High-order point-value enhanced finite volume method for two-dimensional hyperbolic equations on unstructured meshes

• Published in: Journal of computational physics 2020-12, Vol.423, p.109756, Article 109756
• Main Authors: Xuan, Li-Jun, Majdalani, Joseph
• Format: Article
• Language: English
• Subjects: Approximate delta function; Compact high order; Computational physics; Delta function; Dimensional stability; Discontinuous Galerkin; Euler-Lagrange equation; Finite volume method; Mathematical analysis; Point-value enhanced finite volume; Polynomials
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2020.109756

•The 2D ADF polynomials extend a new point-value enhanced finite volume method (PFV).•PFV proves stable and accurate in solving linear wave and nonlinear Euler equations.•The new PFV substantially reduces the number of DOFs for the same order of accuracy.•Against the DG, the PFV achieves higher orders of accuracy for the same DOF numbers.•Beside stability, it offers more advantages at higher orders and spatial dimensions. Presently, an approximate delta function (ADF) is defined in multi-dimensional space in the form of a finite-order polynomial that holds identical integral properties to the Dirac delta function when used in conjunction with a finite-order polynomial integrand spanning a finite domain. By using this ADF over a two-dimensional unstructured mesh, a compact high-order point-value enhanced finite volume method (PFV) is constructed. The corresponding scheme is capable of storing and updating cell-averaged values inside each element as well as the unknown quantities and their derivatives (when needed) on vertex points. The sharing of vertex information with surrounding elements reduces the number of degrees of freedom relative to other compact and high-order methods of comparable convergence rate. To ensure conservation, cell-averaged values are updated using an identical approach to the one traditionally followed in the finite volume method. To verify the performance of the PFV scheme at successive orders, its accuracy and stability are vetted using benchmark problems associated with the two-dimensional wave and Euler equations encompassing a total of six standard test cases.