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. (English) Zbl 1524.65305
Summary: 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 [R. Abgrall, Commun. Appl. Math. Comput. 2, No. 3, 341–368 (2020; Zbl 1476.65202)], and the active flux formulations [T. A. Eyman and P. L. Roe, “Active flux schemes”, in 49th AIAA Aerospace Science Meeting, Orlando, Florida, January 4–7, 2011. Article ID AIAA 2011-382, 11 p. (2021; doi:10.2514/6.2011-382); T. A. Eyman, Active flux schemes. Detroit: University of Michigan (PhD Thesis) (2013); C. Helzel et al., J. Sci. Comput. 80, No. 3, 1463–1497 (2019; Zbl 1428.65020); W. Barsukow, J. Sci. Comput. 86, No. 1, Paper No. 3, 34 p. (2021; Zbl 1457.65068); P. Roe, J. Sci. Comput. 73, No. 2–3, 1094–1114 (2017; Zbl 1381.65073)]. 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 non-linear stability. We illustrate the behaviour on several benchmarks, some quite challenging.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
Software:
MOODReferences:
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