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. Colombo, A.; Crivellini, A.; Nigro, A.: On the entropy conserving/stable implicit DG discretization of the Euler equations in entropy variables (2022)
  2. Sandu, Adrian; Günther, Michael; Roberts, Steven: Linearly implicit GARK schemes (2021)
  3. Wang, Yazhou; Xue, Tao; Tamma, Kumar K.; Maxam, Dean; Qin, Guoliang: An accurate and simple universal a posteriori error estimator for GS4-1 framework: adaptive time stepping in first-order transient systems (2021)
  4. Noventa, G. M.; Massa, Francesco Carlo; Rebay, Stefano; Bassi, Francesco; Ghidoni, Antonio: Robustness and efficiency of an implicit time-adaptive discontinuous Galerkin solver for unsteady flows (2020)
  5. Wang, Lai; Yu, Meilin: Comparison of ROW, ESDIRK, and BDF2 for unsteady flows with the high-order flux reconstruction formulation (2020)
  6. Franciolini, Matteo; Crivellini, Andrea; Nigro, Alessandra: On the efficiency of a matrix-free linearly implicit time integration strategy for high-order discontinuous Galerkin solutions of incompressible turbulent flows (2017)
  7. González-Pinto, S.; Hernández-Abreu, D.; Pérez-Rodríguez, S.: W-methods to stabilize standard explicit Runge-Kutta methods in the time integration of advection-diffusion-reaction PDEs (2017)
  8. Tranquilli, Paul; Glandon, S. Ross; Sarshar, Arash; Sandu, Adrian: Analytical Jacobian-vector products for the matrix-free time integration of partial differential equations (2017)
  9. Weller, Stephan; Bänsch, Eberhard: Time discretization for capillary flow: beyond backward Euler (2017)
  10. Bassi, F.; Botti, L.; Colombo, A.; Crivellini, A.; Ghidoni, A.; Massa, F.: On the development of an implicit high-order discontinuous Galerkin method for DNS and implicit LES of turbulent flows (2016)
  11. Noventa, G.; Massa, F.; Bassi, F.; Colombo, A.; Franchina, N.; Ghidoni, A.: A high-order discontinuous Galerkin solver for unsteady incompressible turbulent flows (2016)
  12. Rang, Joachim: The Prothero and Robinson example: convergence studies for Runge-Kutta and Rosenbrock-Wanner methods (2016)
  13. Uzunca, Murat; Karasözen, Bülent; Sarıaydın-Filibelioğlu, Ayşe: Time-space adaptive method of time layers for the advective Allen-Cahn equation (2016)
  14. Bassi, F.; Botti, L.; Colombo, A.; Ghidoni, A.; Massa, F.: Linearly implicit Rosenbrock-type Runge-Kutta schemes applied to the discontinuous Galerkin solution of compressible and incompressible unsteady flows (2015)
  15. Debrabant, Kristian; Lang, Jens: On asymptotic global error estimation and control of finite difference solutions for semilinear parabolic equations (2015)
  16. Liao, Wenyuan: A strongly A-stable time integration method for solving the nonlinear reaction-diffusion equation (2015)
  17. Li, Tongxing (ed.); Diblík, Josef (ed.); Domoshnitsky, Alexander (ed.); Rogovchenko, Yuriy V. (ed.); Sadyrbaev, Felix (ed.); Wang, Qi-Ru (ed.): Qualitative analysis of differential, difference equations, and dynamic equations on time scales (2015)
  18. Rang, Joachim: Improved traditional Rosenbrock-Wanner methods for stiff ODEs and DAEs (2015)
  19. 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)
  20. 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)

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