The following are some examples that use explicit, implicit or semi-implicit (IMEX) time integration methods implemented in PETSc (https://petsc.org/release/). To run them, HyPar needs to be compiled with PETSc. Familiarity with using PETSc is assumed.

Note:

In general, any example or simulation can use PETSc time-integrators (assuming HyPar is compiled with PETSc) by specifying the PETSc inputs through a .petscrc file, similar to the ones in the examples below. The following file is an example of a .petscrc file (with explanatory comments).
- hypar/Examples/PETScInputs/.petscrc_Example
The PETSc example directories have a file .petscrc and a sym link petscrc pointing to .petscrc. The file .petscrc is the actual input file; petscrc is needed to generate this documentation because Doxygen does not seem to include files with names starting with a dot!
The inputs in the .petscrc files (any lines not starting with a #) can also be specified in the command line, for example,
```
/path/to/hypar/bin/HyPar -use-petscts -ts_type rk -ts_rk_type 4 ...
```

Explicit time integration:

1D Linear Advection - Sine Wave
1D Linear Advection - Discontinuous Waves (with local truncation error-based adaptive time-step)

Implicit time integration:

1D Linear Diffusion - Sine Wave
2D Linear Diffusion - Sine Wave (with local truncation error-based adaptive time-step)
2D Euler Equations - Isentropic Vortex Convection
2D Navier-Stokes Equations - Lid-Driven Square Cavity (with local truncation error-based adaptive time-step)
2D Navier-Stokes Equations - Laminar Flow over Flat Plate

Implicit-Explicit (IMEX) time integration:

2D Euler Equations - Low-Mach Isentropic Vortex Convection
2D Euler Equations (with gravitational force) - Inertia-Gravity Waves
2D Euler Equations (with gravitational force) - Rising Thermal Bubble (with local truncation error-based adaptive time-step)
2D Navier-Stokes Equations - Lid-Driven Square Cavity
3D Navier-Stokes Equations - Rising Thermal Bubble