References
- G. Avalos and I. Lasiecka, “Differential Riccati equation for the active control of a problem in structural acoustics,” J. Optim. Theory Appl., vol. 91, no. 3, pp. 695–728, 1996. DOI: https://doi.org/10.1007/BF02190128.
- A. Metaxas and R. J. Meredith, Industrial Microwave Heating. London, UK: IET, 1983.
- C. Bailly and D. Juve, “Numerical solution of acoustic propagation problems using linearized Euler equations,” AIAA J., vol. 38, no. 1, pp. 22–29, 2000. DOI: https://doi.org/10.2514/2.949.
- V. Gopal, R. Mohanty, and N. Jha, “New nonpolynomial spline in compression method of for the solution of 1D wave equation in polar coordinates,” Adv. Numer. Anal., vol. 2013, pp. 1–8, 2013. DOI: https://doi.org/10.1155/2013/470480.
- A. Brandt and I. Livshits, “Wave-ray multigrid method for standing wave equations,” Electron. Trans. Numer. Anal., vol. 6, pp. 162–181, 1997.
- V. Devi, R. K. Maurya, S. Singh, and V. K. Singh, “Lagrange’s operational approach for the approximate solution of two-dimensional hyperbolic telegraph equation subject to Dirichlet boundary conditions,” Appl. Math. Comput., vol. 367, pp. e124717, 2020. DOI: https://doi.org/10.1016/j.amc.2019.124717.
- M. Dehghan and A. Mohebbi, “The combination of collocation, finite difference, and multigrid methods for solution of the two-dimensional wave equation,” Numer. Methods Partial Differ. Equ., vol. 24, no. 3, pp. 897–910, 2008. DOI: https://doi.org/10.1002/num.20295.
- M. A. Rincon and N. Quintino, “Numerical analysis and simulation for a nonlinear wave equation,” J. Comput. Appl. Math., vol. 296, pp. 247–264, 2016. DOI: https://doi.org/10.1016/j.cam.2015.09.024.
- H. M. Baskonus, H. Bulut, and T. A. Sulaiman, “New complex hyperbolic structures to the lonngren-wave equation by using sine-gordon expansion method,” Appl. Math. Nonlinear Sci., vol. 4, no. 1, pp. 129–138, 2019. DOI: https://doi.org/10.2478/AMNS.2019.1.00013.
- H. De Sterck, S. Friedhoff, A. J. Howse, and S. P. MacLachlan, “Convergence analysis for parallel-in-time solution of hyperbolic systems,” Numer. Linear Algebra Appl., vol. 27, no. 1, pp. e2271, 2020. DOI: https://doi.org/10.1002/nla.2271.
- M. J. Gander, L. Halpern, J. Rannou, and J. Ryan, “A direct time parallel solver by diagonalization for the wave equation,” SIAM J. Sci. Comput., vol. 41, no. 1, pp. A220–A245, 2019. DOI: https://doi.org/10.1137/17M1148347.
- M. J. Gander, F. Kwok, and B. C. Mandal, “Dirichlet–Neumann waveform relaxation methods for parabolic and hyperbolic problems in multiple subdomains,” BIT Numer. Math., vol. 61, no. 1, pp. 173–135, 2021. DOI: https://doi.org/10.1007/s10543-020-00823-2.
- J. A. Cuminato and M. Meneguette, Discretization of Partial Differential Equations: Finite Difference Techniques (in Portuguese). Rio de Janeiro, Brazil: Brazilian Mathematical Society, 2013.
- U. Trottenberg, C. W. Oosterlee, and A. Schuller, Multigrid. London, UK: Elsevier, 2000.
- R. Wienands and C. W. Oosterlee, “On three-grid fourier analysis for multigrid,” SIAM J. Sci. Comput., vol. 23, no. 2, pp. 651–671, 2001. DOI: https://doi.org/10.1137/S106482750037367X.
- W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial. Philadelphia, PA: SIAM, 2000.
- N. Umetani, S. P. MacLachlan, and C. W. Oosterlee, “A multigrid-based shifted Laplacian preconditioner for a fourth-order Helmholtz discretization,” Numer. Linear Algebra Appl., vol. 16, no. 8, pp. 603–626, 2009. DOI: https://doi.org/10.1002/nla.634.
- J. Van Lent , Multigrid Methods for Time-Dependent Partial Differential Equations. Leuven: Katholieke Universiteit Leuven, 2006.
- B. Ji, H.-L. Liao, Y. Gong, and L. Zhang, “Adaptive second-order Crank–Nicolson time-stepping schemes for time-fractional molecular beam epitaxial growth models,” SIAM J. Sci. Comput., vol. 42, no. 3, pp. B738–B760, 2020. DOI: https://doi.org/10.1137/19M1259675.
- H.-L. Liao, T. Tang, and T. Zhou, “A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations,” J. Comput. Phys., vol. 414, pp. e109473, 2020. DOI: https://doi.org/10.1016/j.jcp.2020.109473.
