Designing CFD methods for bandwidth—A physical approach

Abstract

A hoped-for outcome from high-order CFD methods, in addition to achieving accurate solutions on fine grids, is to achieve useful answers on coarse grids. In this paper we note that this outcome is by no means automatic, and consider how the different quests of accuracy and bandwidth may be achieved. Although accuracy is exclusively a low-frequency property and so amenable to Taylor series analysis, bandwidth is concerned with higher frequencies and may not be approachable by this route. It is argued that an important aspect should be the physically correct flow of information, especially for hyperbolic problems, and that this is strongly influenced by the choice of computational stencil. This is confirmed by applying von Neumann analysis to the linear advection equation, giving a strong preference to odd-order fully discrete schemes using upwind-biased stencils. However, it is pointed out that these schemes do not always generalize to higher dimensions, and a method is shown that achieves very accurate directionality even on unstructured grids. This is accomplished by taking information from the exact solution to the multidimensional IVP for acoustics.