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A robust hyperviscosity formulation for stable RBF-FD discretizations of advection-diffusion-reaction equations on manifolds. (English) Zbl 1478.65098

The present article is devoted to the discretization of advection-reaction-diffusion equations on manifolds using the higher-order finite difference (FD) methods based on the radial basis functions (RBF) technique. These methods have spectral-like accuracy but they may also suffer from instabilities. One of the main issues addressed in this study is the hyperviscosity formulation aiming to stabilize the resulting discretization. The main idea behind is to add artificial hyper-viscosity to damp out substantially the spurious modes arising in FD discretization matrices. This approach is a more or less straightforward generalization of the corresponding techniques available in the flat (i.e. Euclidean) setting. The convergence rates of the proposed method have been studied numerically. The manuscript is quite long but it contains a lot of technical details. It should become the standard text on hyperviscosity RBF-FD methods for advection-diffusion equations on manifolds.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] K. A. Aiton, A Radial Basis Function Partition of Unity Method for Transport on the Sphere, master’s thesis, Boise State University, Boise, ID, 2014.
[2] U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton, Implicit-explicit methods for time-dependent PDEs, SIAM J. Numer. Anal., 32 (1997), pp. 797-823. · Zbl 0841.65081
[3] V. Bayona, N. Flyer, and B. Fornberg, On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries, J. Comput. Phys., 380 (2019), pp. 378-399, https://doi.org/10.1016/j.jcp.2018.12.013. · Zbl 1451.65012
[4] V. Bayona, N. Flyer, B. Fornberg, and G. A. Barnett, On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs, J. Comput. Phys., 332 (2017), pp. 257-273. · Zbl 1380.65144
[5] V. Bayona, M. Moscoso, M. Carretero, and M. Kindelan, RBF-FD formulas and convergence properties, J. Comput. Phys., 229 (2010), pp. 8281-8295. · Zbl 1201.65038
[6] J. Behrens and A. Iske, Grid-free adaptive semi-Lagrangian advection using radial basis functions, Comput. Math. Appl., 43 (2002), pp. 319-327. · Zbl 0999.65104
[7] O. Davydov and D. T. Oanh, Adaptive meshless centres and RBF stencils for Poisson equation, J. Comput. Phys., 230 (2011), pp. 287-304. · Zbl 1207.65136
[8] G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences 6, World Scientific, Singapore, 2007. · Zbl 1123.65001
[9] G. E. Fasshauer and M. J. McCourt, Stable evaluation of Gaussian radial basis function interpolants, SIAM J. Sci. Comput., 34 (2012), pp. A737-A762. · Zbl 1252.65028
[10] N. Flyer, G. A. Barnett, and L. J. Wicker, Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations, J. Comput. Phys., 316 (2016), pp. 39-62. · Zbl 1349.76460
[11] N. Flyer, B. Fornberg, V. Bayona, and G. A. Barnett, On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy, J. Comput. Phys., 321 (2016), pp. 21-38. · Zbl 1349.65642
[12] N. Flyer, E. Lehto, S. Blaise, G. B. Wright, and A. St-Cyr, A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere, J. Comput. Phys., 231 (2012), pp. 4078-4095. · Zbl 1394.76078
[13] N. Flyer and G. B. Wright, Transport schemes on a sphere using radial basis functions, J. Comput. Phys., 226 (2007), pp. 1059-1084. · Zbl 1124.65097
[14] N. Flyer and G. B. Wright, A radial basis function method for the shallow water equations on a sphere, Proc. Roy. Soc. A, 465 (2009), pp. 1949-1976. · Zbl 1186.76664
[15] B. Fornberg, E. Larsson, and N. Flyer, Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33(2) (2011), pp. 869-892. · Zbl 1227.65018
