High-order, multidimensional, and conservative coarse–fine interpolation for adaptive mesh refinement

The author presents a polynomial-based algorithm for high-order multidimensional interpolation at the coarse–fine interface in the context of adaptive mesh refinement on structured Cartesian grids. The proposed algorithm reduces coarse–fine interpolation to matrix–vector products by exploiting the static mesh geometry and a family of nonsingularity-preserving stencil transformations. As such, no linear system is solved at the runtime and the ill-conditioning of Vandermonde matrix is avoided. The algorithm is also generic in that D, the dimensionality of the computational domain, and p, the degree of the interpolating polynomial, are both arbitrary positive integers. Stability and accuracy are verified by interpolating simple functions, and by applying the proposed method to adaptively solving Poisson’s equation and the convection–diffusion equation. The companion MATLAB® package, AMRCFI, is also freely available for convenience and more implementation details.

► The proposed method, AMRCFI, satisfies automatic conservation.
► AMRCFI linearly maps coarse-cell averages in a poised stencil to fine-ghost averages.
► AMRCFI is generic in dimensionality and interpolation order.
► AMRCFI only entails matrix–vector products by compile-time precomputation.
► The algorithm of principal stencil transformation benefits least squares.