Dgm-fd: a finite difference scheme based on the discontinuous galerkin method

Abstract

Accurate and efficient numerical wave propagation is important in many areas of study such as computational aero-acoustics (CAA). While dissipation and dispersion errors influence the accuracy of a method, efficiency can be assessed by convergence rates and effective adaptability to different mesh structures. Finite difference and finite element methods are commonly used numerical schemes in CAA. Finite difference methods have the advantages of ease of use as well as high order convergence, but often require a uniform grid, and stable boundary closure can be non-trivial. Finite element methods adapt well to different mesh structures but can become difficult to implement as the order of approximation increases. In this research we formulate a numerical method that has high-order convergence, with strong accuracy for numerical wave numbers, and is adaptive to non-uniform grids. Such a method is developed based on the Discontinuous Galerkin Method (DGM) applied to the hyperbolic equation. Finite difference type schemes applicable to non-uniform grids are proposed. The schemes will be referred to as DGM-FD schemes. These schemes inherit, naturally, some features of the DGM, such as high-order approximations, applicability to non-uniform grids and super-accuracy for wave propagations. Two grid structures are studied. In the first structure, a regular, but non-uniform, finite difference type grid is assumed. In the second structure, some grid points are double-valued and the derivative scheme has a shortened stencil. Fourth-order upwind and third order central schemes are presented as examples of the first grid structure. Fifth-order upwind schemes are derived for the second structure. For non-linear equations, flux finite difference formula are given where no explicit upwind and downwind split of the flux is needed. This is in contrast to existing upwind finite difference schemes in the literature. Stability of the schemes with boundary closures and the super-accuracy for wave propagation problems are investigated and validated. The new schemes are demonstrated by numerical examples including the linearized acoustic waves, the solution of non-linear Burger's equation and the flat-plate boundary layer problem.