Conservative and Stable Degree Preserving SBP Operators for Non-conforming Meshes
- 268 Downloads
Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP–SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new degree preserving discretizations require an ansatz that the norm matrix of the SBP operator is of a degree \(\ge 2p\), in contrast to, for example, existing finite difference SBP operators, where the norm matrix is \(2p-1\) accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.
KeywordsFirst derivative Summation-by-parts Simultaneous-approximation-term Conservation Energy stability Finite difference methods Non-conforming methods Intermediate grids
The work of Lucas Friedrich, Andrew Winters and Gregor Gassner was funded by the Deutsche Forschungsgemeinschaft (DFG) Grant TA 2160/1-1. The work of Jason Hicken was partially funded by the Air Force Office of Scientific Research Award FA9550-15-1-0242 under Dr. Jean-Luc Cambier. This work was partially performed on the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) at the Regionales Rechenzentrum Köln (RRZK).
- 4.Carpenter, M.H., Kennedy, C.A.: Fourth-order 2N-storage Runge–Kutta schemes. Tech. Report, NASA TM-109112. NASA Langley Research Center (1994)Google Scholar
- 8.Del Rey Fernández, D.C., Hicken, J.E., Zingg, D.W.: Simultaneous approximation terms for multidimensional summation-by-parts operators. J. Sci. Comput. (see arXiv:1605.03214v2 [math.NA]), (2016)
- 14.Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. In: Pure and Applied Mathematics, 2nd edn. Willey, New York (2013)Google Scholar
- 15.Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Opportunities for efficient high-order methods based on the summation-by-parts property, In: 22nd Conference on AIAA Computational Fluid Dynamics, AIAA Paper 2015-3198, Dallas, Texas, USA, (2015)Google Scholar
- 17.Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Simultaneous approximation terms for multidimensional summation-by-parts operators, In: 46th Conference on AIAA Fluid Dynamics, Washington, DC, USA. (Accepted) (2016)Google Scholar
- 21.Lundquist, T., Nordström, J.: On the suboptimal accuracy of summation-by-parts schemes with non-conforming block interfaces. Tech. Report, LiTH-MAT-R-2015/16-SE. Linköping University (2015)Google Scholar