The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids. (English) Zbl 1157.76372

Summary: Discrete geometric conservation laws (DGCLs) govern the geometric parameters of numerical schemes designed for the solution of unsteady flow problems on moving grids. A DGCL requires that these geometric parameters, which include, among others, grid positions and velocities, be computed so that the corresponding numerical scheme reproduces exactly a constant solution. Sometimes, this requirement affects the intrinsic design of an arbitrary Lagrangian Eulerian (ALE) solution method. In this paper, we show for sample ALE schemes that satisfying the corresponding DGCL is a necessary and sufficient condition for a numerical scheme to preserve the nonlinear stability of its fixed grid counterpart. We also highlight the impact of this theoretical result on practical applications of computational fluid dynamics.


76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)


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[1] S. A. Morton, R. B. Melville, and, M. R. Visbal, Accuracy and coupling issues of aeroelastic Navier-Stokes solutions of deforming meshes, AIAA Paper 97-1085, 38th AIAA Structures, presented at Structural Dynamics and Materials Conference, Kissimmee, Florida, April 7-10, 1997.
[2] Donea, J., An arbitrary Lagrangian-Eulerian finite element method for transient fluid-structure interactions, Comput. meth. appl. mech. eng., 33, 689, (1982) · Zbl 0508.73063
[3] Farhat, C.; Lesoinne, M.; Maman, N., Mixed explicit/implicit time integration of coupled aeroelastic problems: three-field formulation, geometric conservation and distributed solution, J. numer. meth. fluids, 21, 807, (1995) · Zbl 0865.76038
[4] J. T. Batina, Unsteady Euler airfoil solutions using unstructured dynamic meshes, AIAA Paper 89-0115, presented at AIAA 27th Aerospace Sciences Meeting, Reno, Nevada, January 9-12, 1989.
[5] Guillard, H.; Farhat, C., On the significance of the geometric conservation law for flow computations on moving meshes, Comput. meth. appl. mech. eng., 190, 1467, (2000) · Zbl 0993.76049
[6] Thomas, P.D.; Lombard, C.K., Geometric conservation law and its application to flow computations on moving grids, Aiaa j., 17, 1030, (1979) · Zbl 0436.76025
[7] NKonga, B.; Guillard, H., Godunov type method on non-structured meshes for three-dimensional moving boundary problems, Comp. meth. appl. mech. eng., 113, 183, (1994) · Zbl 0846.76060
[8] Lesoinne, M.; Farhat, C., Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations, Comput. meth. appl. mech. eng., 34, 71, (1996) · Zbl 0896.76044
[9] Koobus, B.; Farhat, C., Second-order time-accurate and geometrically conservative implicit schemes for flow computations on unstructured dynamic meshes, Comput. meth. appl. mech. eng., 170, 103, (1999) · Zbl 0943.76055
[10] Formaggia, L.; Nobile, F., A stability analysis for the arbitrary Lagrangian Eulerian formulation with finite elements, East-west J. numer. math., 7, 105, (1999) · Zbl 0942.65113
[11] Boris, J.P.; Book, D.L., Flux-corrected transport I. SHASTA, a fluid transport algorithm that works, J. comput. phys., 11, 38, (1973) · Zbl 0251.76004
[12] E. Godlewski, and, P. A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer-Verlag, New York, 1996. · Zbl 0860.65075
[13] C. B. Laney, Computational Gas Dynamics, Cambridge Univ. Press, Cambridge, UK, 1998. · Zbl 0947.76001
[14] Kruzkhov, S.N., Generalized solutions of the Cauchy problem in the large for nonlinear equations of first order, Dokl. akad. nauk. SSSR, 187, 29, (1969)
[15] C. Farhat, P. Geuzaine, and, C. Grandmont, The discrete geometric conservation law and its effects on nonlinear stability and accuracy. AIAA Paper 2001-2607, presented at 15th, AIAA Computational Fluid Dynamics Conference, Anaheim, California, June 11-14, 2001.
[16] Roe, P.L., Approximate Riemann solvers, parameters vectors and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[17] Van Leer, B., Towards the ultimate conservative difference scheme V: A second-order sequel to Godunov’s method, J. comput. phys., 32, 361, (1979) · Zbl 1364.65223
[18] Sod, G.A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. comput. phys., 27, 1, (1978) · Zbl 0387.76063
[19] Lesoinne, M.; Farhat, C., A higher-order subiteration free staggered algorithm for nonlinear transient aeroelastic problems, Aiaa j., 36, 1754, (1998)
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