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

Abstract

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 * Corresponding author, Email address: J.M.Turner@soton.ac.uk * * Current address: Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glassgow, G1 1XQ, UK Preprint submitted to Computers & Fluids August 16, 2016 efficiency improvements while maintaining stable solutions and fourth-order accuracy.

Optimized Runge-Kutta (LDDRK) schemes with non-constant-amplitude waves

“…This is the result given in (10). Requiring this to be less than zero gives the stability conditions following (10).…”

“…for some givenη s , with the first constraint taken from (10). Many optimization parameters were considered, and only three of the better performing schemes are described here.…”

“…Finite difference schemes to calculate spatial derivatives, and Runge-Kutta and Adams-Bashforth schemes to step forwards in time, have all been optimized to attempt to accurately propagate acoustic perturbations with few points per wavelength and steps per period respectively, and such schemes are commonly used in modern CAA simulations. Examples of optimized spatial derivatives include: the bynow classic 7-point 4th order explicit DRP schemes [2,3]; optimized implicit/compact schemes of up to 6th order [4,5]; prefactored implicit MacCormack schemes [6]; trigonometrically optimized schemes [7]; 2nd and 4th order 9, 11, and 13 point schemes [8]; and asymmetric optimized schemes for use near boundaries [9,10]. Examples of optimized timestepping schemes include: an optimized Adams-Bashforth scheme [2]; Low Dispersion and Dissipation Runge-Kutta (LDDRK) 5-stage, 6-stage and alternating 4/6-and 5/6-stage schemes [11]; and optimized 5-and 6-stage Runge-Kutta schemes [8].…”