Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations.

*(English)*Zbl 1218.65085Summary: We propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge-Kutta (RK) method in time (WENO-RK) by a high order coupling. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy.

We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes.

In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes.

In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

##### Keywords:

adaptive mesh refinement; multiscale simulations; numerical examples; finite difference method; hyperbolic conservation laws; weighted essentially non-oscillatory method; total variation diminishing Runge-Kutta method
PDF
BibTeX
XML
Cite

\textit{C. Shen} et al., J. Comput. Phys. 230, No. 10, 3780--3802 (2011; Zbl 1218.65085)

Full Text:
DOI

##### References:

[1] | Baeza, A.; Mulet, P., Adaptive mesh refinement techniques for high-order shock capturing schemes for multi-dimensional hydrodynamic simulations, International journal for numerical methods in fluids, 52, 455-471, (2006) · Zbl 1370.76116 |

[2] | Barad, M.; Colella, P., A fourth-order accurate local refinement method for Poisson equation, Journal of computational physics, 209, 1-18, (2005) · Zbl 1073.65126 |

[3] | Bell, J.; Berger, M.; Saltzman, J.; Welcome, M., Three-dimensional adaptive mesh refinement for hyperbolic conservation laws, SIAM journal on scientific computing, 15, 127, (1994) · Zbl 0793.65072 |

[4] | M.J. Berger, Adaptive mesh refinement for hyperbolic partial differential equations, Phd. Thesis, Standford University, 1982. |

[5] | Berger, M.J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, Journal of computational physics, 82, 64-84, (1989) · Zbl 0665.76070 |

[6] | Berger, M.J.; Leveque, R.J., Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM journal on numerical analysis, 35, 2298-2316, (1998) · Zbl 0921.65070 |

[7] | Carpenter, M.; Gottlieb, D.; Abarbanel, S.; Don, W., The theoretical accuracy of runge – kutta time discretizations for the initial boundary value problem: a careful study of the boundary error, SIAM journal on scientific computing, 16, 1241-1252, (1995) · Zbl 0839.65098 |

[8] | Chang, S., The method of space-time conservation element and solution element – a new approach for solving the navier – stokes and Euler equations, Journal of computational physics, 119, 295-324, (1995) · Zbl 0847.76062 |

[9] | Chang, S.; Wang, X.; Chow, C., The space-time conservation element and solution element method: a new high-resolution and genuinely multidimensional paradigm for solving conservation laws, Journal of computational physics, 156, 89-136, (1999) · Zbl 0974.76060 |

[10] | Cockburn, B.; Johnson, C.; Shu, C.-W.; Tadmor, E., Advanced numerical approximation of nonlinear hyperbolic equations, (1998), Springer New York · Zbl 0904.00047 |

[11] | Cockburn, B.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of computation, 52, 411-435, (1989) · Zbl 0662.65083 |

[12] | Cockburn, B.; Shu, C.-W., The runge – kutta discontinuous Galerkin method for conservation laws V∗ 1:: multidimensional systems, Journal of computational physics, 141, 199-224, (1998) · Zbl 0920.65059 |

[13] | Gottlieb, S.; Ketcheson, D.; Shu, C.-W., High order strong stability preserving time discretizations, Journal of scientific computing, 38, 251-289, (2009) · Zbl 1203.65135 |

[14] | Hartmann, R.; Houston, P., Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations, Journal of computational physics, 183, 508-532, (2002) · Zbl 1057.76033 |

[15] | Hesthaven, J.; Gottlieb, S.; Gottlieb, D., Spectral methods for time-dependent problems, (2007), Cambridge University Press · Zbl 1111.65093 |

[16] | Houston, P.; Senior, B.; Suli, E., Hp-discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity, International journal for numerical methods in fluids, 40, 153-169, (2002) · Zbl 1021.76027 |

[17] | Jameson, L., AMR vs high order schemes, Journal of scientific computing, 18, 1-24, (2003) · Zbl 1112.76050 |

[18] | Jiang, G.S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 126, 202-228, (1996) · Zbl 0877.65065 |

[19] | Karni, S.; Kurganov, A.; Petrova, G., A smoothness indicator for adaptive algorithms for hyperbolic systems, Journal of computational physics, 178, 323-341, (2002) · Zbl 0998.65092 |

[20] | Leveque, R.J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press New York, NY · Zbl 1010.65040 |

[21] | R.J. LeVeque, e. M.J. Berger, Clawpack software 4.3. Available from: <www.clawpack.org>, 2009 (Accessed 28.04.2009). |

[22] | S.T. Li and J. Mac Hyman, Adaptive mesh refinement for finite difference WENO schemes, 2003 <http://math.lanl.gov/shenli/publications/amrweno.pdf>. |

[23] | D. Mavriplis, A. Jameson, Multigrid solution of the Euler equations on unstructured and adaptive meshes, in: Copper Mountain Conference on Multigrid Methods, 1987. · Zbl 0643.76079 |

[24] | T. Oliver, A high-order, adaptive, discontinuous Galerkin finite element method for the Reynolds-averaged Navier-Stokes equations, Phd Thesis, Citeseer, 2008. |

[25] | Pan, M.; Wood, E.F.; McLaughlin, D.B.; Entekhabi, D.; Luo, L., A multiscale ensemble filtering system for hydrologic data assimilation. part I: implementation and synthetic experiment, Journal of hydrometeorology, 10, 794-806, (2009) |

[26] | T. Plewa, T. Linde, e. Gregory Weirs, V., Adaptive mesh refinement – theory and applications, in: Proceedings of the Chicago workshop on adaptive mesh refinement methods, Springer, 2003. · Zbl 1053.65002 |

[27] | Ray, J.; Kennedy, C.A.; Lefantzi, S.; Najm, H.N., Using high-order methods on adaptively refined block-structured meshes: derivatives, interpolations, and filters, SIAM journal on scientific computing, 29, 139-181, (2007) · Zbl 1133.65068 |

[28] | Sebastian, K.; Shu, C.-W., Multidomain WENO finite difference method with interpolation at subdomain interfaces, Journal of scientific computing, 19, 405-438, (2003) · Zbl 1081.76577 |

[29] | Shu, C.-W., High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD, International journal of computational fluid dynamics, 17, 107-118, (2003) · Zbl 1034.76044 |

[30] | Shu, C.-W., High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM review, 51, 82-126, (2009) · Zbl 1160.65330 |

[31] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of computational physics, 77, 439-471, (1988) · Zbl 0653.65072 |

[32] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes.2, Journal of computational physics, 83, 32-78, (1989) · Zbl 0674.65061 |

[33] | Tam, A.; Ait-Ali-Yahia, D.; Robichaud, M.; Moore, M.; Kozel, V.; Habashi, W., Anisotropic mesh adaptation for 3D flows on structured and unstructured grids, Computer methods in applied mechanics and engineering, 189, 1205-1230, (2000) · Zbl 1005.76061 |

[34] | Wang, L.; Mavriplis, D., Adjoint-based hp adaptive discontinuous Galerkin methods for the 2D compressible Euler equations, Journal of computational physics, 228, 7643-7661, (2009) · Zbl 1391.76367 |

[35] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of computational physics, 54, 115-173, (1984) · Zbl 0573.76057 |

[36] | Xing, Y.-L.; Shu, C.-W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations, Journal of computational physics, 208, 206-227, (2005) · Zbl 1114.76340 |

[37] | Xu, Z.-F.; Shu, C.-W., Anti-diffusive flux corrections for high order finite difference WENO schemes, Journal of computational physics, 205, 458-485, (2005) · Zbl 1087.76080 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.