The stability of numerical boundary treatments for compact high-order finite-difference schemes
Journal of Computational Physics
Summation by parts for finite difference approximations for d/dx
Journal of Computational Physics
A stable and conservative interface treatment of arbitrary spatial accuracy
Journal of Computational Physics
Journal of Computational Physics
Designing an efficient solution stragety for fluid flows
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
Journal of Scientific Computing
Accurate and stable grid interfaces for finite volume methods
Applied Numerical Mathematics
Journal of Computational Physics
Local adaptive mesh refinement for shock hydrodynamics
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
High Order Stable Finite Difference Methods for the Schrödinger Equation
Journal of Scientific Computing
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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.