ANALYSIS OF A MESHFREE METHOD FOR THE COMPRESSIBLE EULER EQUATIONS

Title & Authors
ANALYSIS OF A MESHFREE METHOD FOR THE COMPRESSIBLE EULER EQUATIONS
Kim, Yong-Sik; Pahk, Dae-Hyeon;

Abstract
Mathematical analysis is made on a mesh free method for the compressible Euler equations. In particular, the Moving Least Square Reproducing Kernel (MLSRK) method is employed for space approximation. With the backward-Euler method used for time discretization, existence of discrete solution and it's $\small{L^2-error}$ estimate are obtained under a regularity assumption of the continuous solution. The result of numerical experiment made on the biconvex airfoil is presented.
Keywords
meshfree method;MLSRK;Euler equations;error estimate;
Language
English
Cited by
References
1.
H. J. Choe, D. W. Kim, H. H. Kim, and Y. S. Kim, Meshless method for the stationary incompressible Navier-Stokes equations, Discrete Contin. Dyn. Syst. Series B 1 (2001), no. 4, 495-526

2.
H. J. Choe, D. W. Kim, and Y. S. Kim, Meshfree method for the non-stationary incompressible Navier-Stokes equations, Discrete Contin. Dyn. Syst. Series B 6 (2006), no. 1, 17-39

3.
L. Demkowicz, J. T. Oden, W. Rachowicz, and O. Hardy, An h - p Taylor- Garlerkin finite element method for compressible Euler equations, Comput. Methods Appl. Mech. Eng 88 (1991), no. 3, 363-396

4.
R. A. Gingold and J.J. Monaghan, Smoothed Particle Hydrodynamics : theory and application to non-spherical stars, Mon. Not. R. Astron. Soc. 181 (1977), 275-389

5.
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equa- tions, Springer-Verlag, Berlin, 1986

6.
F. C. Gunther and W. K. Liu, Implementation of boundary conditions for mesh- less methods, Comput. Methods Appl. Mech. Engrg. 163 (1998), no. 1-4, 205- 230

7.
T. Kato, Quasi-linear equations of evolution with applications to partial differ- ential equations, Lecture Notes in Math. 448, Springer-Verlag, Berlin, 1975

8.
D. W. Kim and Y. S. Kim, Point collocation methods using the fast moving least- square reproducing kernel approximation, Internat. J. Numer. Methods Engrg. 56 (2003), no. 10, 1445-1464

9.
P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conf. Series in Appl. Math. 11, SIAM, 197

10.
S. Li and W. K. Liu, Reproducing Kernel Hierarchical Partition of Unity, Part I{Formulation and Theory, Internat. J. Numer. Methods Engrg. 45 (1999), no. 3, 251-288

11.
S. Li and W. K. Liu, Reproducing Kernel Hierarchical Partition of Unity, Part II{Appli- cations, Internat. J. Numer. Methods Engrg. 45 (1999), no. 3, 289-317

12.
L. D. Libersky, A. G. Petschek, T. C. Carney, J. R. Hipp, and F. A. Allahdadi, High Strain Lagrangian Hydrodynamics-a Three Dimensional SPH code for Dy- namic Material Response, J. Comput. Phys. 109 (1993), 67-75

13.
W. K. Liu, S. Jun, and Y. F. Zhang, Reproducing Kernel Particle Methods, Internat. J. Numer. Methods Fluids 20 (1995), no. 8-9, 1081-1106

14.
W. K. Liu, S. Jun, S. Li, J. Adee, and T. Belytschko, Reproducing Kernel Particle Methods for Structural Dynamics, Internat. J. Numer. Methods Engrg. 38 (1995), no. 10, 1655-1679

15.
W. K. Liu, S. Li, and T. Belytschko, Moving Least Square Reproducing Ker- nel Methods (I) Methodology and Convergence, Comput. Methods Appl. Mech. Engrg. 143 (1997), no. 1-2, 113{154

16.
W. K. Liu and Y. Chen, Wavelet and Multiple Scale Reproducing Kernel Methods, Internat. J. Numer. Methods Fluids 21 (1995), no. 10, 901-931

17.
Y. Y. Lu, T. Belytschko, and L. Gu, A New Implementation of the Element Free Galerkin Method, Comput. Methods Appl. Mech. Engrg. 113 (1994), no. 3-4, 397-414

18.
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, Berlin, 1984

19.
J. M. Melenk and I. Babu.ska, The Partition of Unity Finite Element Method : basic theory and applications, Comput. Methods Appl. Mech. Engrg. 139 (1996), no. 1-4, 289-314

20.
B. Nayroles, G. Touzot, and P. Villon, Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements, Comput. Mech. 10 (1992), 307- 318