×

zbMATH — the first resource for mathematics

A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels. (English) Zbl 1442.65324
Summary: Recent developments have made it possible to overcome grid-based limitations of finite difference (FD) methods by adopting the kernel-based meshless framework using radial basis functions (RBFs). Such an approach provides a meshless implementation and is referred to as the radial basis-generated finite difference (RBF-FD) method. In this paper, we propose a stabilized RBF-FD approach with a hybrid kernel, generated through a hybridization of the Gaussian and cubic RBF. This hybrid kernel was found to improve the condition of the system matrix, consequently, the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard RBF-FD methods coupled with present stable algorithms. Unlike other RBF-FD approaches, the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity, and the size of the stencils. As an application, we solve the frequency-domain acoustic wave equation in a 2D half-space. In order to suppress spurious reflections from truncated computational boundaries, absorbing boundary conditions have been effectively implemented.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Software:
Matlab; rbf_qr; GaussQR
PDF BibTeX XML Cite
Full Text: DOI
References:
[2] Tolstykh, I. A.; Shirobokov, A. D., On using radial basis functions in a “finite difference mode” with applications to elasticity problems, Comput. Mech., 33, 1, 68-79, (2003) · Zbl 1063.74104
[3] Wright, G. B., Radial basis function interpolation: numerical and analytical developments, (2003), University of Colorado Boulder, (Ph.D. thesis)
[4] Wright, G. B.; Fornberg, B., Scattered node compact finite difference-type formulas generated from radial basis functions, J. Comput. Phys., 212, 1, 99-123, (2006) · Zbl 1089.65020
[5] Bayona, V.; Moscoso, M.; Carretero, M.; Kindelan, M., RBF-FD formulas and convergence properties, J. Comput. Phys., 229, 22, 8281-8295, (2010) · Zbl 1201.65038
[6] Fornberg, B.; Lehto, E., Stabilization of rbf-generated finite difference methods for convective PDEs, J. Comput. Phys., 230, 6, 2270-2285, (2011) · Zbl 1210.65154
[7] Fornberg, B.; Flyer, N., Solving pdes with radial basis functions, Acta Numer., 24, 215-258, (2015) · Zbl 1316.65073
[8] Chandhini, G.; Sanyasiraju, Y. V.S. S., Local rbf-fd solutions for steady convection-diffusion problems, Internat. J. Numer. Methods Engrg., 72, 3, 352-378, (2007) · Zbl 1194.76174
[9] Stevens, D.; Power, H.; Lees, M.; Morvan, H., The use of pde centres in the local RBF Hermitian method for 3D convective-diffusion problems, J. Comput. Phys., 228, 12, 4606-4624, (2009) · Zbl 1167.65447
[10] Chinchapatnam, P.; Djidjeli, K.; Nair, P.; Tan, M., A compact rbf-fd based meshless method for the incompressible Navier-Stokes equations, Proc. Inst. Mech. Eng. M, 223, 3, 275-290, (2009)
[11] Flyer, N.; Lehto, E.; bastien Blaise, S.; Wright, G. B.; St-Cyr, A., A guide to rbf-generated finite differences for nonlinear transport: Shallow water simulations on a sphere, J. Comput. Phys., 231, 11, 4078-4095, (2012) · Zbl 1394.76078
[12] Shankar, V.; Wright, G. B.; Kirby, R. M.; Fogelson, A. L., A Radial basis function (RBF)-finite difference (FD) method for diffusion and reaction-diffusion equations on surfaces, J. Sci. Comput., 63, 3, 745-768, (2015) · Zbl 1319.65079
[13] Shankar, V., The overlapped radial basis function-finite difference (rbf-fd) method: A generalization of RBF-FD, J. Comput. Phys., 342, 211-228, (2017) · Zbl 1380.65180