- M. Chen, S. Zhang, S. Li, X. Ma, X. Zhang, and Y. Zou, “An explicit algorithm for modeling planar 3D hydraulic fracture growth based on a super-time-stepping method,” Int. J. Solids Struct., vol. 191–192, pp. 370–389, 2020. DOI: https://doi.org/10.1016/j.ijsolstr.2020.01.011.
- J. C. Tannehill , D. A. Anderson and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer. Philadelphia, PA: Taylor & Francis,1997.
- P. J. Olver, Introduction to Partial Differential Equations. New York, USA: Springer, 2014.
- R. Burden and J. Faires, Numerical Analysis. Boston: Brooks/Cole Cengage Learning, 2016.
- A. Brandt and O. E. Livne, Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, revised ed. SIAM, 2011. DOI: https://doi.org/10.1137/1.9781611970753.
- P. Wesseling, “Introduction to Multigrid Methods,” Technical Report. Institute for Applications in Science and Engineering, Hampton, VA, 1995.
- S. R. Franco, F. J. Gaspar, M. A. V. Pinto, and C. Rodrigo, “Multigrid method based on a space-time approach with standard coarsening for parabolic problems,” Appl. Math. Comput., vol. 317, pp. 25–34, 2018. DOI: https://doi.org/10.1016/j.amc.2017.08.043.
- M. A. V. Pinto, C. Rodrigo, F. J. Gaspar, and C. Oosterlee, “On the robustness of ilu smoothers on triangular grids,” Appl. Numer. Math., vol. 106, pp. 37–52, 2016. DOI: https://doi.org/10.1016/j.apnum.2016.02.007.
- H. C. Elman, O. G. Ernst, and D. P. O’leary, “A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations,” SIAM J. Sci. Comput., vol. 23, no. 4, pp. 1291–1315, 2001. DOI: https://doi.org/10.1137/S1064827501357190.
- T. C. Clevenger, T. Heister, G. Kanschat, and M. Kronbichler, “A flexible, parallel, adaptive geometric multigrid method for fem,” arXiv preprint arXiv:1904.03317, 2019.
- M. L. Oliveira, M. A. V. Pinto, S. F. T. Gonçalves, and G. V. Rutz, “On the robustness of the xy-zebra-gauss-seidel smoother on an anisotropic diffusion problem,” Comput. Model. Eng. Sci., vol. 117, no. 2, pp. 251–270, 2018.
- S. R. Franco, C. Rodrigo, F. J. Gaspar, and M. A. V. Pinto, “A multigrid waveform relaxation method for solving the poroelasticity equations,” Comput. Appl. Math., vol. 37, no. 4, pp. 4805–4820, 2018. DOI: https://doi.org/10.1007/s40314-018-0603-9.
- M. Adams, M. Brezina, J. Hu, and R. Tuminaro, “Parallel multigrid smoothing: polynomial versus Gauss–Seidel,” J. Comput. Phys., vol. 188, no. 2, pp. 593–610, 2003. DOI: https://doi.org/10.1016/S0021-9991(03)00194-3.
- C. H. Marchi and A. F. C. Silva, “Multi-dimensional discretization error estimation for convergent apparent order,” J. Braz. Soc. Mech. Sci. Eng., vol. 27, no. 4, pp. 432–439, 2005. DOI: https://doi.org/10.1590/S1678-58782005000400012.
- G. Horton and S. Vandewalle, “A space-time multigrid method for parabolic partial differential equations,” SIAM J. Sci. Comput., vol. 16, no. 4, pp. 848–864, 1995. DOI: https://doi.org/10.1137/0916050.
- C.-A. Thole and U. Trottenberg , “Basic smoothing procedures for the multigrid treatment of elliptic 3D-operators,” in Advances in multi-grelha Methods ,Springer, Wiesbaden, 1985, pp.102–111.
- G. R. Stroher and C. D. Santiago, “Numerical two-dimensional thermal analysis of the human skin using the multigrid method,” Acta Sci. Technol., vol. 42, pp. e40992, 2019. DOI: https://doi.org/10.4025/actascitechnol.v42i1.40992.
- P. Kumar, C. Rodrigo, F. J. Gaspar, and C. W. Oosterlee, “A parametric acceleration of multilevel monte carlo convergence for nonlinear variably saturated flow,” Comput. Geosci., vol. 24, no. 1, pp. 311–331, 2020. DOI: https://doi.org/10.1007/s10596-019-09922-8.
- F. J. Gaspar, F. J. Lisbona, C. W. Oosterlee, and R. Wienands, “A systematic comparison of coupled and distributive smoothing in multigrid for the poroelasticity system,” Numer. Linear Algebra Appl., vol. 11, no. 2–3, pp. 93–113, 2004. DOI: https://doi.org/10.1002/nla.372.
- P. Luo, C. Rodrigo, F. J. Gaspar, and C. Oosterlee, “On an uzawa smoother in multigrid for poroelasticity equations,” Numer. Linear Algebra Appl., vol. 24, no. 1, pp. e2074, 2017. DOI: https://doi.org/10.1002/nla.2074.