5. Transient Problems and Ode Solvers

  1. Pavel Šolín

Published Online: 30 NOV 2005

DOI: 10.1002/0471764108.ch5

Partial Differential Equations and the Finite Element Method

Partial Differential Equations and the Finite Element Method

How to Cite

Šolín, P. (2005) Transient Problems and Ode Solvers, in Partial Differential Equations and the Finite Element Method, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/0471764108.ch5

Publication History

  1. Published Online: 30 NOV 2005
  2. Published Print: 4 NOV 2005

ISBN Information

Print ISBN: 9780471720706

Online ISBN: 9780471764106

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Summary

Chapter 5 introduces the reader into modern ODE solvers for transient problems solved by the Method of lines. Emphasis is on higher-order one-step methods which are most suitable for the combination with the spatially-adaptive finite element methods. Discussed are the explicit and implicit Euler methods, the concept of stiffness, the explicit higher-order Runge-Kutta methods, embedded Runge-Kutta methods and adaptivity, and general implicit Runge-Kutta methods. Emphasis is given to an introduction to stability of the ODE schemes. Introduced are the notions of the stability function and stability domain, and A- and L-stability. These concepts are applied to the previously-introduced higher-order one-step schemes. We also discuss the collocation methods, and the construction of higher-order implicit (Gauss and Radau) Runge-Kutta methods via the higher-order quadrature rules. This chapter is closed with the discussion of efficient Newton-type iterations for the application in nonlinear ODE systems.