Abstract
In previous work, a new adaptive meshfree advection scheme for numerically solving linear transport equations has been proposed. The scheme, being a combination of an adaptive semi-Lagrangian method and local radial basis function interpolation, is essentially a method of backward characteristics. The adaptivity of the meshfree advection scheme relies on customized rules for the refinement and coarsening of scattered nodes. In this paper, the method is extended to nonlinear transport equations. To this end, in order to be able to model shock propagation, an artificial viscosity term is added to the scheme. Moreover, the local interpolation method and the node adaption rules are modified accordingly. The good performance of the resulting method is finally shown in the numerical examples by using two specific nonlinear model problems: Burgers equation and the Buckley-Leverett equation, the latter describing a two-phase fluid flow in a porous medium.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Behrens J. (1996) An adaptive semi-Lagrangian advection scheme and its parallelization. Mon. Wea. Rev. 124, 2386–2395.
Behrens J. (1998) Atmospheric and ocean modeling with an adaptive finite element solver for the shallow-water equations. Applied Numerical Mathematics 26, 217–226.
Behrens J., and A. Iske (2001) Grid-free adaptive semi-Lagrangian advection using radial basis functions. To appear in Computers and Mathematics with Applications.
Behrens J., A. Iske, and St. Pöhn (2001) Effective node adaption for grid-free semi-Lagrangian advection. Discrete Modelling and Discrete Algorithms in Continuum Mechanics, Th. Sonar and I. Thomas (eds.), Logos Verlag, Berlin, 110–119.
Buckley, J. M., and M. C. Leverett (1942) Mechanism of fluid displacement in sands. Trans. AIME 146, 107–116.
Burgers, J. M. (1940) Application of a model system to illustrate some points of the statistical theory of free turbulence. Proc. Acad. Sci. Amsterdam 43, 2–12.
Courant, R., E. Isaacson, and M. Rees (1952) On the solution of nonlinear hyperbolic differential equations by finite differences. Comm. Pure Appl. Math. 5, 243–255.
Duchon, J. (1977) Splines minimizing rotation-invariant semi-norms in Sobolev spaces. Constructive Theory of Functions of Several Variables, W. Schempp and K. Zeller (eds.), Springer, Berlin, 85–100.
Durran, D. R. (1999) Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer, New York.
Falcone, M., and R. Ferretti (1998) Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J. Numer. Anal. 35:3, 909–940.
Fürst, J., and Th. Sonar (2001) On meshless collocation approximations of conservation laws: positive schemes and dissipation models. ZAMM 81, 403–415.
Gustafsson, B., H.-O. Kreiss, and J. Oliger (1995) Time Dependent Problems and Difference Methods, John Wiley and Sons, New York.
Gutzmer, T., and A. Iske (1997) Detection of discontinuities in scattered data approximation. Numerical Algorithms 16:2, 155–170.
Iske, A. (2001) Scattered data modelling using radial basis functions. Principles of Multiresolution in Geometric Modelling, M.S. Floater, A. Iske, and E. Quak (eds.), Summer School Lecture Notes, Munich University of Technology, 271–301.
Preparata, F. P., and M. I. Shamos (1985) Computational Geometry, Springer, New York.
Robert, A. (1981) A stable numerical integration scheme for the primitive meteorological equations. Atmosphere-Ocean, 19, 35–46.
Welge, H. J. (1952) A simplified method for computing oil recovery by gas or water drive. Trans. AIME 195, 97–108.
Wu, Z., and R. Schaback (1993) Local error estimates for radial basis function interpolation of scattered data. IMA J. of Numerical Analysis 13, 13–27.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Behrens, J., Iske, A., Käser, M. (2003). Adaptive Meshfree Method of Backward Characteristics for Nonlinear Transport Equations. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56103-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-56103-0_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43891-5
Online ISBN: 978-3-642-56103-0
eBook Packages: Springer Book Archive