Towards a genuinely stable boundary closure for pentadiagonal compact finite difference schemes

A new optimisation strategy to develop high-order and genuinely stable boundary closure schemes for a pentadiagonal, seven-point stencil, compact finite difference system is proposed. Previous approaches to developing boundary compact schemes often yielded either potential instabilities or non-optimal accuracy in the numerical solutions. In the present optimisation, the numerical accuracy and stability of the compact differencing system are addressed simultaneously by maximising the global accuracy of the system under stability constraints that are enhanced by a non-Dirichlet boundary condition. A unified extrapolation method is proposed to assist in the derivation of boundary schemes. A set of optimised boundary compact schemes is obtained for use along with an existing interior compact scheme. The resultant fourth-order compact finite difference system is applied to flow and acoustic benchmark problems as well as a large-eddy simulation, demonstrating superior performance in numerical stability, accuracy, and spatial resolution.