Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods

Matlab functions and scripts used in the paper Robust Approaches to Handling Complex Geometries with Galerkin Difference Methods by Jeremy E. Kozdon, Lucas C. Wilcox, Thomas Hagstrom, and Jeffrey W. Banks.

Note: These are research codes written for the above paper, and it will take some work in order to understand and apply these codes to other problems.

In general files in directory src may be useful for other applications whereas those in acoustic2D are specific files for the acoustic wave equation.

Weight Adjusted Projection Test

The weight adjusted projection tests can be run using the script wa_error_test.m in subdirectory wa_error

Acoustic2D run scripts

GD only scripts

The following scripts are in the directory acoustic2D

• GDConservation :: GD conservation test
• GDConservation_central :: GD conservation test with central flux
• GDConstPres :: GD constant-state preservation test
• GDBox_curved :: GD ghost basis curved box test
• GDBox_curved_extrapolation :: GD extrapolation basis curved box test
• GDDisk2D :: Disk with curved GD elements using ghost basis
• GDDisk2D_extrapolation :: Disk with curved GD elements using extrapolation
• GDDisk2D_dt :: Compute the maximum stable time step for the curved ghost basis GD disk mesh
• GDDisk2D_extrapolation_dt :: Compute the maximum stable time step for the curved extrapolation GD disk mesh

GD and simplicial scripts

The following scripts require the installation of the nodal DG codes from HesthavenWarburton2008; bibtex reference below, and please cite if you use these codes! Installation instructions for these codes are given below.

The following scripts are in the directory acoustic2D

• box_run :: script to check error and compute time step for simple box problem (runs simplicial, GD ghost basis, and GD extrapolation)
• box_spectrum :: script to compute the eigenvalue spectrum for simplicial, GD ghost basis, and GD extrapolation (data cached on git LFS)
• compute_max_timestep :: Compute the maximum stable time step for the Taylor time stepping method for each of the schemes (uses spectrums from box_spectrum)
• DGDisk2D :: Disk with curved simplicial elements
• DGDisk2D_dt :: compute the maximum stable time step with curved simplicial elements
• GDDGDisk2D :: Disk with coupled ghost basis GD and curved simplicial elements
• GDDGDisk2D_dt :: compute the maximum stable time step with coupled ghost basis GD and curved simplicial elements ### Installing nodal-dg codes
• GDDGDisk2D_extrapolation :: Disk with coupled extrapolation GD and curved simplicial elements
• GDDGDisk2D_dt :: compute the maximum stable time step with coupled extrapolation GD and curved simplicial elements ### Installing nodal-dg codes
• GDDGInclusion :: Run inclusion test problem
• GDDGInclusion_error_interp :: Do the convergence analysis to GDDGInclusion runs (see note below)
• GDDGInclusion_geometry :: Check to make sure the geometry interpolations are working

Note on GDDGInclusion_error_interp: The data needed by this script must be generated prior to running. The data is stored in the directory acoustic2D/data/ using GIT LFS.

If you do not want to install GIT LFS, you can download the data directly with either the following wget or curl. If you use this method you may wish to locally untrack these files since they will make your respository larger if they raw binary is added. This can be done with the commands:

git rm --cached -r acoustic2D/data/
git commit -m "untracking data files"

wget data download

wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n5_r0_t15.mat -O acoustic2D/data/data_n5_r0_t15.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n5_r1_t15.mat -O acoustic2D/data/data_n5_r1_t15.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n5_r2_t15.mat -O acoustic2D/data/data_n5_r2_t15.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n7_r0_t15.mat -O acoustic2D/data/data_n7_r0_t15.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n7_r1_t15.mat -O acoustic2D/data/data_n7_r1_t15.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n7_r2_t15.mat -O acoustic2D/data/data_n7_r2_t15.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p1.mat -O acoustic2D/data/box_spectrum_p1.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p2.mat -O acoustic2D/data/box_spectrum_p2.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p3.mat -O acoustic2D/data/box_spectrum_p3.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p4.mat -O acoustic2D/data/box_spectrum_p4.mat
wget https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p5.mat -O acoustic2D/data/box_spectrum_p5.mat

curl data download

curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n5_r0_t15.mat -o acoustic2D/data/data_n5_r0_t15.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n5_r1_t15.mat -o acoustic2D/data/data_n5_r1_t15.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n5_r2_t15.mat -o acoustic2D/data/data_n5_r2_t15.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n7_r0_t15.mat -o acoustic2D/data/data_n7_r0_t15.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n7_r1_t15.mat -o acoustic2D/data/data_n7_r1_t15.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/data_n7_r2_t15.mat -o acoustic2D/data/data_n7_r2_t15.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p1.mat -o acoustic2D/data/box_spectrum_p1.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p2.mat -o acoustic2D/data/box_spectrum_p2.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p3.mat -o acoustic2D/data/box_spectrum_p3.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p4.mat -o acoustic2D/data/box_spectrum_p4.mat
curl https://media.githubusercontent.com/media/bfam/GDComplexGeometries/master/acoustic2D/data/box_spectrum_p5.mat -o acoustic2D/data/box_spectrum_p5.mat

Installing nodal-dg codes

As noted above, the nodal-dg codes from HesthavenWarburton2008 are used to handle the simplicial domain. The codes in this repository assume that these have have been installed in the base directory of this repository; if this is not the case then you should set the variable NODAL_DG_ROOT in the file src/Globals2D_gdgd.m

To install the nodal-dg code you can use the git command:

git clone https://github.com/tcew/nodal-dg
cd nodal-dg
git checkout 50c09d1 -b gddg
patch -p1 < ../nodal-dg.patch

The third command will ensure that you are on the commit that is known to work with the codes and the fourth command patches the nodal-dg codes so that they work with the gddg codes

References:

@BOOK{HesthavenWarburton2008,
  title = {Nodal Discontinuous {G}alerkin Methods: {A}lgorithms, Analysis, and
           Applications},
  publisher = {Springer},
  year = {2008},
  author = {Hesthaven, Jan S. and Warburton, Tim},
  volume = {54},
  series = {Texts in Applied Mathematics},
  doi = {10.1007/978-0-387-72067-8}
}