Very high‐order accurate finite volume scheme for the convection‐diffusion equation with general boundary conditions on arbitrary curved boundaries
Résumé
Obtaining very high-order accurate solutions in curved domains is a challenging task as the accuracy of numerical methods may dramatically reduce without an appropriate treatment of the boundary condition. The classical techniques to preserve the optimal order of accuracy, proposed in the context of Finite Element and Finite Volume methods, rely on curved mesh elements to fit the curved boundaries. Such techniques often demand sophisticated meshing algorithms, cumbersome quadrature rules for integration, and complex nonlinear transformations to map the locally curved element onto the reference polygonal mesh elements. In this regard, the Reconstruction for Off-site Data method was proposed in Costa et al. (2018) to provide very high-order accurate polynomial reconstructions for curved boundaries, enabling the integration of the governing equations on polygonal mesh elements, and, therefore, without the need of complex integration quadrature rules or nonlinear transformations. The method was introduced with Dirichlet boundary conditions and the present article proposes an extension for general boundary conditions, which represents an important advance for real context applications. To achieve that, a generic framework to compute the polynomial reconstructions is also proposed based on the Least-squares method, enabling the method to handle general constraints and further improving the algorithm. A comprehensive numerical benchmark test suite is provided to validate and assess the accuracy, convergence rates, robustness, and efficiency, which proves that boundary conditions for curved domains are properly satisfied and the optimal very high-order convergence rates are successfully achieved.
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