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High-Order Finite-Volume Method with Block-Based AMR for Magnetohydrodynamics Flows

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Abstract

A high-order central essentially non-oscillatory (CENO) finite volume scheme combined with a block-based adaptive mesh refinement (AMR) algorithm is proposed for the solution of the ideal magnetohydrodynamics equations. The high-order CENO finite-volume scheme is implemented with fourth-order spatial accuracy within a flexible multi-block, body-fitted, hexahedral grid framework. An important feature of the high-order adaptive approach is that it allows for anisotropic refinement, which can lead to large computational savings when anisotropic flow features such as isolated propagating fronts and/or waves, shocks, shear surfaces, and current sheets are present in the flow. This approach is designed to handle complex multi-block grid configurations, including cubed-sphere grids, where some grid blocks may have degenerate edges or corners characterized by missing neighboring blocks. A procedure for building valid high-order reconstruction stencils, even at these degenerate block edges and corners, is proposed, taking into account anisotropic resolution changes in a systematic and general way. Furthermore, a non-uniform or heterogeneous block structure is used where the ghost cells of a block containing the solution content of neighboring blocks are stored directly at the resolution of the neighbors. A generalized Lagrange multiplier divergence correction technique is applied to achieve numerically divergence-free magnetic fields while preserving high-order accuracy on the anisotropic AMR grids. Parallel implementation and local grid adaptivity are achieved by using a hierarchical block-based domain partitioning strategy in which the connectivity and refinement history of grid blocks are tracked using a flexible binary tree data structure. Physics-based refinement criteria as well as the CENO smoothness indicator are both used for directing the mesh refinement. Numerical results, including solution-driven anisotropic refinement of cubed-sphere grids, are presented to demonstrate the accuracy and efficiency of the approach.

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References

  1. Adams, M., Colella, P., Graves, D.T., Johnson, J.N., Keen, N.D., Ligocki, T.J., Martin, D.F., McCorquodale, P.W., Modiano, D., Schwartz, P.O., Sternberg T. D. Van Straalen, B.: Chombo: Software package for AMR applications—design document. Lawrence Berkeley National Technical Report LBNL-6616E

  2. Balsara, D.: Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics. J. Comput. Phys. 228(14), 5040–5056 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barth, T.: Recent developments in high order k-exact reconstruction on unstructured meshes. In: 31st Aerospace Sciences Meeting (1993)

  4. Bell, J., Berger, M., Saltzman, J., Welcome, M.: Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput. 15(1), 127–138 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berger, M.: On conservation at grid interfaces. SIAM J. Numer. Anal. 24, 967–984 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berger, M., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53(3), 484–512 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brackbill, J., Barnes, D.: The effect of nonzero \(\nabla \cdot \mathbf{B}\) on the numerical solution of the magnetohydrodynamics equations. J. Comput. Phys. 35(3), 426–430 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burstedde, C., Wilcox, L.C., Ghattas, O.: p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM J. Sci. Comput. 33(3), 1103–1133 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Charest, M.R.J.: Effect of variables choices on Godunov-type high-order finite-volume methods (to be submitted)

  10. Charest, M.R.J., Groth, C.P.T.: A high-order central ENO finite-volume scheme for three-dimensional low-speed viscous flows on unstructured mesh. Commun. Comput. Phys. 17, 615–656 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Charest, M.R.J., Groth, C.P.T.: A high-order central ENO finite-volume scheme for three-dimensional turbulent flows on unstructured mesh. AIAA Paper (June 2013)

  12. Charest, M.R.J., Groth, C.P.T., Gülder, Ö.L.: A computational framework for predicting laminar reactive flows with soot formation. Combust. Theory Model. 14(6), 793–825 (2010)

    Article  MATH  Google Scholar 

  13. Chen, Y., Toth, G., Gombosi, T.: A fifth-order finite difference scheme for hyperbolic equations on block-adaptive curvilinear grids. J. Comput. Phys. 305, 604–621 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Christlieb, A.J., Rossmanith, J.A., Tang, Q.: Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics. J. Comput. Phys. 268, 302–325 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Clauer, C.R., Gombosi, T.I., De Zeeuw, D.L., Ridley, A.J., Powell, K.G., Van Leer, B., Stout, Q.F., Groth, C.P.T.: High performance computer methods applied to predictive space weather simulations. IEEE Trans. Plasma Sci. 28, 1931–1937 (2000)

