1. High-order, finite-volume methods in mapped coordinates
- Author
-
Phillip Colella, Jeffrey Hittinger, Daniel F. Martin, and M. R. Dorr
- Subjects
Numerical Analysis ,Partial differential equation ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Discretization ,Applied Mathematics ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Order of accuracy ,Computer Science Applications ,law.invention ,Quadrature (mathematics) ,Computational Mathematics ,law ,Modeling and Simulation ,Cartesian coordinate system ,Coordinate space ,Hyperbolic partial differential equation ,Mathematics - Abstract
We present an approach for constructing finite-volume methods for flux-divergence forms to any order of accuracy defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method developed for Cartesian grids to the case of mapped grids. The second is a method for computing the averages of the metric terms on faces such that freestream preservation is automatically satisfied. We derive detailed formulas for the cases of fourth-order accurate discretizations of linear elliptic and hyperbolic partial differential equations. For the latter case, we combine the method so derived with Runge-Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients. For both elliptic and hyperbolic problems, we demonstrate that the resulting methods are fourth-order accurate for smooth solutions.
- Published
- 2011