LPRH — Local Polynomial Regression Hydrodynamics
Local Polynomial Regression (LPR) is a weighted local least-squares method for fitting a curve to data. LPR provides a local Taylor series fit of the data at any spatial location. LPR provides estimates not only to the data, but also to derivatives of the data. This method is contrasted to the method of Moving Least Squares (MLS) which only provides a functional fit for the data. To obtain derivatives using MLS, one would be required to take analytic derivatives of the MLS functional fit. Since differentiation is known to be an unsmoothing operation, the derivatives obtained in MLS are thus less smooth than LPR derivatives. This fact has great implications for the stability of numerical methods based on MLS and LPR.
MLS and LPR can be directly used in a differential equation to provide a numerical scheme that mimics finite-differences. LPR was found to be much more stable than MLS in such a setting. However, these numerical methods cannot accurately solve nonlinear PDE’s in this fashion.
Particle or mesh-free methods for hydrodynamics typically use artificial viscosity to stabilize themselves when shocks are present. LPR can be used to solve the equations of hydrodynamics (Euler equations) without artificial viscosity. The Van Leer flux splitting scheme is used in conjunction with LPR to provide a stable and robust solution to the Euler equations. Numerical solutions are computed on both fixed and moving particle distributions.
KeywordsShock Tube Smooth Particle Hydrodynamic Meshless Method Move Little Square High Velocity Impact
Unable to display preview. Download preview PDF.
- 3.Dilts, G.A.: Equivalence of the SPH Method and a Space-Time Galerkin Moving Particle Method. Los Alamos National Laboratory Unlimited Release LA-UR 96–134 (1996)Google Scholar
- 4.Dilts, G.A.: Moving-Least-Squares-Particle Hydrodynamics I. Los Alamos National Laboratory Unlimited Release LA-UR 96–134 (1997)Google Scholar
- 8.Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics — Theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181 (1977) 867–871Google Scholar
- 11.Hultman, J., Kalellander, D.: An SPH code for galaxy formation problems. Presentation of the code. Astron. Astrophys. 324 (1997) 534 1997 AA 324, 534Google Scholar
- 12.Laney, C.B.: Computational Gasdynamics. (New York: Cambridge) 1998Google Scholar
- 14.Lhner, R., Sacco, C., Onate, E., Idelsohn, S.: A Finite Point Method for Compressible Flow, paper presented at ECCOMAS 2000, Barcelona, September (2000)Google Scholar
- 16.Morris, J.P.: A study of stability properties of smooth particle hydrodynamics. Publ. Astron. Soc. Aust. 13 (1996) 97–102Google Scholar
- 19.Steinmetz, M., Müller, E.: On the capabilities and limits of smoothed particle hydrodynamics. Astron. Astrophy. 268 (1993) 391Google Scholar