Nondissipative and energy-stable high-order finite-difference interface schemes for 2-D patch-refined grids

R. M.J. Kramer, C. Pantano, D. I. Pullin

Research output: Contribution to journalArticlepeer-review


A class of finite-difference interface schemes suitable for two-dimensional cell-centered grids with patch-refinement and step-changes in resolution is presented. Grids of this type are generated by adaptive mesh refinement methods according to resolution needs dictated by the physics of the problem being modeled. For these grids, coarse and fine nodes are not aligned at the mesh interfaces, resulting in hanging nodes. Three distinct geometries are identified at the interfaces of a domain with interior patch-refinement: edges, concave corners and convex corners. Asymptotic stability in time of the numerical scheme is achieved by imposing a summation-by-parts condition on the interface closure, which is thus also nondissipative. Interface stencils corresponding to an explicit fourth-order finite-difference scheme are presented for each geometry. To preserve stability, a reduction in local accuracy is required at the corner geometries. It is also found that no second-order accurate solution exists that satisfies the summation-by-parts condition. Tests using the 2-D scalar advection equation and an inviscid compressible vortex support the stability and accuracy of these stencils for both linear and nonlinear problems.

Original languageEnglish (US)
Pages (from-to)5280-5297
Number of pages18
JournalJournal of Computational Physics
Issue number14
StatePublished - Aug 1 2009


  • Adaptive mesh refinement
  • High-order finite difference
  • Mesh interface
  • Stable stencil
  • Summation by parts

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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