scholarly journals Avoiding the order reduction when solving second-order in time PDEs with Fractional Step Runge–Kutta–Nyström methods

2016 ◽  
Vol 71 (7) ◽  
pp. 1425-1447 ◽  
Author(s):  
M.J. Moreta ◽  
B. Bujanda ◽  
J.C. Jorge
Keyword(s):  
Order Reduction ◽  
Second Order ◽  
Fractional Step ◽  
Nyström Methods ◽  
Second Order In Time ◽  
Runge Kutta

scholarly journals Rosenbrock Type Methods for Solving Non-Linear Second-Order in Time Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2225
Author(s):  
Maria Jesus Moreta
Keyword(s):  
Computational Cost ◽  
Intermediate Stage ◽  
Second Order ◽  
First Order ◽  
New Class ◽  
Nyström Methods ◽  
Non Linear ◽  
Second Order In Time ◽  
Linear Problems ◽  
Runge Kutta

In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they can be seen as a generalization of these methods when they are applied to second-order in time problems which have been previously transformed into first-order in time problems. As they also follow the ideas of Runge–Kutta–Nyström methods when solving second-order in time problems, we have called them Rosenbrock–Nyström methods. When solving non-linear problems, Rosenbrock–Nyström methods present less computational cost than implicit Runge–Kutta–Nyström ones, as the non-linear systems which arise at every intermediate stage when Runge–Kutta–Nyström methods are used are replaced with sequences of linear ones.

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