Reduced Order Modelling

[donovan97]

Pull request to incorporate the addition of BurgersRewienski derived from Burgers, for reduced-order modelling, as well as set up the basic structure for POD reduced order modelling.

The changes include:

• Creation of a new PDE type named BurgersRewienski derived from Burgers which is defined as eq. (18) of Carlberg 2011 (The GNAT nonlinear model reduction method and its application to fluid dynamics problems)
• Creation of a new parameter subsection named "reduced order" which specifies parameters used for the reduced order model
• Creation of a new test type named "burgers_rewienski_snapshot" which runs BurgersRewienski using the parameters "rewienski_a", "rewienski_b" and "final_time" specified in the reduced order parameter subsection. (See test here)
• Addition of a new parameter "output_solution_vector_modulo" in the subsection ODE solver which outputs the solution vector every "output_solution_vector_modulo" iterations of the nonlinear solver. (See here)
• Addition of a bash script snapshot_generator.sh which creates multiple parameter files for the parameters specified and outputs results for each test in a separate sub-directory
• Creation of a new class POD. Currently only has 2 functions. get_full_pod_basis() extracts the output files generated by snapshot_generator.sh and output U_svd. get_reduced_pod_basis retains only n columns of U_svd.
• Creation of a new test type named "reduced_order" which uses POD. This currently outputs the result of get_reduced_pod_basis to a text file, but will eventually be a POD reduced order model solver. (See test here)
• Addition of a manufactured solution test to ensure that BurgersRewienski converges at the proper order.

[dougshidong]

Looking good. Running the test suite overnight.