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

Abstract

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 those 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.