FLOW: Constructing non-reflecting boundaries using multiple Penalty terms
For any difference method the boundary conditions must be implemented so the problem is stable, which can be done by adding a single penalty term at the first and/or last grid point for the onedimensional problem. By expanding this penalty domain to several grid points one can still maintain stability by choosing the penalty coefficients in an appropriate way. Once the stability condition is satisfied, the remaining coefficients can be used to • increase the accuracy at the boundaries • speed up the convergence • produce non-reflecting boundaries The first two problems has been solved successfully and this project will aim to produce the nonreflecting boundaries.
SeRC Project at LiU
Stable Boundary Conditions for In and Outgoing Waves
Finite difference operators which satisfy summation-by-parts (SBP) property  in combination with weak well-posed boundary conditions always give energy stability [2, 3, 4, 9]. One such boundary treatment is the simultaneous approximation term (SAT) method , which linearly combines the partial differential equation with the well-posed boundary conditions [1, 7, 10]. We will extend the application of this technique by applying weak boundary conditions in an extended domain. We aim for higher accuracy from the difference schemes, the possibility to modify the spectrum of the resulting operator and the possibility to construct better non-reflecting properties at the boundaries. All conditions will be stable by construction. The boundary procedure technique has clear connections and many similarities to methods refereed to as fringe region, sponge layers and buffer layers [8, 11, 12, 13] but is more general. Typical applications where this would be beneficial includes all wave propagation problems which require accurate ingoing signals as well as non-reflecting boundaries for outgoing signals as well as receptivity studies in for boundary layer instability.
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