    Article  Google Scholar 

  16. Clawpack Development Team: Clawpack software (2017). https://doi.org/10.5281/zenodo.262111. http://www.clawpack.org. Version 5.4.0

  17. Colella, P., Dorr, M., Hittinger, J.A., Martin, D.: High-order finite-volume methods in mapped coordinates. J. Comput. Phys. 230, 2952–2976 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Davis, B.N., LeVeque, R.J.: Adjoint methods for guiding adaptive mesh refinement in tsunami modeling. Pure Appl. Geophys. 173, 4055–4074 (2016)

    Article  Google Scholar 

  19. De Sterck, H.: Multi-dimensional upwind constrained transport on unstructured grid for shallow water magnetohydrodynamics. AIAA (2001)

  20. De Sterck, H., Poedts, S.: Intermediate shocks in three-dimensional magnetohydrodynamic bow-shock flows with multiple interacting shock fronts. Phys. Rev. Lett. 84(24), 5524–5527 (2000)

    Article  Google Scholar 

  21. De Zeeuw, D., Gombosi, T., Groth, C.P.T., Powell, K., Stout, Q.: An adaptive MHD method for global space weather simulations. IEEE Trans. Plasma Sci. 105, 1956–1965 (2000)

    Article  Google Scholar 

  22. Dedner, A., Kemm, F., Kroner, D., Munz, C., Schnitzer, T., Wesenberg, M.: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175(2), 645–673 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Einfeldt, B.: On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25(2), 294–318 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Evans, C.R., Hawley, J.F.: Simulation of magnetohydrodynamics flows: a constrained transport method. Astrophys. J. 332, 659–677 (1988)

    Article  Google Scholar 

  25. Freret, L., Groth, C.P.T.: Anisotropic non-uniform block-based adaptive mesh refinement for three-dimensional inviscid and viscous flows. In: 22nd AIAA Computational Fluid Dynamics Conference (2015)

  26. Freret, L., Groth, C.P.T.: A parallel high-order CENO finite-volume scheme with AMR for three-dimensional ideal MHD flows. In: International Conference On Spectral and High-Order Methods (2016)

  27. Freret, L., Groth, C.P.T.: A high-order finite-volume method with anisotropic AMR for ideal MHD flows. In: 55th AIAA Aerospace Science Meeting (2017)

  28. Gao, X., Groth, C.P.T.: A parallel adaptive mesh refinement algorithm for predicting turbulent non-premixed combusting flows. Int. J. Comput. Fluid Dyn. 20(5), 349–357 (2006)

    Article  MATH  Google Scholar 

  29. Gao, X., Groth, C.P.T.: A parallel solution adaptive method for three-dimensional turbulent non-premixed combusting flows. J. Comput. Phys. 229(9), 3250–3275 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gao, X., Northrup, S.A., Groth, C.P.T.: Parallel solution-adaptive method for two-dimensional non-premixed combusting flows. Int. J. Prog. Comput. Fluid Dyn. 11(2), 76–95 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Groth, C.P.T., De Zeeuw, D., Powell, K., Gombosi, T., Stout, Q.: A parallel adaptive 3D MHD scheme for modeling coronal and solar wind plasma flows, pp. 193–198 (1999)

  32. Groth, C.P.T., De Zeeuw, D.L., Gombosi, T.I., Powell, K.G.: Global three-dimensional MHD simulation of a space weather event: CME formation, interplanetary propagation, and interaction with the magnetosphere. J. Geophys. Res. 105(A11), 25053–25078 (2000)

    Article  Google Scholar 

  33. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high-order accurate essentially non-oscillatory scheme III. J. Comput. Phys. 131(1), 3–47 (1997)

