AMF-Runge–Kutta formulas and error estimates for the time integration of advection diffusion reaction PDEs

• S. Gonzalez-Pinto ^[1] ; D. Hernandez-Abreu ^[1] ; S. Perez-Rodriguez ^[1]
  1. [1] Universidad de La Laguna

     Universidad de La Laguna

     San Cristóbal de La Laguna, España

     
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 289, Nº 1 (1 December 2015), 2015, págs. 3-21
• Idioma: inglés
• DOI: 10.1016/j.cam.2015.03.048
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• Resumen

  • The convergence of a family of AMF-Runge–Kutta methods (in short AMF-RK) for the time integration of evolutionary Partial Differential Equations (PDEs) of Advection Diffusion Reaction type semi-discretized in space is considered. The methods are based on very few inexact Newton Iterations applied to Implicit Runge–Kutta formulas by combining the use of a natural splitting for the underlying Jacobians and the Approximate Matrix Factorization (AMF) technique. This approach allows a very cheap implementation of the Runge–Kutta formula under consideration. Particular AMF-RK methods based on Radau IIA formulas are considered. These methods have given very competitive results when compared with important formulas in the literature for multidimensional systems of non-linear parabolic PDE problems. Uniform bounds for the global time-space errors on semi-linear PDEs when simultaneously the time step-size and the spatial grid resolution tend to zero are derived. Numerical illustrations supporting the theory are presented.