Very high‐order accurate finite volume scheme for the convection‐diffusion equation with general boundary conditions on arbitrary curved boundaries

Abstract

Summary Obtaining very high‐order accurate solutions in curved domains is a challenging task as the accuracy of discretization methods may dramatically reduce without an appropriate treatment of boundary conditions. The classical techniques to preserve the nominal convergence order of accuracy, proposed in the context of finite element and finite volume methods, rely on curved mesh elements, which fit curved boundaries. Such techniques often demand sophisticated meshing algorithms, cumbersome quadrature rules for integration, and complex nonlinear transformations to map the curved mesh elements onto the reference polygonal ones. In this regard, the reconstruction for off‐site data method, proposed in the work of Costa et al, provides very high‐order accurate polynomial reconstructions on arbitrary smooth curved boundaries, enabling integration of the governing equations on polygonal mesh elements, and therefore, avoiding the use of complex integration quadrature rules or nonlinear transformations. The method was introduced for Dirichlet boundary conditions and the present article proposes an extension for general boundary conditions, which represents an important advance for real context applications. A generic framework to compute polynomial reconstructions is also developed based on the least‐squares method, which handles general constraints and further improves the algorithm. The proposed methods are applied to solve the convection‐diffusion equation with a finite volume discretization in unstructured meshes. A comprehensive numerical benchmark test suite is provided to verify and assess the accuracy, convergence orders, robustness, and efficiency, which proves that boundary conditions on arbitrary smooth curved boundaries are properly fulfilled and the nominal very high‐order convergence orders are effectively achieved.

Very high-order method on immersed curved domains for finite difference schemes with regular Cartesian grids

“…A new way to impose the boundary conditions is presented, via the use of ghost points with constrained least squares polynomial fitting. The methodology proposed in this work could be extended to unstructured grids using the framework presented in [54,55]. Moreover, this methodology can be easily extended to systems of conservation laws such as the Euler equations and to arbitrarily distributed curvilinear grids, if necessary.…”

“…The Reconstruction Off-site Data (ROD) method was initially introduced for the finite volume method on unstructured meshes where the numerical flux computed on the computational domain faces takes into account the data localized on the physical boundary. In the original method, [54,55], no ghost cells are required but the algorithm has to check if we are dealing with a cell close to the boundary or not, leading to additional tests and reconstruction matrices. The key idea of this work is to use the same scheme for any cell.…”

“…In this work, we propose a new high-order procedure of imposing boundary conditions on arbitrary shaped boundaries. This technique is based on the work of S. Clain et al [54,55] in finite volumes, where it was applied to the steady Convection-Diffusion equation. Here, we have extended their formulation to time-dependent equations and we develop a new, more flexible way of imposing the boundary conditions via a constrained least-squares polynomial fitting.…”

Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

“…For the convection-diffusion models, the piecewise-analytical method [9], high-order ADI method [10], leastsquares homotopy perturbation method [11], multigrid solver [12], lattice Boltzmann model [13], stabilized finiteelement method [14], second kind Chebyshev wavelets [15], finite element [16], finite difference method [17], decomposition method [18], compact finite difference method [19], meshless local Petrov-Galerkin method [20], discrete duality finite volume scheme [21], discontinuous Galerkin (DG) schemes [22], and high-order finite volume scheme [23] have been used in the literature.…”

Similarity Solution and Heat Transfer Characteristics for a Class of Nonlinear Convection‐Diffusion Equation with Initial Value Conditions

“…For convection-diffusion equations, only few cases with special initial or boundary value conditions have analytical solutions. Therefore, most of the main concerns were the study of the qualitative properties of the solutions [5][6][7][8][9][10][11][12][13] and numerical study [14][15][16][17][18][19][20][21][22][23] for convection-diffusion equations. However, the very important approximate solution of convection-diffusion equation has not been well solved.…”