An Essentially Non-Oscillatory Spectral Deferred Correction Method for Conservation Laws

We present a computational method based on the Spectral Deferred Corrections (SDC) time integration technique and the Essentially Non-Oscillatory (ENO) finite volume method for the conservation laws (one-dimensional Euler equations). The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (Piece-wise Parabolic Method (PPM)) for solving the conservation laws is first carried out by Layton et al. in [Layton, A. T. and Minion, M. L. [2004] “Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics,” J. Comput. Phys. 194(2), 697–714]. Issues about this approach have been addressed and some improvements have been added to it in [Kadioglu et al. [2012] “A gas dynamics method based on the spectral deferred corrections (SDC) time integration technique and the piecewise parabolic method (PPM),” Am. J. Comput. Math. 1–4, 303–317]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM methods). We have tested the new SDC-ENO technique by solving several test problems involving moderate to strong shock waves and smooth/complex flow structures. Our numerical results show that we have numerically achieved the formally fourth-order convergence of the new method for smooth problems. Our numerical results also indicate that the newly proposed technique performs very well providing highly resolved shock discontinuities and fairly good contact solutions. More importantly, the discontinuities in the flow test problems are captured with essentially no-oscillations. We have numerically compared the fourth-order SDC-ENO scheme to the fourth-order SDC-PPM method for the same test problems. The results are similar for most of the test problems except in some cases the SDC-PPM method suffers from minor oscillations compared to SDC-ENO scheme being completely oscillation free.