A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids

Abstract

A recent work of Li et al. [Numer. Math. Theor. Meth. Appl., 1(2008), pp. 92-112] proposed a finite volume solver to solve 2D steady Euler equations. Although the Venkatakrishnan limiter is used to prevent the non-physical oscillations nearby the shock region, the overshoot or undershoot phenomenon can still be observed. Moreover, the numerical accuracy is degraded by using Venkatakrishnan limiter. To fix the problems, in this paper the WENO type reconstruction is employed to gain both the accurate approximations in smooth region and non-oscillatory sharp profiles near the shock discontinuity. The numerical experiments will demonstrate the efficiency and robustness of the proposed numerical strategy.

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

“…The main difficulties lie on the non-differentiability of the numerical scheme leading to the creation of slight oscillations nearby shock discontinuities. More adapted WENO schemes then have been developed [52,23,21] to overcome the problem of convergence toward the steady-state solution. We also mention some interesting developments for the steady-state case based on the Discontinuous Galerkin method [6,7,38], the Spectral Volume technique [10,18], and the Residual Distribution schemes [12,2,22].…”

Addressing the Challenges of Implementation of High-Order Finite-Volume Schemes for Atmospheric Dynamics on Unstructured Meshes

“…In general, to compute the degrees of freedom a k we need at least K cells in the stencil, different from the target cell E 0 . However, the use of the minimum possible number of cells in the stencil M ≡ K results in a scheme which may become unstable on general meshes [9,21,22,49]. It is, therefore, recommended to use more cells in the stencil than the minimum required number.…”

Оne-dimensional gas dynamics problems and their solution based on high-resolution finite difference schemes

“…Несмотря на многочисленные примеры использования ENO-и WENO-схем [5][6][7][8][9][10][11], вопросы их практического исполь-зования и тестирования на модельных задачах газовой динамики остаются недостаточно освещенными.…”