DG Code

Overview

DG code is the temporary name for this code which solves the 1D advection equation du/dt+adu/dx=0 for arbitrary order and grid size along with a python script to plot results.

Features

• Solves 1X0V advection equation to arbitrary order
• Uniform, structured mesh
• Periodic boundary conditions
• Simple minmod slope limiter
• Lax-Friedrichs (Rusanov) numerical flux
• Input parameters specified via input.txt
• Results can be animated using plot_results.py

Requirements

• C++ compiler (supporting C++17 or higher)
• Python

Usage

1. Compile the code using the provided Makefile:

   make

2. Run the solver executable:

   ./dg1d

   The time it took the simulation to run and error (if test = true in input.txt) are output

3. After running the solver, animate the results using the python script:

   python3 plot_results.py

   This will generate an animation of the solution.

Input File Format

The input file (input.txt) must contain the following parameters:

• Advection speed (a)
• Number of cells (jMax)
• Order of solution (lMax)
• Number of time steps (tMax)
• Order of Gaussian Quadrature (quadratureOrder)
• Length of domain (length)
• Time step (dt)
• Basis function (basis)
  • Only options are legendre, quadratic (lMax = 2), linear (lMax = 1)
• If the error should be output (test)
• What value alpha to use for Lax-Friedrichs numerical flux
  • alpha = 0 corresponds to upwind flux
  • alpha = 1 corresponds to centered flux
• Initial function (input)
  • Only options are sin (a sine wave), pulse (a Gaussian pulse), topHat (a discontinuous top hat function)
• If slope limiter should be used (slopeLimiter)
• Number of outputed time steps (nout)

Code Structure

• dg1dAdvection.cxx: Entry point of the program, reads input parameters and orchestrates the simulation.
• Solver.cxx: Defines the solver class responsible for solving the advection equation and other relevant tasks
• Mesh.cxx: Defines the mesh representing the computational grid
• Matrix/Vector.cxx: Defines the Matrix/Vector types and operator definitions
• SpecialFunctions.cxx: Defines all non-native functions
• GaussianQuadrature.cxx: Performs Gaussian quadrature numerical integration
• FunctionMapper.cxx: Maps string of function name to function

Author

-Jack Gabriel