Verification of the unified coordinate system by the method of manufactured solutions

Although computational fluid dynamics has revolutionized aerodynamic analysis, it has remained less useful for preliminary design because of the dificulties associated with generating a computational grid. Hui's unified coordinate system has the potential to alleviate this concern by virtue of its automatically generated grid, but the method needs more study and development. We present a formal verification of the unified coordinate system in one dimension using both exact and manufactured solutions, including an integrative extension of the method of manufactured solutions which allows for discontinuities. This will provide the framework necessary to evaluate algorithmic improvements to the method, and provide confidence in the unified coordinate system as a tool for aerodynamic design, where its unique characteristics make it an extremely valuable alternative to traditional methods. First-order convergence is observed with a first-order method, while convergence is less for discontinuous solutions, as expected. The integrative method of manufactured solutions performs well, providing formal verification with a discontinuous, manufactured, solution.