Development and Verification of a Higher-Order Computational Fluid Dynamics Solver

Abstract

Over the past two decades, higher-order methods have gained much broader use in computational science and engineering as these schemes are often more efficient per degree-of-freedom at achieving a prescribed error tolerance than lower-order methods. During this time, higher-order variants of most discretization schemes, such as finite-difference methods, finite-volume methods, and finite-element methods, have emerged. The finite-volume method is arguably the most widely used discretization technique in production-level computational fluid dynamics solvers due to its robustness and conservation properties. However, most finite-volume solvers only employ a conventional second-order scheme. To leverage the benefits of higher-order methods, the higher-order finite-volume method seems the most natural for those seeking to extend their legacy solvers to higher-order. Nonetheless, ensuring higher-order accuracy is maintained is quite challenging as the implementation requirements for a higher-order scheme are much greater than that of a lower-order scheme. In this work, a methodology for verifying higher-order finite-volume codes is presented. The higher-order finite-volume method is outlined in detail. Order verification tests are proposed for all major components, including the treatment of curved boundaries and the higher-order solution reconstruction. System-level verification tests are performed using the weak form of the Method of Manufactured Solutions. Several canonical verification cases are also presented for the Euler and laminar Navier-Stokes equations.