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A new stiffly accurate Rosenbrock-Wanner method for solving the incompressible Navier-Stokes equations. (English) Zbl 1382.65286
Ansorge, Rainer (ed.) et al., Recent developments in the numerics of nonlinear hyperbolic conservation laws. Lectures presented at a workshop at the Mathematical Research Institute Oberwolfach, Germany, January 15–21, 2012. Berlin: Springer (ISBN 978-3-642-33220-3/hbk; 978-3-642-33221-0/ebook). Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM) 120, 301-315 (2013).
Summary: One possibility to solve stiff ODEs like the example of A. Prothero and A. Robinson [Math. Comput. 28, 145–162 (1974; Zbl 0309.65034)] or differential algebraic equations are Runge-Kutta methods (RK methods) [E. Hairer and G. Wanner, Solving ordinary differential equations. II: Stiff and differential-algebraic problems. 2nd rev. ed. Berlin: Springer (1996; Zbl 0859.65067)] and [K. Strehmel and R. Weiner, Linear-implizite Runge-Kutta-Methoden und ihre Anwendung. Leipzig: Teubner (1992; Zbl 0759.65047)]. Explicit RK methods may not be a good choice since for getting a stable numerical solution a stepsize restriction should be accepted, i.e. the problem should be solved with very small timesteps. Therefore it might be better to use implicit or linear implicit RK methods, so-called Rosenbrock-Wanner methods. Fully implicit RK methods may be ineffective for solving high dimensional ODEs since they need a high computational effort to solve the huge nonlinear system. Therefore we consider in this note diagonally implicit RK methods (DIRK methods).
For the entire collection see [Zbl 1254.65002].

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35Q30 Navier-Stokes equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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