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Isogeometric semi-Lagrangian analysis for transport problems. (English) Zbl 1502.76078

Summary: Isogeometric analysis (IGA) is combined with the semi-Lagrangian scheme to develop a stable and highly accurate method for the numerical solution of transport problems. An \(L^2\) projection using the non-uniform rational B-splines (NURBS) is proposed for the approximation of the solution at the departure points. The proposed method maintains the advantages of the semi-Lagrangian scheme in reducing the truncation errors and allowing for large CFL numbers in the simulations while the IGA guarantees the exact representation of the geometry of the computational domain. The performance of the isogeometric semi-Lagrangian analysis is demonstrated for a deformational flow problem and the benchmark of a single vortex flow.

MSC:

76M99 Basic methods in fluid mechanics
76R99 Diffusion and convection
65D07 Numerical computation using splines
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs

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References:

[1] Bazilevs, Y.; da Veiga, L. B.; Cottrell, J. A.; Hughes, T. J.R.; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes, Math. Models Methods Appl. Sci., 16, 07, 1031-1090 (2006) · Zbl 1103.65113
[2] El-Amrani, M.; Seaid, M., An \(\operatorname{L}^2\)-projection for the Galerkin-characteristic solution of incompressible flows, SIAM J. Sci. Comput., 33, 6, 3110-3131 (2011) · Zbl 1232.65143
[3] Nguyen, V. P.; Anitescu, C.; Bordas, S. P.A.; Rabczuk, T., Isogeometric analysis: an overview and computer implementation aspects, Math. Comput. Simulation, 117, 89-116 (2015) · Zbl 07313396
[4] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric Analysis: Toward Integration of CaD and FEa (2009), John Wiley & Sons · Zbl 1378.65009
[5] Chernov, A.; Schwab, C., Exponential convergence of Gauss-Jacobi quadratures for singular integrals over simplices in arbitrary dimension, SIAM J. Numer. Anal., 50, 3, 1433-1455 (2012) · Zbl 1252.65209
[6] Amestoy, P. R.; Duff, I. S.; L’Excellent, J. Y.; Koster, J., A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23, 1, 15-41 (2001) · Zbl 0992.65018
[7] Kou, J.; Hurtado-de Mendoza, A.; Joshi, S.; Le Clainche, S.; Ferrer, E., Eigensolution analysis of immersed boundary method based on volume penalization: Applications to high-order schemes, J. Comput. Phys., Article 110817 pp. (2021)
[8] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 39-41, 4135-4195 (2005) · Zbl 1151.74419
[9] Hólm, E. V., A fully two-dimensional, non-oscillatory advection scheme for momentum and scalar transport equations, Mon. Wea. Rev., 123, 536-552 (1995)
[10] Enright, D.; Losasso, F.; Fedkiw, R., A fast and accurate semi-Lagrangian particle level set method, Comput. Struct., 83, 6-7, 479-490 (2005)
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