We show how to combine in a natural way (i.e., without any test nor switch) the conservative and non-conservative formulations of an hyperbolic system that has a conservative form. This is inspired from two different classes of schemes: the residual distribution one (Abgrall in Commun Appl Math Comput 2(3): 341–368, 2020), and the active flux formulations (Eyman and Roe in 49th AIAA Aerospace Science Meeting, 2011; Eyman in active flux. PhD thesis, University of Michigan, 2013; Helzel et al. in J Sci Comput 80(3): 35–61, 2019; Barsukow in J Sci Comput 86(1): paper No. 3, 34, 2021; Roe in J Sci Comput 73: 1094–1114, 2017). The solution is globally continuous, and as in the active flux method, described by a combination of point values and average values. Unlike the “classical” active flux methods, the meaning of the point-wise and cell average degrees of freedom is different, and hence follow different forms of PDEs; it is a conservative version of the cell average, and a possibly non-conservative one for the points. This new class of scheme is proved to satisfy a Lax-Wendroff-like theorem. We also develop a method to perform nonlinear stability. We illustrate the behaviour on several benchmarks, some quite challenging.
R. Abgrall
. A Combination of Residual Distribution and the Active Flux Formulations or a New Class of Schemes That Can Combine Several Writings of the Same Hyperbolic Problem: Application to the 1D Euler Equations[J]. Communications on Applied Mathematics and Computation, 2023
, 5(1)
: 370
-402
.
DOI: 10.1007/s42967-021-00175-w
1. Abgrall, R.: Some remarks about conservation for residual distribution schemes. Comput. Methods Appl. Math. 18(3), 327–351 (2018)
2. Abgrall, R.: The notion of conservation for residual distribution schemes (or fluctuation splitting schemes), with some applications. Commun. Appl. Math. Comput. 2(3), 341–368 (2020)
3. Abgrall, R., Bacigaluppi, P., Tokareva, S.: High-order residual distribution scheme for the timedependent Euler equations of fluid dynamics. Comput. Math. Appl. 78(2), 274–297 (2019)
4. Abgrall, R., Ivanova, K.: High order schemes for compressible flow problems with staggered grids (2021) (in preparation)
5. Abgrall, R., Lipnikov, K., Morgan, N., Tokareva, S.: Multidimensional staggered grid residual distribution scheme for Lagrangian hydrodynamics. SIAM J. Sci. Comput. 42(1), A343–A370 (2020)
6. Abgrall, R., Tokareva, S.: Staggered grid residual distribution scheme for Lagrangian hydrodynamics. SIAM J. Sci. Comput. 39(5), A2317–A2344 (2017)
7. Barsukow, W.: The active flux scheme for nonlinear problems. J. Sci. Comput. 86(1), 3 (2021)
8. Clain, S., Diot, S., Loubère, R.: A high-order finite volume method for systems of conservation laws— multi-dimensional optimal order detection (MOOD). J. Comput. Phys. 230(10), 4028–4050 (2011)
9. Dakin, G., Després, B., Jaouen, S.: High-order staggered schemes for compressible hydrodynamics. Weak consistency and numerical validation. J. Comput. Phys. 376, 339–364 (2019)
10. Dobrev, V.A., Kolev, T., Rieben, R.N.: High-order curvilinear finite element methods for Lagrangian hydrodynamics. SIAM J. Sci. Comput. 34(5), B606–B641 (2012)
11. Eyman, T.A.: Active flux schemes. PhD thesis, University of Michigan, USA (2013)
12. Eyman, T.A., Roe, P.L.: Active flux schemes for systems. In: 20th AIAA Computationa Fluid Dynamics Conference, AIAA 2011-3840, AIAA, USA (2011)
13. Eyman, T.A., Roe, P.L.: Active flux schemes. In: 49th AIAA Aerospace Science Meeting including the New Horizons Forum and Aerospace Exposition, AIAA 2011-382, AIAA, USA (2011)
14. Godlewski, E., Raviart, P.-A.: Hyperbolic systems of conservation laws. In: Mathématiques and Applications (Paris), vol. 3/4. Ellipses, Paris (1991)
15. Helzel, C., Kerkmann, D., Scandurra, L.: A new ADER method inspired by the active flux method. J. Sci. Comput. 80(3), 35–61 (2019)
16. Herbin, R., Latché, J.-C., Nguyen, T.T.: Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations. ESAIM Math. Model. Numer. Anal. 52(3), 893–944 (2018)
17. Hou, T.Y., Le Floch, P.G.: Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comput. 62(206), 497–530 (1994)
18. Iserles, A.: Order stars and saturation theorem for first-order hyperbolics. IMA J. Numer. Anal. 2, 49–61 (1982)
19. Karni, S.: Multicomponent flow calculations by a consistent primitive algorithm. J. Comput. Phys. 112(1), 31–43 (1994)
20. Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 381–394 (1960)
21. Loubère, R.: Validation test case suite for compressible hydrodynamics computation (2005). http:// loube re. free. fr/ images/ test_ suite. PDF
22. Ramani, R., Reisner, J., Shkoller, S.: A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, part 2: the 2-D case. J. Comput. Phys. 387, 45–80 (2019)
23. Roe, P.L.: Is discontinuous reconstruction really a good idea? J. Sci. Comput. 73, 1094–1114 (2017)
24. Vilar, F.: A posteriori correction of high-order discontinuous Galerkin scheme through subcell finite volume formulation and flux reconstruction. J. Comput. Phys. 387, 245–279 (2019)
