1
$\begingroup$

A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients:

  parameter( ROWcc= 0.43586652150845900 )

  parameter( ROWb1 =(/ 0.0021103008548132443, &
                        .88607515441580453, &
                       -0.32405197677907682, &
                        0.43586652150845900 /) )

  parameter( ROWb2 =(/ 0.50000000000000000, &
                       0.38752422953298199, &
                      -0.20949226315045236, &
                       0.32196803361747034 /) )

  parameter( ROWa=reshape( (/ 0., 0.5, 0.5, 0.5,  &
                              0., 0., 0.5, 0.5,  &
                              0., 0., 0., 0. /), &
                           (/ ROWns,ROWns-1 /) ) )

  parameter( ROWc=reshape( (/ 0., -0.5, -0.79156480420464204, -0.49788969914518677 , &
                              0., 0.,   0.35244216792751432, 0.38607515441580453, &
                              0., 0., 0., -0.32405197677907682 /), &
                           (/ ROWns,ROWns-1 /) ) )

How can one determine what type of a Rosenbrock-Wanner (RK ROW) method does this code use solely from the coefficients? Search engine and universit library queries have not yielded successful results.

$\endgroup$
3
$\begingroup$

A search for the specific coefficients listed led me to the method ROS3PRL from

J. Sieber, Konvergenzanalyse und Numerische Tests für die Prothero–Robinson–Gleichung (Master thesis), TU Darmstadt, 2014.

I can't seem to find this thesis online, but the method is mentioned in the following which may be of interest.

Rang, Joachim. "Improved traditional Rosenbrock–Wanner methods for stiff ODEs and DAEs." Journal of Computational and Applied Mathematics 286 (2015): 128-144.

Ghasemi, Maryam. An Error Controlled Time Adaptive Numerical Scheme for Nonlinear Degenerate Diffusion-reaction Biofilm Models with Applications in Microbiology and Bioengineering. Diss. 2017.

To answer your more general question on how to track down a method from coefficients, he are some things to keep in mind:

  • The $\gamma$ coefficient is usually closely tied to the order of accuracy and stability. The choice $\gamma \approx .4358665$ is frequently seen on third order, L-stable Runge-Kutta and Rosenbrock methods
  • Based on your tag, it's used to solve a parabolic PDE. Rosenbrock methods for DAEs, SPPs, and parabolic PDEs must satisfy additional order conditions on top of the classical ones to avoid order reduction. This can be used to narrow down the search.
| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.