Barycentric rational interpolation method of the Helmholtz equation with irregular domain
Abstract
In the work, a numerical method of the 2D Helmholtz equation with meshless interpolation collocation method is developed, which is defined in arbitrary domain with irregular shape. In our numerical method, based on the Chebyshev points, the partial derivatives and the spatial variables are discretized by the barycentric rational form basis function. After that the differential equations are simplified by employing differential matrix. To verify the the accuracy, effectiveness and stability in our method, some numerical tests based on the three types of different test points are adopted. Moreover, we can also verify that present method can be applied to both variable wave number problems and high wave number problems.
Keyword : barycentric rational interpolation, meshless method, irregular domain, Helmholtz equation, variable wave number
How to Cite
Yang, M., Ma, W., & Ge, Y. (2023). Barycentric rational interpolation method of the Helmholtz equation with irregular domain. Mathematical Modelling and Analysis, 28(2), 330–351. https://doi.org/10.3846/mma.2023.16408
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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T.T. Wu and R.M. Xu. An optimal compact sixth-order finite difference scheme for the Helmholtz equation. Computers & Mathematics with Applications. An International Journal, 75(7):2520–2537, 2018. https://doi.org/10.1016/j.camwa.2017.12.023
M. Yang, X. Du and Y. Ge. Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method. Engineering Computations, 38(10):3964–3994, 2021. https://doi.org/10.1108/EC-09-2020-0516
M. Yang, W. Ma and Y. Ge. A meshless collocation method with barycentric Lagrange interpolation for solving the Helmholtz equation. Computer Modeling in Engineering and Sciences, 126(1):25–54, 2021. https://doi.org/10.32604/cmes.2021.012575
X. You, Y. Chai and W. Li. A coupled FE-meshfree method for Helmholtz problems using point interpolation shape functions and edge-based gradient smoothing technique. Computers & Structures, 213:1–22, 2019. https://doi.org/10.1016/j.compstruc.2018.07.011
Y. Zhuang and X.H. Sun. A high order ADI method for separable generalized Helmholtz equations. Advances in Engineering Software, 31:585–591, 2020. https://doi.org/10.1016/S0965-9978(00)00026-0
R. Baltensperger. Barycentric rational interpolation with asymptotically monitored poles. Numerical Algorithms, 57(1):67–81, 2011. https://doi.org/10.1007/s11075-010-9415-8
J.P. Berrut, A. Hosseini and G. Klein. The linear barycentric rational quadrature method for Volterra integral equations. SIAM Journal on Scientific Computing, 36(1):A105–A123, 2014. https://doi.org/10.1137/120904020
J.P. Berrut and G. Klein. Recent advances in linear barycenteic rational interpolation. Journal of Computational and Applied Mathematics, 259:95–107, 2014. https://doi.org/10.1016/j.cam.2013.03.044
J.P. Berrut and L.N. Trefethen. Barycentric Lagrange interpolation. SIAM Review, 46:501–517, 2004. https://doi.org/10.1137/S0036144502417715
B. Bialecki and A. Karageorghis. Legendre Gauss spectral collocation for the Helmholtz equation on a rectangle. Numerical Algorithms, 36(3):203–227, 2004. https://doi.org/10.1023/B:NUMA.0000040056.52424.49
S. Britt, S. Petropavlovsky, S. Tsynkov and E. Turkel. Computation of singular solutions to the Helmholtz equation with high order accuracy. Applied Numerical Mathematics. An IMACS Journal, 93:215–241, 2015. https://doi.org/10.1016/j.apnum.2014.10.006
H.X. Chen and W.F. Qiu. A first order system least squares method for the Helmholtz equation. Journal of Computational and Applied Mathematics, 309:145–162, 2017. https://doi.org/10.1016/j.cam.2016.06.019
L.C. Chen and X.L. Li. A complex variable boundary element-free method for the Helmholtz equation using regularized combined field integral equations. Applied Mathematics Letters. An International Journal of Rapid Publication, 101:Paper No. 106067, 7, 2020. https://doi.org/10.1016/j.aml.2019.106067
L.C. Chen and X.L. Li. An efficient meshless boundary point interpolation method for acoustic radiation and scattering. Computers & Structures, 229:Paper No. 106182, 2020.
M.S. Floater and K. Hormann. Barycentric rational interpolation with no poles and high rates of approximation. Numerische Mathematik, 107(2):315–331, 2007. https://doi.org/10.1007/s00211-007-0093-y
S. Güttel and G. Klein. Convergence of linear barycentric rational interpolation for analytic functions. SIAM Journal on Numerical Analysis, 50(5):2560–2580, 2012. https://doi.org/10.1137/120864787
E. Haber and S. Maclachlan. A fast method for the solution of the Helmholtz equation. Journal of Computational Physics, 230(12):4403–4418, 2011. https://doi.org/10.1016/j.jcp.2011.01.015
R. Jiwari. Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional sine-Gordon equation. Numerical Methods for Partial Differential Equations, 37(3):1965–1992, 2021. https://doi.org/10.1002/num.22636
G. Klein and J.P. Berrut. Linear rational finite differences from derivatives of barycentric rational interpolants. SIAM Journal on Numerical Analysis, 50(2):643–656, 2012. https://doi.org/10.1137/110827156
K.H. Kumar and R. Jiwari. A note on numerical solution of classical Darboux problem. Mathematical Methods in the Applied Sciences, 44:12998–13007, 2021. https://doi.org//10.1002/mma.7602
K.H. Kumar and R. Jiwari. A hybrid approach based on Legendre wavelet for numerical simulation of Helmholtz equation with complex solution. International Journal of Computer Mathematics, pp. 1–16, 2022.
