Summary
It is shown that the interpretation of the Galerkin-Characteristic method for the scalar advection equation in the framework of particle methods yields a computationally efficient algorithm. Such an algorithm consists of updating the dependent variable at the grid points by cubic spline interpolation at the feet of the characteristic curves. The algorithm is unconditionally stable. The error analysis in the maximum norm shows that for sufficiently smooth functions the feet of the characteristic curves are points of high order convergence.
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References
Bardos, C., Bercovier, M., Pironneau, O. (1981): The vortex method with finite elements. Math. Comput.36, 119–136
Behforooz, G.H., Papamichael, N. (1979): End Conditions for Cubic Spline Interpolation. J. Inst. Maths. Applics.23, 355–366
Benque, J.P., Ibler, B., Keramsi, A., Labadie, G. (1980): A Finite Element Method for Navier-Stokes Equations. Proceedings of the third international conference on finite elements in flow problems. Banff, Alberta, Canada
de Boor, C. (1978): A Practical Guide to Splines. Springer, Berlin Heidelberg New York
Boris, J.P., Book, D.L. (1973): Flux corrected transport. SHASTA. J. Comput. Phys.11, 36–39
Ciarlet, P. (1978): The finite element method for elliptic problems. North-Holland, Amsterdam
Douglas, J., Russell, T.F. (1982): Numerical methods for convection dominated diffusion problems based on combining the method of the characteristics with finite elements or finite differences. SIAM J. Numer. Anal.19, 871–885
Girault, V., Raviart, P.-A. (1986): Finite Element Methods for Navier-Stokes Equations. Springer, Berlin Heidelberg New York
Hale, J.K. (1980): Ordinary Differential Equations, 2nd ed. Krieger, Malabar, Florida
Hasbani, Y., Livne, E., Bercovier, M. (1983): Finite elements and characteristics applied to advection-diffusion equations. Comput. Fluids11, 71–83
Hughes, T.J.R., Tezduyar, T.E., Brooks, A. (1982): Streamline upwind formulation for advection-diffusion, Navier-Stokes and first order hyperbolic equations. Fourth Internat. Conf. on Finite Element Methods in Fluids, Tokyo
Johnson, C., Navaert, U., Pitkaranta, J. (1984): Finite element methods for linear hyperbolic problems. Comput. Meth. Appl. Mech. Engrg.45, 285–312
Mas-Gallic, S., Raviart, P-A. (1987): A particle method for first order symmetric systems. Numer. Math.51, 323–352
Maz'ja, V.G. (1979): Sobolev Spaces. Springer, Berlin Heidelberg New York
Morton, K.W., Priestley, A., Suli, E. (1988): Stability of the Lagrange-Galerkin method with non-exact intergration. RAIRO4, 225–250
Munz, C-D. (1988): On the numerical dissipation of high resolution schemes for the hyperbolic conservation laws. J. Comput. Phys.77, 18–39
Pironneau, O. (1982): On the transport diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math.38, 309–332
Raviart, P-A. (1985): An analysis of particle methods. In: F. Brezzi, ed., Numerical Methods in Fluid Dynamics. Lecture Notes in Mathematics, Vol. 1127. Springer, Berlin Heidelberg New York
Suli, E. (1988): Convergence and Nonlinear Stability of the Lagrange-Galerkin Method for the Navier-Stokes Equations. Numer. Math.53, 459–483
Temperton, C., Staniforth, A. (1987): An efficient two-time level semi-Lagrangian semi-implicit integration scheme. Q.J.R. Meteorl. Soc.113, 1025–1039
Widlund, O. (1977): On best error bounds for approximation by piecewise polynomial functions. Numer. Math.27, 327–338
Zalesak, S.T. (1979): Fully multidimensional flux-corrected transport. J. Comput. Phys.31, 335–362
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Bermejo, R. Analysis of an algorithm for the Galerkin-characteristic method. Numer. Math. 60, 163–194 (1991). https://doi.org/10.1007/BF01385720
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DOI: https://doi.org/10.1007/BF01385720