[14] Martin, B.; Fornberg, B.; St-Cyr, A., Seismic modeling with radial-basis-function-generated finite differences, Geophysics, 80, 4, T137-T146, (2015)
[15] Martin, B.; Fornberg, B., Using radial basis function-generated finite differences (rbf-fd) to solve heat transfer equilibrium problems in domains with interfaces, Eng. Anal. Bound. Elem., 79, 38-48, (2017) · Zbl 1403.80035
[16] Driscoll, T.; Fornberg, B., Interpolation in the limit of increasingly flat radial basis functions, Comput. Math. Appl., 43, 35, 413-422, (2002) · Zbl 1006.65013
[17] Fornberg, B.; Wright, G.; Larsson, E., Some observations regarding interpolants in the limit of flat radial basis functions, Comput. Math. Appl., 47, 1, 37-55, (2004) · Zbl 1048.41017
[18] Fasshauer, G. F., Meshfree approximation methods with matlab, (2007), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ, USA · Zbl 1123.65001
[19] Larsson, E.; Fornberg, B., Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions, Comput. Math. Appl., 49, 1, 103-130, (2005) · Zbl 1074.41012
[20] Kansa, E.; Hon, Y., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Comput. Math. Appl., 39, 78, 123-137, (2000) · Zbl 0955.65086
[21] Fornberg, B.; Wright, G., Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl., 48, 5, 853-867, (2004) · Zbl 1072.41001
[22] Fornberg, B.; cile Piret, C., A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30, 1, 60-80, (2007) · Zbl 1159.65307
[23] Fornberg, B.; Larsson, E.; Flyer, N., Stable computations with gaussian Radial Basis Functions, SIAM J. Sci. Comput., 33, 2, 869-892, (2011) · Zbl 1227.65018
[24] Fasshauer, G. E.; McCourt, M. J., Stable evaluation of gaussian Radial Basis Function Interpolants, SIAM J. Sci. Comput., 34, 2, A737-A762, (2012) · Zbl 1252.65028
[25] Fornberg, B.; Lehto, E.; Powell, C., Stable calculation of gaussian-based RBF-FD stencils, Comput. Math. Appl., 65, 4, 627-637, (2013) · Zbl 1319.65011
[26] De Marchi, S.; Santin, G., A new stable basis for radial basis function interpolation, J. Comput. Appl. Math., 253, 1-13, (2013) · Zbl 1288.65013
[27] Kindelan, M.; Moscoso, M.; González-Rodríguez, P., Radial basis function interpolation in the limit of increasingly flat basis functions, J. Comput. Phys., 307, 225-242, (2016) · Zbl 1351.41013
[28] Wright, G. B.; Fornberg, B., Stable computations with flat radial basis functions using vector-valued rational approximations, J. Comput. Phys., 331, 137-156, (2017) · Zbl 1378.65045
[29] Flyer, N.; Barnett, G. A.; Wicker, L. J., Enhancing finite differences with radial basis functions: experiments on the navier-stokes equations, J. Comput. Phys., 316, 39-62, (2016) · Zbl 1349.76460
[30] Flyer, N.; Fornberg, B.; Bayona, V.; Barnett, G. A., On the role of polynomials in rbf-fd approximations: I. Interpolation and accuracy, J. Comput. Phys., 321, 21-38, (2016) · Zbl 1349.65642
[31] Bayona, V.; Flyer, N.; Fornberg, B.; Barnett, G. A., On the role of polynomials in rbf-fd approximations: II. Numerical solution of elliptic PDEs, J. Comput. Phys., 332, 257-273, (2017) · Zbl 1380.65144
[32] Fornberg, B.; Flyer, N., A Primer on Radial Basis Functions with Applications to the Geosciences, (2015), SIAM: SIAM Philadelphia · Zbl 1358.86001
[33] Mishra, P. K.; Nath, S. K.; Sen, M. K.; Fasshauer, G. E., Hybrid gaussian-cubic radial basis functions for scattered data interpolation, Computat. Geosci., 22, 5, 1203-1218, (2018) · Zbl 1406.65010
[34] Mishra, P. K.; Nath, S. K.; Kosec, G.; Sen, M. K., An improved radial basis-pseudospectral scheme with hybrid gaussian-cubic kernels, Eng. Anal. Bound. Elem., 80, 162-171, (2017) · Zbl 1403.65171