    Article  MATH  Google Scholar 

  34. Helzel, C., Rossmanith, J.A., Taetz, B.: A high-order unstaggered constrained transport method for the three-dimensional ideal magnetohydrodynamics equations based on the method of lines. J. Sci. Comput. 35(2), 623–651 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Helzel, C., Rossmanith, J.A., Taetz, B.: An unstaggered constrained transport method for the 3d ideal magnetohydrodynamic equations. J. Comput. Phys. 230, 3803–3829 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ivan, L., De Sterck, H., Northrup, S.A., Groth, C.P.T.: Multi-dimensional finite-volume scheme for hyperbolic conservation laws on three-dimensional solution-adaptive cubed-sphere grids. J. Comput. Phys. 255, 205–227 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ivan, L., De Sterck, H., Susanto, A., Groth, C.P.T.: High-order central ENO finite-volume scheme for hyperbolic conservation laws on three-dimensional cubed-sphere grids. J. Comput. Phys. 282, 157–182 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ivan, L., Groth, C.P.T.: High-order solution-adaptive central essentially non-oscillatory CENO method for viscous flows. J. Comput. Phys. 257, 830–862 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Jiang, B., Lin, T., Povinelli, L.: Large-scale computation of incompressible viscous flow by least-squares finite element method. Comput. Methods Appl. Mech. Eng. 144, 213–231 (1994)

    Article  MathSciNet  Google Scholar 

  40. Jiang, G., Shu, C.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  41. Keppens, R., Maliani, Z., Van Marle, A.J., Delmont, P., Vlasis, A., van der Holst, B.: Parallel, grid-adaptive approaches for relativistic hydro and magnetohydrodynamics. J. Comput. Phys. 231(1), 718–744 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1992)

    Book  MATH  Google Scholar 

  43. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  44. MacNeice, P., Olson, K., Mobarry, C., de Fainchtein, R., Packer, C.: Paramesh: a parallel adaptive mesh refinement community toolkit. Comput. Phys. Commun. 126, 330–354 (2000)

    Article  MATH  Google Scholar 

  45. McCorquodale, P., Colella, P.: A high-order finite volume method for conservation laws on locally refined grids. Commun. Appl. Math. Comput. Sci. 6(1), 1–25 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. McCorquodale, P., Dorr, M., Hittinger, J., Colella, P.: High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids. J. Comput. Phys. 288, 181–195 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. McDonald, J.G., Sachdev, J.S., Groth, C.P.T.: Application of gaussian moment closure to micron-scale flows with moving embedded boundaries. AIAA J. 52(9), 1839–1857 (2014)

    Article  Google Scholar 

  48. Mignone, A., Tzeferacos, P., Bodo, G.: High-order conservative finite difference GLM-MHD schemes for cell-centered MHD. J. Comput. Phys. 229, 5896–5920 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Mocz, P., Pakmor, R., Springel, V., Vogelsberger, M., Marinacci, F., Hernquist, L.: A moving mesh unstaggered constrained transport scheme for magnetohydrodynamics. Mon. Not. R. Astron. Soc. 463(1), 477–488 (2016)

    Article  Google Scholar 

  50. Mocz, P., Vogelsberger, M., Hernquist, L.: A constrained transport scheme for MHD on unstructured static and moving meshes. Mon. Not. R. Astron. Soc. 442(1), 43–55 (2014)

    Article  Google Scholar 

  51. Narechania, N., Freret, L., Groth, C.P.T.: Block-based anisotropic AMR with A Posteriori adjoint-based error estimation for three-dimensional inviscid and viscous flows. In: 23rd AIAA Computational Fluid Dynamics (2017)

  52. Olsson, F., Petersson, N.: Stability of interpolation on overlapping grids. Comput. Fluids 25, 583–605 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  53. Pärt-Enander, E., Sjörgreen, B.: Conservative and non-conservative interpolation between overlapping grids for finite volume solutions of hyperbolic problems. Comput. Fluids 23, 551–574 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  54. Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I., De Zeeuw, D.L.: A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys. 154, 284–309 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  55. Sachdev, J.S., Groth, C.P.T., Gottlieb, J.J.: A parallel solution-adaptive scheme for multi-phase core flows in solid propellant rocket motors. Int. J. Comput. Fluid Dyn. 19(2), 159–177 (2005)

