Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form

Journal Article · Tue May 19 00:00:00 EDT 2020 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2020.109545· OSTI ID:1630859

Toro, E. F.  ^[1]; Saggiorato, B. ^[2]; Tokareva, S.  ^[3]; Hidalgo, A.  ^[4]

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1. Univ. of Trento (Italy)
2. Delft Univ. of Technology (Netherlands)
3. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
4. Univ. Politécnica de Madrid (Spain)

We propose a one-parameter family of low-dissipation centred numerical schemes for solving hyperbolic equations in conservative or non-conservative form, using finite volume or discontinuous Galerkin finite element methods. The new schemes spring out from the multi-dimensional FORCE method and are determined by a single parameter α ≥ 1. Given an increasing sequence of real numbers 1 ≤ α₁ < α₂ < . . . < α_K, there corresponds a sequence of numerical schemes with stability restriction associated to a decreasing sequence of Courant numbers 1 > c₁ > c₂ > . . . > c_K > 0 and a decreasing sequence of corresponding numerical viscosity functions d₁ > d₂ > . . . > d_K. For a given Courant number c_k ≤ 1 there is a real number α_k > 0 and a corresponding stable scheme with minimal numerical viscosity d_k. The proposed schemes suit very well the family of high-order discontinuous Galerkin finite element methods and the recently proposed class of ADER-TR schemes, whose orders of accuracy define decreasing sequences of Courant numbers, as the order of accuracy increases. The centred methods of this paper are stable in 2D and 3D in the frame of simultaneous updating formulae, unlike other centred methods, such as 1D FORCE, which are not. Furthermore, the schemes are highly accurate for slowly-moving waves, which is precisely the kind of waves that traditional centred methods smear disastrously. Additional features of the proposed schemes include ease of implementation and applicability to any hyperbolic system either in conservative or non-conservative form. Here, the proposed schemes are analysed and computationally assessed through a suite of test problems for a linear model system, for the Euler equations in one and two space dimensions, and for the Baer-Nunziato equations for compressible two-phase flow.

• Research Organization: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE Laboratory Directed Research and Development (LDRD) Program
• Grant/Contract Number: 89233218CNA000001; LA-UR-19-29379
• OSTI ID: 1630859
• Alternate ID(s): OSTI ID: 1701948
• Report Number(s): LA-UR-19-29379; TRN: US2200948
• Journal Information: Journal of Computational Physics, Vol. 416, Issue C; ISSN 0021-9991
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

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Mathematics