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 [5] 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 [9], 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.

[1] M.H. Carpenter, J. Nordstrom, D. Gottlieb, A stable and conservative interface treatment of arbitrary spatial accuracy, Journal of Computational Physics 148 (1999)
[2] B. Gustafsson, H.O. Kreiss, J. Oliger, Time dependent Problems and Difference Methods, Wiley-Interscience, New York, 1995.
[3] P. Olsson, Summation by parts, projections, and stability, I. Math. Comp. 64 (211) (1995a)
[4] P. Olsson, Summation by parts, projections, and stability, II. Math. Comp. 64(212) (1995b) 1473
[5] Bo Strand, Summation by parts for finite difference approximations for d/dx, Journal of Computational Physics 110 (1994).
[6] M.H. Carpenter, D. Gottlieb, S. Abarbanel, Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, Journal of Computational Physics 111 (2) (1994)
[7] J. Nordstrom, M.H. Carpenter, Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier- Stokes equations, Journal of Computational Physics 148 (1999)
[8] J. Nordstrom, N. Nordin, D. Henningson, The fringe region technique and the Fourier method used in the direct numerical simulation of spatially evolving viscous flows, SIAM Journal of Scientific Computing 20 (4) (1999) 1365
[9] K. Mattsson, Boundary procedures for summation-by-parts operators. Journal of Scientific Computing 18 (2003) 133
[10] D. J. Bodony, Accuracy of the simultaneous-approximation-term boundary condition for time-dependent problems, Journal of Scientific Computing 43 (2010) 118
[11] D. J. Bodony, Analysis of sponge zones for computational fluid mechanics, Journal of Computational Physics 212 (2) (2006) 681
[12] T. Colonius, Modeling artificial boundary conditions for compressible flow, Ann. Rev. Fluid Mech. 36 (2004), pp. 315
[13] T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numerica 8 (1999) 47