AENO: a Novel Reconstruction Method in Conjunction with ADER Schemes for Hyperbolic Equations

doi:10.1007/s42967-021-00147-0

Abstract

Abstract: In this paper, we present a novel spatial reconstruction scheme, called AENO, that results from a special averaging of the ENO polynomial and its closest neighbour, while retaining the stencil direction decided by the ENO choice. A variant of the scheme, called m-AENO, results from averaging the modified ENO (m-ENO) polynomial and its closest neighbour. The concept is thoroughly assessed for the one-dimensional linear advection equation and for a one-dimensional non-linear hyperbolic system, in conjunction with the fully discrete, high-order ADER approach implemented up to fifth order of accuracy in both space and time. The results, as compared to the conventional ENO, m-ENO and WENO schemes, are very encouraging. Surprisingly, our results show that the L₁-errors of the novel AENO approach are the smallest for most cases considered. Crucially, for a chosen error size, AENO turns out to be the most efficient method of all five methods tested.

Key words: Hyperbolic equations, High-order ADER, ENO/m-ENO/WENO, Novel reconstruction technique AENO/m-AENO

CLC Number:

35L50
65M08

Eleuterio F. Toro, Andrea Santacá, Gino I. Montecinos, Morena Celant, Lucas O. Müller. AENO: a Novel Reconstruction Method in Conjunction with ADER Schemes for Hyperbolic Equations[J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 776-852.

