ROS3P -- An accurate third-order Rosenbrock solver designed for parabolic problems WWe present a new Rosenbrock solver which is third-order accurate for nonlinear parabolic problems. Since Rosenbrock methods suffer from order reduction when they are applied to partial differential equations, additional order conditions have to be satisfied. Although these conditions have been known for a longer time, from the practical point of view only little has been done to construct new methods. G. Steinebach [Order-reduction of ROW-methods for DAEs and method of lines applications, Preprint 1741, Technische Hochschule Darmstadt, Germany (1995)] modified the well-known solver RODAS of E. Hairer and G. Wanner [Solving ordinary differential equations. II: Stiff and differential-algebraic problems. 2nd rev. ed. (1996; Zbl 0859.65067)] to preserve its classical order four for special problem classes including linear parabolic equations. His solver RODASP, however, drops down to order three for nonlinear parabolic problems. Our motivation here was to derive an efficient third-order Rosenbrock solver for the nonlinear situation. Such a method exists with three stages and two function evaluations only. A comparison with other third-order methods shows the substantial potential of our new method.

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  1. Rang, Joachim: The Prothero and Robinson example: convergence studies for Runge-Kutta and Rosenbrock-Wanner methods (2016)
  2. Liao, Wenyuan: A strongly A-stable time integration method for solving the nonlinear reaction-diffusion equation (2015)
  3. Rang, Joachim: Improved traditional Rosenbrock-Wanner methods for stiff ODEs and DAEs (2015)
  4. Becker, Urs; Simeon, Bernd; Burger, Michaek: On Rosenbrock methods for the time integration of nearly incompressible materials and their usage for nonlinear model reduction (2014)
  5. Bott, Stefanie; Clever, Debora; Lang, Jens; Ulbrich, Stefan; Ziems, Jan; Schröder, Dirk: On a fully adaptive SQP method for PDAE-constrained optimal control problems with control and state constraints (2014)
  6. Rang, Joachim: An analysis of the Prothero-Robinson example for constructing new DIRK and ROW methods (2014)
  7. Crivellini, Andrea; D’Alessandro, Valerio; Bassi, Francesco: Assessment of a high-order discontinuous Galerkin method for incompressible three-dimensional Navier-Stokes equations: benchmark results for the flow past a sphere up to Re=500 (2013)
  8. Bänsch, Eberhard; Weller, Stephan: A comparison of several time discretization methods for free surface flows (2012)
  9. Nagaiah, Ch.; Rüdiger, S.; Warnecke, G.; Falcke, M.: Adaptive space and time numerical simulation of reaction-diffusion models for intracellular calcium dynamics (2012)
  10. Jebens, Stefan; Knoth, Oswald; Weiner, Rüdiger: Partially implicit peer methods for the compressible Euler equations (2011)
  11. Goldstein, R.J.; Ibele, W.E.; Patankar, S.V.; Simon, T.W.; Kuehn, T.H.; Strykowski, P.J.; Tamma, K.K.; Heberlein, J.V.R.; Davidson, J.H.; Bischof, J.; Kulacki, F.A.; Kortshagen, U.; Garrick, S.; Srinivasan, V.; Ghosh, K.; Mittal, R.: Heat transfer -- a review of 2005 literature (2010)
  12. Hartmann, Stefan; Hamkar, Ahmad-Wahadj: Rosenbrock-type methods applied to finite element computations within finite strain viscoelasticity (2010)
  13. John, Volker; Rang, Joachim: Adaptive time step control for the incompressible Navier-Stokes equations (2010)
  14. Gerisch, A.; Lang, J.; Podhaisky, H.; Weiner, R.: High-order linearly implicit two-step peer - finite element methods for time-dependent PDEs (2009)
  15. Clemens, M.; Lang, D.; Teleaga, D.; Wimmer, G.: Space and time adaptive calculation of transient 3D magnetic fields (2008)
  16. Nagaiah, Ch.; Rüdiger, S.; Warnecke, G.; Falcke, M.: Adaptive numerical simulation of intracellular calcium dynamics using domain decomposition methods (2008)
  17. Novati, Paolo: Some secant approximations for Rosenbrock $W$-methods (2008)
  18. Rang, Joachim; Angermann, Lutz: New Rosenbrock methods of order 3 for PDAEs of index 2 (2008)
  19. Bassi, Francesco; Crivellini, Andrea; Di Pietro, Daniele A.; Rebay, Stefano: An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows (2007)
  20. Hartmann, Stefan; Wensch, Jörg: Finite element analysis of viscoelastic structures using Rosenbrock-type methods (2007)

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