High-resolution compact upwind finite difference methods for linear wave phenomena

A new family of high-resolution compact upwind finite difference schemes has been recently developed by the author for resolving discontinuities in the solution of linear convection problems. The improved method are based on fairly different approaches. By the modified equation analysis, all free parameters have been rigorously determined for obtaining physically more accurate solutions. In addition, by means of the characteristic analysis of the operators, the group velocity of wave packets has been controlled to avoid spurious oscillations while keeping the formal accuracy of the scheme. As for other classical methods, the application of the flux-corrected transport technique has been examined to eliminate or reduce the appearance of over- and under-shoots in discrete solutions. Finally, we have build a class of adaptive compact algorithms based on the non-linear combination of low- and high-order upwind schemes. We present a wide series of experiments for which well-established analytical and numerical results are available. The proposed numerical algorithms are compared with one another and with some conventional and shock-capturing schemes and very good agreement is achieved in computer calculations. The quality of the results is very encouraging and demonstrated the potential of the methods. The numerical experiments are also consistent with the expectations resulting from the theoretical analysis.