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References

1. Ben-Artzi, M., Falcovitz, J.:A second order Godunov-type scheme for compressible fluid dynamics. J. Comput. Phys. 55, 1-32 (1984)
2. Castro, C.E., Toro, E.F.:Solvers for the high-order Riemann problem for hyperbolic balance laws. J. Comput. Phys. 227, 2481-2513 (2008)
3. Dematté, R., Titarev, V.A., Motecinos, G.I., Toro, E.F.:ADER methods for hyperbolic equations with a time-reconstruction solver for the generalized Riemann problem. The scalar case. Commun. Appl. Math. Comput. 2, 369-402 (2020)
4. Dumbser, M., Enaux, C., Toro, E.F.:Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227(8), 3971-4001 (2008)
5. Dumbser, M., Käser, M.:Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 221(2), 693-723 (2007)
6. Dumbser, M., Käser, M., Titarev, V.A., Toro, E.F.:Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys. 226(8), 204-243 (2007)
7. Dumbser, M., Munz, C.D.:ADER discontinuous Galerkin schemes for aeroacoustics. Comptes Rendus Mécanique 333, 683-687 (2005)
8. Godunov, S.K.:A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 357-393 (1959)
9. Götz, C.R., Dumbser, M.:A novel solver for the generalized Riemann problem based on a simplified LeFloch-Raviart expansion and a local space-time discontinuous Galerkin formulation. J. Sci. Comput. (2016). https://doi.org/10.1007/s10915-016-0218-5
10. Götz, C.R., Iske, A.:Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws. Math. Comput. 85, 35-62 (2016)
11. Harten, A.:High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357-393 (1983)
12. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.:Uniformly high order accuracy essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231-303 (1987)
13. Harten, A., Osher, S.:Uniformly high-order accurate nonoscillatory schemes, I. SIAM J. Numer. Anal. 24(2), 279-309 (1987)
14. Jiang, G.S., Shu, C.-W.:Efficient Implementation of Weighted ENO Schemes. Technical Report ICASE 95-73. NASA Langley Research Center, Hampton (1995)
15. Jiang, G.S., Shu, C.-W.:Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202-228 (1996)
16. Kolgan, V.P.:Application of the principle of minimum derivatives to the construction of difference schemes for computing discontinuous solutions of gas dynamics. Uch. Zap. TsaGI, Russia 3(6), 68-77 (1972) (in Russian)
17. Laney, C.B.:Computational Gasdynamics. Cambridge University Press, Cambridge (1998)
18. Le Floch, P.:Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form. Commun. Part. Differ. Equ. 13(6), 669-727 (1988)
19. Le Floch, P., Raviart, P.A.:An asymptotic expansion for the solution of the generalized Riemann problem. Part 1:general theory. Ann. Inst. Henri Poincaré. Analyse non Lineáre 5(2), 179-207 (1988)
20. Montecinos, G.I., Castro, C.E., Dumbser, M., Toro, E.F.:Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms. J. Comput. Phys. 231, 6472-6494 (2012)
21. Montecinos, G.I., Toro, E.F.:Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes. J. Comput. Phys. 275, 415-442 (2014)
22. Müller, Lucas O., Toro, Eleuterio F.:A global multi-scale model for the human circulation with emphasis on the venous system. Int. J. Numer. Methods Biomed. Eng. 30(7), 681-725 (2014)
23. Roe, P.L.:Numerical Algorithms for the Linear Wave Equation. Technical Report 81047. Royal Aircraft Establishment, Bedford (1981)
24. Santaca. High-order ADER schemes for hyperbolic equations. Selected topics. Master Thesis, Department of Mathematics, University of Trento (2017)
25. Schwartzkopff, T., Munz, C.D., Toro, E.F.:ADER:high-order approach for linear hyperbolic systems in 2D. J. Sci. Comput. 17, 231-240 (2002)
26. Shu, C.-W.:Numerical experiments on the accuracy of ENO and modified ENO schemes. J. Sci. Comput. 5, 127-149 (1990)
27. Shu, C.-W., Osher, S.:Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439-471 (1988)
28. Shu, C.-W., Osher, S.:Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83, 32-78 (1989)
29. Sweby, P.K.:High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995-1011 (1984)
30. Titarev, V.A., Toro, E.F.:ADER:arbitrary high order Godunov approach. J. Sci. Comput. 17, 609-618 (2002)
31. Titarev, V.A., Toro, E.F.:Finite volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201(1), 238-260 (2004)
32. Titarev, V.A., Toro, E.F.:ENO and WENO schemes based on upwind and centred TVD fluxes. Comput. Fluids 3, 705-720 (2005)
33. Toro, E.F.:Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (1997)
34. Toro, E.F.:Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, New York (2001)
35. Toro, E.F.:Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2009)
36. Toro, E.F.:The ADER path to high-order Godunov methods. In:Continuum Mechanics, Applied Mathematics and Scientific Computing:Godunov's Legacy-A Liber Amicorum to Professor Godunov, pp. 359-366. Springer, Berlin (2020)
37. Toro, E.F., Billett, S.J.:Centred TVD schemes for hyperbolic conservation laws. IMA J. Numer. Anal. 20, 47-79 (2000)
38. Toro, E.F., Millington, R.C., Nejad, L.A.M.:Towards very high-order Godunov schemes. In:Toro, E.F. (ed.) Godunov Methods:Theory and Applications. Edited Review, pp. 905-937. Kluwer Academic/Plenum Publishers (2001)
39. Toro, E.F., Siviglia, A.:Flow in collapsible tubes with discontinuous mechanical properties:mathematical model and exact solutions. Commun. Comput. Phys. 13(2), 361-385 (2013)
40. Toro, E.F., Montecinos, G.I.:Implicit, semi-analytical solution of the generalised Riemann problem for stiff hyperbolic balance laws. J. Comput. Phys. 303, 146-172 (2015)
41. Toro, E.F., Titarev, V.A.:Solution of the generalised Riemann problem for advection-reaction equations. Proc. R. Soc. Lond. A 458, 271-281 (2002)
42. Toro, E.F., Titarev, V.A.:TVD fluxes for the high-order ADER schemes. J. Sci. Comput. 24, 285-309 (2005)
43. van Leer, B.:Towards the ultimate conservative difference scheme I. The quest for monotonicity. Lect. Notes Phys. 18, 163-168 (1973)
44. van Leer, B.:On the relation between the upwind-differencing schemes of Godunov, Enguist-Osher and Roe. SIAM J. Sci. Stat. Comput. 5(1), 1-20 (1985)
45. Woodward, P., Colella, P.:The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115-173 (1984)

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