Abstract
We present a 3D finite volume generalization of the 1-dimensional Lax–Friedrichs and Nessyahu–Tadmor schemes for hyperbolic equations on Cartesian grids. The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is by-passed thanks to the use of the staggered Lax–Friedrichs scheme, is extended here to a two-step, 3-dimensional non-oscillatory centered scheme in finite volume formulation. Piecewise linear cell interpolants using several van Leer-type limiting techniques to estimate the gradient (van Leer, van Albada, SuperBee, MinMod) lead to a non-oscillatory spatial resolution of order superior to 1. The fact that the expected second-order resolution is not fully attained in 3D is investigated first by considering an alternate dual grid (in 2D), and by using the original van Albada limiter in 3D. Numerical results for a linear advection problem with continuous and discontinuous initial conditions in 2D and 3D show the accuracy and stability of the method. A comparison is made between the 2D Arminjon–Stanescu–Viallon and Jiang–Tadmor formulations and the new one. A new simple projection method is used for the gradients in the new 2D scheme. We also include results for the 3D Euler system (channel with a forward facing step).
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