of the Euler equations

“…High-order methods for unstructured grids are also relatively well established. Initial efforts in this area in computational aerodynamics used finite-volume discretizations [3][4][5]. The discontinuous Galerkin finite-element method has also been successfully applied to the high-order accurate solution of compressible flow (see, for instance, [6][7][8][9]), as has the spectral volume method [10][11][12].…”