LOCA is a new software library for performing bifurcation analysis of large-scale applications. The source code has been available for download since April 19, 2002. The algorithms in LOCA are being designed as part of an ongoing research project at Sandia National Laboratories in Albuquerque into scalable stability analysis algorithms. When implemented with an application code, LOCA enables the tracking of solution branches as a function of system parameters and the direct tracking of bifurcation points. LOCA (which is written in ”C”) is designed to drive application codes that use Newton’s method to locate steady-state solutions to nonlinear problems. The algorithms are chosen to work for large problems, such as those that arise from discretizations of partial differential equations, and to run on distributed memory parallel machines. The approach in LOCA for locating and tracking bifurcations begins with augmenting the residual equations defining a steady state with additional equations that describe the bifurcation. A Newton method is then formulated for this augmented system; however, instead of loading up the Jacobian matrix for the entire augmented system (a task that involved second derivatives and dense matrix rows), bordering algorithms are used to decompose the linear solve into several solves with smaller matrices. Almost all of the algorithms just require multiple solves of the Jacobian matrix for the steady state problem to calculate the Newton updates for the augmented system. This greatly simplifies the implementation, since this is the same linear system that an application code using Newton’s method will already have invested in. Only the Hopf tracking algorithm requires the solution of a larger matrix, which is the complex matrix involving the Jacobian matrix and an imaginary multiple of the mass matrix.

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  1. Uecker, Hannes: Continuation and bifurcation in nonlinear PDEs - algorithms, applications, and experiments (2022)
  2. Douglas, Christopher M.; Emerson, Benjamin L.; Lieuwen, Timothy C.: Nonlinear dynamics of fully developed swirling jets (2021)
  3. Sánchez Umbría, J.; Net, M.: Continuation of double Hopf points in thermal convection of rotating fluid spheres (2021)
  4. Thies, Jonas; Wouters, Michiel; Hennig, Rebekka-Sarah; Vanroose, Wim: Towards scalable automatic exploration of bifurcation diagrams for large-scale applications (2021)
  5. Parker, Jeremy P.; Caulfield, C. P.; Kerswell, R. R.: Kelvin-Helmholtz billows above Richardson number (1/4) (2019)
  6. Wouters, Michiel; Vanroose, Wim: Automatic exploration techniques of numerical bifurcation diagrams illustrated by the Ginzburg-Landau equation (2019)
  7. Groh, R. M. J.; Avitabile, D.; Pirrera, A.: Generalised path-following for well-behaved nonlinear structures (2018)
  8. Kelley, C. T.: Numerical methods for nonlinear equations (2018)
  9. López, Vanessa: Numerical continuation of invariant solutions of the complex Ginzburg-Landau equation (2018)
  10. Canton, J.; Auteri, F.; Carini, M.: Linear global stability of two incompressible coaxial jets (2017)
  11. Dhillon, Daljit Singh J.; Milinkovitch, Michel C.; Zwicker, Matthias: Bifurcation analysis of reaction diffusion systems on arbitrary surfaces (2017)
  12. Canton, Jacopo; Schlatter, Philipp; Örlü, Ramis: Modal instability of the flow in a toroidal pipe (2016)
  13. Deremble, Bruno: Convective plumes in rotating systems (2016)
  14. Draelants, Delphine; Kłosiewicz, Przemysław; Broeckhove, Jan; Vanroose, Wim: Solving general auxin transport models with a numerical continuation toolbox in Python: PyNCT (2015)
  15. Net, M.; Sánchez, J.: Continuation of bifurcations of periodic orbits for large-scale systems (2015)
  16. Sharma, Sanjiv; Coetzee, Etienne B.; Lowenberg, Mark H.; Neild, Simon A.; Krauskopf, Bernd: Numerical continuation and bifurcation analysis in aircraft design: an industrial perspective (2015)
  17. Laing, Carlo R.: Numerical bifurcation theory for high-dimensional neural models (2014)
  18. Thompson, J. Michael T.; Sieber, Jan: Predicting climate tipping as a noisy bifurcation: a review (2011)
  19. Ciani, A.; Kreutner, W.; Frouzakis, C. E.; Lust, K.; Coppola, G.; Boulouchos, K.: An experimental and numerical study of the structure and stability of laminar opposed-jet flows (2010)
  20. Sánchez, Juan; Net, Marta: On the multiple shooting continuation of periodic orbits by Newton-Krylov methods (2010)

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