Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs

  • Authors:

  • S. Blanes;E. Ponsoda

  • Affiliations:

  • Instituto de Matemática Multidisciplinar, Building 8G, second floor, Universitat Politècnica de València, 46022 Valencia, Spain;Instituto de Matemática Multidisciplinar, Building 8G, second floor, Universitat Politècnica de València, 46022 Valencia, Spain

  • Venue:
  • Applied Numerical Mathematics
  • Year:
  • 2012

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Abstract

We consider time-averaging methods based on the Magnus series expansion jointly with exponential integrators for the numerical integration of general linear non-homogeneous differential equations. The schemes can be considered as averaged methods which transform, for one time step, a non-autonomous problem into an autonomous one whose flows agree up to a given order of accuracy at the end of the time step. The problem is reformulated as a particular case of a matrix Riccati differential equation and the Mobius transformation is considered, leading to a homogeneous linear problem. The methods proposed can be used both for initial value problems (IVPs) as well as for two-point boundary value problems (BVPs). In addition, they allow to use different approximations for different parts of the equation, e.g. the homogeneous and non-homogeneous parts, or to use adaptive time steps. The particular case of separated boundary conditions using the imbedding formulation is also considered. This formulation allows us to transform a stiff and badly conditioned BVP into a set of well conditioned IVPs which can be integrated using some of the previous methods. The performance of the methods is illustrated on some numerical examples.