A high-order immersed boundary method to approximate flow problems in domains with curved boundaries

High-order h/p solvers in computational fluid dynamics offer scalability, efficiency, and superior error reduction compared to traditional low-order methods. Immersed boundary methods eliminate the need for body-fitted meshes but often degrade the order of the solution near boundaries, which can damage the overall accuracy of the high-order solver. This paper presents a new approach to impose boundary conditions in high-order finite element or finite volume flow solvers that retain high-order P+1 convergence, where P is the polynomial order. Furthermore, the methodology takes into account curved boundary conditions without loss in accuracy. It introduces a surrogate boundary that eliminates instabilities due to badly cut elements. We test the methodology using a high-order discontinuous Galerkin framework to solve purely elliptic problems and the compressible Navier-Stokes equations (2D and 3D), to show that we retain the formal order of convergence P+1. Finally, we compare the results with a volume penalization approach and show that spurious pressure oscillations on the immersed boundary are eliminated when the proposed methodology is used.