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A comment on the computation of non-conservative products. (English) Zbl 1188.65134
Summary: We are interested in the solution of non-conservative hyperbolic systems, and consider in particular the so-called path-conservative schemes (see e.g. [C. Parés, SIAM J. Numer. Anal. 44, No. 1, 300–321 (2006; Zbl 1130.65089) and Manuel J. Castro, Philippe G. LeFloch, M. L. Muñoz-Ruiz and C. Parés, J. Comput. Phys. 227, No. 17, 8107–8129 (2008; Zbl 1176.76084)] which rely on the theoretical work of G. Dal Maso, P. G. LeFloch, and F. Murat [J. Math. Pures Appl., IX. Sér. 74, No. 6, 483–548 (1995; Zbl 0853.35068)]. The example of the standard Euler equations for a perfect gas is used to illuminate some computational issues and shortcomings of this approach.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
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[1] Dal Maso, G.; LeFloch, P.; Murat, F., Definition and weak stability of non-conservative products, Journal of math pures application., 74, 483-548, (1995) · Zbl 0853.35068
[2] Parés, C., Numerical methods for non-conservative hyperbolic systems: a theoretical framework, SIAM journal of numerical analysis., 44, 300-321, (2006) · Zbl 1130.65089
[3] Castro, M.J.; LeFloch, P.; Muñoz-Ruiz, M.L.; Parés, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, Journal of computational physics, 227, 17, 8107-8129, (2008) · Zbl 1176.76084
[4] Karni, S., Viscous shock profiles and primitive formulations, SIAM journal on numerical analysis, 29, 1592-1609, (1992) · Zbl 0764.76032
[5] Munz, C.D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM journal on numerical analysis, 31, 17-42, (1994) · Zbl 0796.76057
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