High-order ENO and WENO schemes for unstructured grids.

*(English)*Zbl 1388.76217Summary: This work describes the implementation and analysis of high-order accurate schemes applied to high-speed flows on unstructured grids. The class of essentially non-oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third- and fourth-order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2-D Euler equations in a cell centred finite volume context. High-order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge-Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high-order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high-speed flow simulations are presented with the objective of assessing the implemented capability.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

##### Software:

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\textit{W. R. Wolf} and \textit{J. L. F. Azevedo}, Int. J. Numer. Methods Fluids 55, No. 10, 917--943 (2007; Zbl 1388.76217)

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