Anisotropic goal‐oriented error analysis for a third‐order accurate CENO Euler discretization

Summary In this paper, a central essentially non‐oscillatory approximation based on a quadratic polynomial reconstruction is considered for solving the unsteady 2D Euler equations. The scheme is third‐order accurate on irregular unstructured meshes. The paper concentrates on a method for a metric‐based goal‐oriented mesh adaptation. For this purpose, an a priori error analysis for this central essentially non‐oscillatory scheme is proposed. It allows us to get an estimate depending on the polynomial reconstruction error. As a third‐order error is not naturally expressed in terms of a metric, we propose a least‐square method to approach a third‐order error by a quadratic term. Then an optimization problem for the best mesh metric is obtained and analytically solved. The resulting mesh optimality system is discretized and solved using a global unsteady fixed‐point algorithm. The method is applied to an acoustic propagation benchmark. The paper concentrates on a method for metric‐based goal‐oriented anisotropic mesh adaptation for a third‐order accurate scheme of essentially non‐oscillatory type. The scheme applies to unstructured triangulations. Applications to nonlinear acoustics are presented.