Marcel Klinge (Martin Luther University Halle-Wittenberg), Domingo Hernández-Abreu, Rüdiger Weiner

In this talk, we consider numerical methods for the solution of stiff initial value problems \[\label{ODE} y^{\prime}(t)=f(t,y(t)),\quad y(t_0)=y_0 \in \mathbb{R}^n,\quad t\in\left[t_0,t_e\right].\] Implicit integration methods require the solution of linear systems, which can be very expensive for high dimensional problems . One possibility is to apply an Approximate Matrix Factorization (AMF) technique. The AMF approach uses some splitting of the right-hand side of and exploits special structures of the corresponding Jacobians. We consider linearly-implicit one-step W-methods and two-step W-methods with AMF. Furthermore, we discuss AMF peer methods, which require the application of Newton iteration. We compare these schemes in numerical experiments on a linear model.

References

[1] S. Beck, S. González-Pinto, S. Pérez-Rodríguez and R. Weiner. A comparison of AMF- and Krylov-methods in Matlab for large stiff ODE systems. J. Comput. Appl. Math. 262, 292–303 (2014).

[2] S. González-Pinto, D. Hernández-Abreu, S. Pérez-Rodríguez and R. Weiner. A family of three-stage third order AMF-W-methods for the time integration of advection diffusion reaction PDEs. Appl. Math. Comput. 274, 565–584 (2016).

[3] H. Podhaisky, R. Weiner and B.A. Schmitt. Two-step W-methods for stiff ODE systems. Vietnam J. Math. 30, 591–603 (2002).