A second‐order upwind finite‐volume method for the Euler solution on unstructured triangular meshes

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Abstract
A scheme for the numerical solution of the two‐dimensional (2D) Euler equations on unstructured triangular meshes has been developed. The basic first‐order scheme is a cell‐centred upwind finite‐volume scheme utilizing Roe's approximate Riemann solver. To obtain second‐order accuracy, a new gradient based on the weighted average of Barth and Jespersen's three‐point support gradient model is used to reconstruct the cell interface values. Characteristic variables in the direction of local pressure gradient are used in the limiter to minimize the numerical oscillation around solution discontinuities. An Approximate LU (ALU) factorization scheme originally developed for structured grid methods is adopted for implicit time integration and shows good convergence characterisitics in the test. To eliminate the data dependency which prohibits vectorization in the inversion process, a black‐gray‐white colouring and numbering technique on unstructured triangular meshes is developed for the ALU factorization scheme. This results in a high degree of vectorization of the final code. Numerical experiments on transonic Ringleb flow, transonic channel flow with circular bump, supersonic shock reflection flow and subsonic flow over multielement aerofoils are calculated to validate the methodology.

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