Meshless local discontinuous Petrov-Galerkin method with application to blasting problems

Abstract

A meshless local discontinuous Petrov-Galerkin (MLDPG) method based on the local symmetric weak form (LSWF) is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin (RKDG) method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional nonoverlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method (MM), and it can handle the problems with extremely complicated physics and geometries easily. Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes.

Keywords

meshless method / discontinuous Galerkin method / meshless local discontinuous Petrov-Galerkin (MLDPG) method / finite-volume particle method / convection-dominated flow

References

[1] Monaghan J. J.. Smoothed particle hydrodynamics[J]. Annual Review of Astronomy and Astrophysics, 1992, 30 543-574.

[2] Monaghan J. J.. Smoothed particle hydrodynamics[J]. Reports on Progress in Physics, 2005, 68(8): 1703-1759.

[3] Oñate E., Idelsohn S.. A finite point method in computational mechanics: Application to convective transport and fluid flow[J]. International Journal for Numerical Methods in Engineering, 1996, 39(22): 3839-3866.

[4] Liu W. K., Jun S.. Multiresolution reproducing kernel particle method for computational fluid mechanics[J]. International Journal for Numerical Methods in Fluids, 1997, 24(12): 1391-1415.

[5] Lin H., Atluri S. N.. Meshless local Petrov-Galerkin (MLPG) method for convection-diffusion problems[J]. Computer Modeling in Engineering and Sciences, 2000, 1(2): 45 60

[6] Hietel D., Steiner K.. A finite volume method for compressible flows[J]. Mathematical Models and Methods in Applied Science, 2000, 10(9): 1363-1382.

[7] Keck R.. A projection technique for incompressible flow in the meshless finite volume particle method[J]. Advances in Computational Mathematics, 2005, 23(1/2): 143-169.

[8] Teleaga D.. A Finite Volume Particle Method for Conservation Laws [D]. 2005, Kaiserslautern: Department of Mathematics, Universitat Kaiserslautern.

[9] Cockburn B., Shu C. W.. Rung-Kutta discontinuous Galerkin method for convection-dominated problems[J]. Journal of Scientific Computing, 2001, 16(3): 173-261.

[10] Cockburn B., Shu C. W.. The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws[J]. Mathematical Modelling and Numerical Analysis, 1991, 25 337-361.

[11] Cockburn B., Shu C. W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws(II): General framework[J]. Mathematics of Computation, 1989, 52 411-435.

[12] Cockburn B., Lin S. Y., Shu C. W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws(III): One-dimensional systems[J]. Journal of Computational Physics, 1989, 84(1): 90-113.

[13] Cockburn B., Hou S., Shu C. W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws(IV): The multidimensional case[J]. Mathematics of Computation, 1990, 54 545-581.

[14] Cockburn B., Shu C. W.. TVB Runge-Kutta discontinuous Galerkin method for conservation laws(V): Multidimensional systems[J]. Journal of Computational Physics, 1998, 141(2): 199-224.

[15] Atluri S. N., Zhu T.. A new meshless local Petrov-Galerkin(MLPG) approach in computational mechanics[J]. Computational Mechanics, 1998, 22(2): 117-127.

[16] Duarte C. A., Oden J. T.. An H-P adaptive method using clouds[J]. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1–4): 237-262.

[17] Jaffre J., Johnson C., Szepessy A.. Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws[J]. Mathematical Models and Methods in Applied Sciences, 1995, 5(3): 367-386.

[18] Steger J. L., Warming R. F.. Flux vector splitting of the inviscid geodynamic equations with application to finite-difference method[J]. SIAM Journal on Numerical Analysis, 1968, 5(2): 263 293

[19] Woodward P., Colella P.. The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks[J]. Journal of Computational Physics, 1984, 54(1): 115-173.