On Enforcing Interface Conservation in Computational Fluid Dynamics

The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this paper. An error indicator based on the interface conservation is then designed for h-adaptive grid methods to effectively detect and identify discontinuities, as the interface conservation can never be attained across the discontinuities. A moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) is also developed for solving compressible flow problems with discontinuities using an operator splitting strategy in the spirit of r-adaptation methods, based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. A number of numerical experiments are conducted to assess effectiveness of the interface-based error indicator and performance of the MDG-ICE method. Numerical results for a variety of discontinuous flows obtained indicate that the error indicator based on the interface conservation can effectively detect all types of discontinuities and under-resolved flow regions and the MDG-ICE method is able to reposition grid, align mesh interfaces with all types of discontinuities, and achieve the designed order of both h- and p-convergence even for discontinuous solutions.