    Article  MATH  Google Scholar 

  56. Shen, C., Qiu, J., Christlieb, A.: Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations. J. Comput. Phys. 230, 3780–3802 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  57. Shu, C.W.: High-order weighted non-oscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Susanto, A., Ivan, L., De Sterck, H., Groth, C.P.T.: High-order central ENO finite-volume scheme for ideal MHD. J. Comput. Phys. 250, 141–164 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  59. Tobaldini Neto, L., Groth, C.P.T.: A high-order finite-volume scheme for large-eddy simulation of turbulent premixed flames. AIAA Paper (January 2014)

  60. Toth, G., van der Holst, B., Sokolov, I., De Zeeuw, D., Gombosi, T., Fand, F., Manchester, W., Meng, X., Najib, D., Powell, K., Stout, Q., Glocer, A., Ma, Y., Opher, M.: Adaptive numerical algorithms in space weather modeling. J. Comput. Phys. 231(1), 870–903 (2012)

    Article  MathSciNet  Google Scholar 

  61. Van Leer, B., Tai, C.H., Powell, K.G.: Design of optimally-smoothing multi-stage schemes for the euler equations. Tech. Rep. 89-1933-CP, AIAA (1989)

  62. Venditti, D., Darmofal, D.: Anisotropic grid adaptation for functionnal outputs: application to two-dimensional viscous flows. J. Comput. Phys. 187, 22–46 (2003)

    Article  MATH  Google Scholar 

  63. Venditti, D., Darmofal, D.: Anisotropic adaptation for functionnal outputs of viscous flow simulations. AIAA Paper (June 2003)

  64. Venkatakrishnan, V.: On the accuracy of limiters and convergence to steady state solutions. In: 31st Aerospace Sciences (1993)

  65. Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, F., Hillewaert, K., Huynh, H.T., Kroll, N., May, G., Persson, P., Leer, B.V., Visbal, M.: High-order CFD methods: current status and perspective. Int. J. Numer. Methods Fluids 72, 1–42 (2012)

    MathSciNet  Google Scholar 

  66. Williamschen, M.J., Groth, C.P.T.: Parallel anisotropic block-based adaptive mesh refinement algorithm for three-dimensional flow. In: 21st AIAA Computational Fluid Dynamics Conference (2013)

  67. Zanotti, O., Dumbser, M.: Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables. Comput. Astrophys. Cosmol. 3, 1 (2016)

    Article  Google Scholar 

  68. Zhang, Z.J., Groth, C.P.T.: Parallel high-order anisotropic block-based adaptive mesh refinement finite-volume scheme. Paper 2011-3695, AIAA (2011)

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Acknowledgements

This work was supported by the Canadian Space Agency and by the Natural Sciences and Engineering Research Council (NSERC) of Canada. In particular, the authors would like to acknowledge the financial support received from the Canadian Space Agency through the Geospace Observatory Canada program. Computational resources for performing all of the calculations reported herein were provided by the SciNet High Performance Computing Consortium at the University of Toronto and Compute/Calcul Canada through funding from the Canada Foundation for Innovation (CFI) and the Province of Ontario, Canada.

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  1. L. Ivan

    Present address: Canadian Nuclear Laboratories, 286 Plant Road, Chalk River, ON, K0J 1J0, Canada

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  1. Institute for Aerospace Studies, University of Toronto, 4925 Dufferin Street, Toronto, ON, M3H5T6, Canada

    L. Freret, L. Ivan & C. P. T. Groth

  2. School of Mathematical Sciences, 9 Rainforest Walk, Monash University, Melbourne, VIC, 3800, Australia

    H. De Sterck

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  1. L. Freret
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Freret, L., Ivan, L., De Sterck, H. et al. High-Order Finite-Volume Method with Block-Based AMR for Magnetohydrodynamics Flows. J Sci Comput 79, 176–208 (2019). https://doi.org/10.1007/s10915-018-0844-1

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