Verification of fluid-dynamic codes in the presence of shocks and other discontinuities

The verification that computer codes correctly solve their model equations is critical to the continued success of numerical simulation. The method of manufactured solutions (MMS) is the best method currently available for this kind of verification for differential equations. However, it cannot be used directly with discontinuous solutions, as is required for the verification of high-speed aerodynamic codes with shocks. An integrative approach can extend the applicability of MMS to both discontinuous solutions such as shocks or material interfaces, as well as integral equations. We present an implementation of integrative MMS based on intelligent subdivision of integration domains that is both highly accurate and fast, and results in a rigorous, one-step verification procedure for shock-capturing codes. Numerical integration is found to be accurate to machine precision when tested on exact solutions of the linear heat equation and the Euler equations, even in the presence of discontinuous flow features. Intelligent subdivision of integration domains also improves computational performance by approximately 60× compared to the same algorithm without intelligent subdivisions. We demonstrate the use of MMS in the verification of the BACL-Streamer inviscid gas dynamics code. Integral MMS is found to compute convergence rates that are equivalent to those computed using differential MMS, and comparable to those computed using discontinuous, exact solutions, suggesting integral MMS is a valid method for verification of both integral and shock-capturing codes.