[35] Mishra, P.; Nath, S.; Fasshauer, G.; Sen, M., Frequency-domain meshless solver for acoustic wave equation using a stable radial basis-finite difference (rbf-fd) algorithm with hybrid kernels, (SEG Technical Program Expanded Abstracts 2017, (2017)), 4022-4027
[36] Fasshauer, G.; McCourt, M., Kernel-based approximation methods using MATLAB, (2015), World Scientific: World Scientific Interdisciplinary Mathematical Sciences
[37] Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76, 8, 1905-1915, (1971)
[38] Franke, R., A critical comparison of some methods for interpolation of scattered data, (1979), Defense Technical Information Center
[39] Flyer, N.; Wright, G. B.; Fornberg, B., Radial basis function-generated finite differences: a mesh-free method for computational geosciences, (Handbook of Geomathematics, (2014)), 1-30
[40] Barnett, G. A., A robust rbf-fd formulation based on polyharmonic splines and polynomials, (2015), University of Colorado Boulder (2015), (Ph.D. thesis)
[41] Schaback, R., A practical guide to radial basis functions, Electron. Resour., 11, (2007)
[42] Beatson, R. K.; Light, W.; Billings, S., Fast solution of the radial basis function interpolation equations: domain decomposition methods, SIAM J. Sci. Comput., 22, 5, 1717-1740, (2001) · Zbl 0982.65015
[43] Dablain, M. A., The application of high-order differencing to the scalar wave equation, Geophysics, 51, 1, 54-66, (1986)
[44] Charl-Hyun, J.; Shin, C.; Suh, J. H., An optimal 9-point, finite-difference, frequency-space, 2-d scalar wave extrapolator, Geophysics, 61, 2, 529-537, (1996)
[45] Shin, C.; Sohn, H., A frequency-space 2-d scalar wave extrapolator using extended 25-point finite-difference operator, Geophysics, 63, 1, 289-296, (1998)
[46] Hustedt, B.; Operto, S.p.; Virieux, J., Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling, Geophys. J. Int., 157, 3, 1269-1296, (2004)
[47] Etgen, J. T.; O’Brien, M. J., Computational methods for large-scale 3d acoustic finite-difference modeling: A tutorial, Geophysics, 72, 5, SM223-SM230, (2007)
[48] Amini, N.; Javaherian, A., A matlab-based frequency-domain finite-difference package for solving 2D visco-acoustic wave equation, Waves Random Complex Media, 21, 1, 161-183, (2011) · Zbl 1274.74138
[49] Tao, Y.; Sen, M. K., Frequency-domain full waveform inversion with plane-wave data, Geophysics, 78, 1, R13-R23, (2013)
[50] Moreira, R. M.; Cetale Santos, M. A.; Martins, J. L.; Silva, D. L.F.; Pessolani, R. B.V.; Filho, D. M.S.; Bulcão, A., Frequency-domain acoustic-wave modeling with hybrid absorbing boundary conditions, Geophysics, 79, 5, A39-A44, (2014)
[51] Liu, Y., An optimal 5-point scheme for frequency-domain scalar wave equation, J. Appl. Geophys., 108, 19-24, (2014)
[52] Takekawa, J.; Mikada, H., An absorbing boundary condition for acoustic-wave propagation using a mesh-free method, Geophysics, 81, 4, T145-T154, (2016)
[53] Clayton, R.; Engquist, B., Absorbing boundary conditions for acoustic and elastic wave equations, Bull. Seismol. Soc. Am., 67, 6, 1529-1540, (1977)
[54] Cerjan, C.; Kosloff, D.; Kosloff, R.; Reshef, M., A nonreflecting boundary condition for discrete acoustic and elastic wave equations, Geophysics, 50, 4, 705-708, (1985)
[55] Berenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 2, 185-200, (1994) · Zbl 0814.65129
[56] Liu, Y.; Sen, M. K., A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation, Geophysics, 75, 2, A1-A6, (2010)
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.