N. Kumar and R.K. Dubey. A new development of sixth order accurate compact method for the Helmholtz equation. Journal of Applied Mathematics and Computing, 62:637–662, 2020. https://doi.org/10.1007/s12190-019-01301-x
S. Kumar, R. Jiwari and R.C. Mittal. Radial basis functions based meshfree schemes for the simulation of non-linear extended Fisher-Kolmogorov model. Wave Motion, 109:Paper No. 102863, 19, 2022. https://doi.org/10.1016/j.wavemoti.2021.102863
C.S. Liu and C.L. Kuo. A multiple-scale Pascal polynomial triangle solving elliptic equations and inverse Cauchy problems. Engineering Analysis with Boundary Elements, 62:35–43, 2016. https://doi.org/10.1016/j.enganabound.2015.09.003
W.T. Ma, B.W. Zhang and H.L. Ma. A meshless collocation approach with barycentric rational interpolation for two-dimensional hyperbolic telegraph equation. Applied Mathematics and Computation, 279:236–248, 2016. https://doi.org/10.1016/j.amc.2016.01.022
Ö. Oruҫ. Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically. Engineering Computations, 38(5):2394–2414, 2020. https://doi.org/10.1108/EC-06-2020-0312
Ö. Oruҫ. A meshless multiple-scale polynomial method for numerical solution of 3D convection-diffusion problems with variable coefficients. Engineering with Computers, 36:1215–1228, 2020. https://doi.org/10.1007/s00366-019-00758-5
Ö. Oruҫ. Two meshless methods based on local radial basis function and barycentric rational interpolation for solving 2D viscoelastic wave equation. Computers & Mathematics with Applications, 79(12):3272–3288, 2020. https://doi.org/10.1016/j.camwa.2020.01.025
Ö. Oruҫ. A local radial basis function-finite difference (RBF-FD) method for solving 1D and 2D coupled Schr¨odinger-Boussinesq (SBq) equations. Engineering Analysis with Boundary Elements, 129:55–66, 2021. https://doi.org/10.1016/j.enganabound.2021.04.019
Ö. Oruҫ. Two meshless methods based on pseudo spectral delta-shaped basis functions and barycentric rational interpolation for numerical solution of modified Burgers equation. International Journal of Computer Mathematics, 98(3):461–479, 2021. https://doi.org/10.1080/00207160.2020.1755432
Ö. Oruҫ. Numerical simulation of two-dimensional and three-dimensional generalized Klein-Gordon-Zakharov equations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation. Numerical Methods for Partial Differential Equations, 38(4):1068–1089, 2022. https://doi.org/10.1002/num.22806
Ö. Oruҫ. A strong-form local meshless approach based on radial basis function-finite difference (RBF-FD) method for solving multi-dimensional coupled damped Schrödinger system appearing in Bose-Einstein condensates. Communications in Nonlinear Science and Numerical Simulation, 104:Paper No. 106042, 18, 2022. https://doi.org/10.1016/j.cnsns.2021.106042
I. Singer and E. Turkel. Sixth-order accurate finite difference schemes for the Helmholtz equation. Journal of Computational Acoustics, 14(3):339–351, 2006. https://doi.org/10.1142/S0218396X06003050
K. Wang, Y.S. Wong and J.Z. Huang. Solving Helmholtz equation at high wave numbers in exterior domains. Applied Mathematics and Computation, 298:221–235, 2017. https://doi.org/10.1016/j.amc.2016.11.015
T.T. Wu and R.M. Xu. An optimal compact sixth-order finite difference scheme for the Helmholtz equation. Computers & Mathematics with Applications. An International Journal, 75(7):2520–2537, 2018. https://doi.org/10.1016/j.camwa.2017.12.023
M. Yang, X. Du and Y. Ge. Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method. Engineering Computations, 38(10):3964–3994, 2021. https://doi.org/10.1108/EC-09-2020-0516
M. Yang, W. Ma and Y. Ge. A meshless collocation method with barycentric Lagrange interpolation for solving the Helmholtz equation. Computer Modeling in Engineering and Sciences, 126(1):25–54, 2021. https://doi.org/10.32604/cmes.2021.012575
X. You, Y. Chai and W. Li. A coupled FE-meshfree method for Helmholtz problems using point interpolation shape functions and edge-based gradient smoothing technique. Computers & Structures, 213:1–22, 2019. https://doi.org/10.1016/j.compstruc.2018.07.011
Y. Zhuang and X.H. Sun. A high order ADI method for separable generalized Helmholtz equations. Advances in Engineering Software, 31:585–591, 2020. https://doi.org/10.1016/S0965-9978(00)00026-0