WIAS Preprint No. 2286, (2016)

Uniform second order convergence of a complete flux scheme on unstructured 1D grids for a singularly perturbed advection-diffusion equation and some multidimensional extensions


  • Farrell, Patricio
    ORCID: 0000-0001-9969-6615
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698

2010 Mathematics Subject Classification

  • 65L11 65L20 65N08 65N12


  • singularly perturbed advection-diffusion equation, uniform second-order convergence, finite-volume method, complete flux scheme




The accurate and efficient discretization of singularly perturbed advection-diffusion equations on arbitrary 2D and 3D domains remains an open problem. An interesting approach to tackle this problem is the complete flux scheme (CFS) proposed by G. D. Thiart and further investigated by J. ten Thije Boonkkamp. For the CFS, uniform second order convergence has been proven on structured grids. We extend a version of the CFS to unstructured grids for a steady singularly perturbed advection-diffusion equation. By construction, the novel finite volume scheme is nodally exact in 1D for piecewise constant source terms. This property allows to use elegant continuous arguments in order to prove uniform second order convergence on unstructured one-dimensional grids. Numerical results verify the predicted bounds and suggest that by aligning the finite volume grid along the velocity field uniform second order convergence can be obtained in higher space dimensions as well.

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