Improving the boundary efficiency of a compact finite difference scheme through optimising its composite template

• A set of optimised boundary coefficients are introduced for a compact finite-difference system.
• Analysis is based on a new formulation for composite template properties.
• Numerical stability is ensured by nonlinear constraints in a genetic algorithm.
• The performance of the new finite-difference system is thoroughly tested through benchmark simulations.

This paper presents efforts to improve the boundary efficiency and accuracy of a compact finite difference scheme, based on its composite template. Unlike precursory attempts the current methodology is unique in its quantification of dispersion and dissipation errors, which are only evaluated after the matrix system of equations has been rearranged for the derivative. This results in a more accurate prediction of the boundary performance, since the analysis is directly based on how the derivative is represented in simulations. A genetic algorithm acts as a comprehensive method for the optimisation of the boundary coefficients, incorporating an eigenvalue constraint for the linear stability of the matrix system of equations. The performance of the optimised composite template is tested on one-dimensional linear wave convection and two-dimensional inviscid vortex convection problems, with uniform and curvilinear grids. In all cases, it yields substantial accuracy and efficiency improvements while maintaining stable solutions and fourth-order accuracy.