Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

Abstract

Abstract.We present an explicit second-order finite volume generalization of the onedimensional (1D) Nessyahu-Tadmor schemes for hyperbolic equations on adaptive unstructured tetrahedral grids. The nonoscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is bypassed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-steps scheme. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in three dimensions (3D), where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods to solve Euler's equations. Numerical experimental tests on classical problems are obtained by our method and by the computational fluid dynamics software Fluent. These tests include results for the 3D Euler system (shock tube problem) and flow around an NACA0012 airfoil. 1. Introduction. The history of schemes on staggered grids can at least be traced back to the famous paper of Courant, Friedrichs, and Lewy in 1928 [13] in which they discovered a scheme on staggered grids for the linear wave equation in one-dimensional (1D). For a special system arising in fluid dynamic problems von Neumann and Richtmyer used staggered grids as well [38]. Four years later, Lax introduced the well-known Lax-Friedrichs scheme and analyzed it [29]. In 1990 Tadmor and Nessyahu [41] picked up the idea to use staggered grids, showed the connection to Godunov's method, and proposed a second-order extension to 1D systems.The main advantage of these schemes is that no information about solutions to local Riemann problems is needed. Using staggered grids one can replace the upwind fluxes by central differences. The price one has to pay is the occurrence of excessive numerical viscosity since the resulting scheme can be interpreted as a Lax-Friedrichs scheme. Therefore, a higher order scheme of monotone upstream-centered schemes for conservation laws (MUSCL)-type in one spatial dimension was proposed in [41]. Later in [4,3,6,7,8,36] central schemes were generalized to multidimensional schemes on unstructured grids. For a Cartesian grid, we refer to [9,22,30,28,31,39,40,45,32] for related work.In the case of staggered unstructured multidimensional grids, there exist only a few convergence results. In [5] convergence of a second-order central scheme on twodimensional (2D) grids has been proven for a linear conservation law. Convergence of the first-order Lax-Friedrichs scheme on the same staggered grids for nonlinear scalar problems has been proven in [19].

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

“…Therefore, a higher order scheme of monotone upstream-centered schemes for conservation laws (MUSCL)-type in one spatial dimension was proposed in [41]. Later in [4,3,6,7,8,36] central schemes were generalized to multidimensional schemes on unstructured grids. For a Cartesian grid, we refer to [9,22,30,28,31,39,40,45,32] for related work.…”