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Department of Information Technology

Summation-by-parts operators

High-order finite difference operators that satisfy a summation-by-parts property are combined with weakly implemented penalty boundary conditions to yield a stable numerical approximation. The penalty treatment is generalized to interfaces that couple grid patches, where each patch is discretized with a structured mesh, allowing for structured adaptive mesh refinement. We can show that the loss of accuracy, due to lower order approximations of the operators near the boundaries with respect to the interior scheme, can in part be recovered.

We are currently implementing the framework in an adaptive solver.

Radial basis function discretizations

Since solutions to the TDSE are usually very smooth, high-order methods are convenient for this type of problems. It is very popular among theoretical chemists to use the Fourier-based pseudospectral method for spatial discretization. Changing the basis to radial basis functions (RBF) has two potential advantages: Firstly, one is more flexible in selecting boundary conditions. In this way, it is possible to solve open boundary problems. Moreover, RBF allow for scattered node distributions, a means of adapting the point distribution to the shape of the solution instead of resolving the whole high-dimensional space.

In this project, we analyze radial basis function approximations for the time-dependent Schrödinger equation. Aspects that we are currently interested in include stability of the RBF discretization, Galerkin formulation, parameter choice for the basis function, transfer between basis sets, and conservation properties of the scheme.

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Updated  2017-02-04 16:19:29 by Kurt Otto.