Generalized High-Order Solver Toolbox (GHOST)
GHOST is a simple yet flexible Python implementation of discontinuous Galerkin and flux reconstruction methods for first-order systems of conservation laws within the framework described in:
Tristan Montoya and David W. Zingg, A unifying algebraic framework for discontinuous Galerkin and flux reconstruction methods based on the summation-by-parts property. Journal of Scientific Computing 92, 2022.
As GHOST was developed for the prototyping and demonstration of numerical schemes and to run numerical experiments for simple model problems in support of the theoretical studies in the above manuscript, competitive performance for practical problems is not expected.
Features
Supported element types
- Line segments in 1D
- Triangles in 2D
Supported PDEs
- Linear advection equation (constant advection velocity)
- Euler equations (compressible flow, ideal gas with constant specific heat)
Supported discretization options
- Discontinuous Galerkin methods and energy-stable flux reconstruction schemes using Vincent-Castonguay-Jameson-Huynh (VCJH) correction fields, implemented in strong and weak conservation form in generalized curvilinear coordinates
- Nodal (Lagrange) or modal (orthonormal) bases
- Quadrature-based or collocation-based treatment of nonlinear fluxes
- Fourth-order Runge-Kutta or explicit Euler time marching
Supported numerical flux functions
- Standard upwind/central/blended numerical flux for linear advection
- Roe's approximate Riemann solver for Euler equations
Post-processing and visualization tools
- Calculation of L2 norms of error relative to exact/reference solutions using quadrature rules of arbitrary order
- Evaluation of primary conservation and energy balances
- High-resolution visualization of numerical solutions of arbitrary polynomial degree on general curvilinear meshes
Usage
The basic usage of GHOST is demonstrated in the Jupyter notebooks provided in the examples directory, in which the periodic linear advection equation is solved in one and two dimensions. The Jupyter notebooks used for the computations in the referenced manuscript are also provided (see notebooks/advection_driver.ipynb and notebooks/euler_driver.ipynb).
Results
The tables in the manuscript were generated using the Jupyter notebooks notebooks/make_tables_advection.ipynb and notebooks/make_tables_euler.ipynb, which retrieve data from the results directory for each set of discretization parameters. The subdirectory names for the linear advection and Euler equations are formatted as advection_pAbBcCtD_E and euler_m04pAcCtD_E, respectively, using the following parameters:
A - polynomial degree of the discretization (2, 3, or 4)
B - upwinding parameter for the numerical flux (0 for central flux, 1 for upwind flux)
C - VCJH parameter (0 for c=cDG, p for c=c+)
D - discretization type (1 for Quadrature I, 2 for Collocation, 3 for Quadrature II)
E - residual formulation (strong or weak)
Dependencies
NumPy, scipy, matplotlib, quadpy modepy, meshio, meshzoo
License
This software is released under the GPLv3 license.