[16] B. Fornberg and E. Lehto, Stabilization of RBF-generated finite difference methods for convective PDEs, J. Comput. Phys., 230 (2011), pp. 2270-2285. · Zbl 1210.65154
[17] B. Fornberg, E. Lehto, and C. Powell, Stable calculation of Gaussian-based RBF-FD stencils, Comput. Math. Appl., 65 (2013), pp. 627-637. · Zbl 1319.65011
[18] B. Fornberg and C. Piret, A stable algorithm for flat radial basis functions on a sphere, SIAM J. Sci. Comput., 30 (2007), pp. 60-80. · Zbl 1159.65307
[19] B. Fornberg and G. Wright, Stable computation of multiquadric interpolants for all values of the shape parameter, Comput. Math. Appl., 48 (2004), pp. 853-867. · Zbl 1072.41001
[20] E. J. Fuselier and G. B. Wright, A high-order kernel method for diffusion and reaction-diffusion equations on surfaces, J. Sci. Comput., 56 (2013), pp. 535-565. · Zbl 1275.65056
[21] S. D. Lawley and V. Shankar, Asymptotic and numerical analysis of a stochastic PDE model of volume transmission. submitted, 2018. · Zbl 1445.35346
[22] E. Lehto, V. Shankar, and G. B. Wright, A radial basis function (RBF) compact finite difference (FD) scheme for reaction-diffusion equations on surfaces, SIAM J. Sci. Comput., 39 (2017), pp. A2129-A2151. · Zbl 1371.41018
[23] H. Ma, Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 869-892. · Zbl 0912.35104
[24] H. Ma, Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 893-908. · Zbl 0912.35105
[25] R. D. Nair and P. H. Lauritzen, A class of deformational flow test cases for linear transport problems on the sphere, J. Comput. Phys., 229 (2010), pp. 8868-8887. · Zbl 1282.86012
[26] A. Narayan and D. Xiu, Stochastic collocation methods on unstructured grids in high dimensions via interpolation, SIAM J. Sci. Comput., 34 (2012), pp. A1729-A1752. · Zbl 1246.65029
[27] P.-O. Persson and G. Strang, A simple mesh generator in Matlab, SIAM Rev., 46 (2004), pp. 329-345, https://doi.org/10.1137/S0036144503429121. · Zbl 1061.65134
[28] A. Petras, L. Ling, C. Piret, and S. J. Ruuth, A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces, J. Comput. Phys., 381 (2019), pp. 146-161. · Zbl 1451.65110
[29] A. Petras, L. Ling, and S. J. Ruuth, An RBF-FD closest point method for solving PDEs on surfaces, J. Comput. Phys., 370 (2018), pp. 43-57. · Zbl 1395.65029
[30] C. Piret, The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces, J. Comput. Phys., 231 (2012), pp. 4662-4675. · Zbl 1248.35009
[31] C. Piret and J. Dunn, Fast RBF OGR for solving PDEs on arbitrary surfaces, AIP Conference Proceedings, 1776 (2016).
[32] J. A. Reeger and B. Fornberg, Numerical quadrature over the surface of a sphere, Stud. Appl. Math., 137 (2016), pp. 174-188. · Zbl 1350.65020
[33] J. A. Reeger and B. Fornberg, Numerical quadrature over smooth surfaces with boundaries, J. Comput. Phys., 355 (2018), pp. 176-190. · Zbl 1380.65053
[34] J. A. Reeger, B. Fornberg, and M. L. Watts, Numerical quadrature over smooth, closed surfaces, Proc. Roy. Soc. Lond. A, 472 (2016), https://doi.org/10.1098/rspa.2016.0401, 472 (2016t). · Zbl 1372.65084
[35] V. Shankar, The overlapped radial basis function-finite difference (RBF-FD) method: A generalization of RBF-FD, J. Comput. Phys., 342 (2017), pp. 211-228. · Zbl 1380.65180
[36] V. Shankar and A. L. Fogelson, Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion equations, J. Comput. Phys., 372 (2018), pp. 616-639. · Zbl 1415.65199
[37] V. Shankar, R. M. Kirby, and A. L. Fogelson, Robust node generation for mesh-free discretizations on irregular domains and surfaces, SIAM J. Sci. Comput., 40 (2018), pp. A2584-A2608. · Zbl 1393.68177
[38] V. Shankar, A. Narayan, and R. M. Kirby, RBF-LOI: Augmenting radial basis functions (RBFs) with least orthogonal interpolation (LOI) for solving PDEs on surfaces, J. Comput. Phys., 373 (2018), pp. 722-735. · Zbl 1416.65382
[39] V. Shankar and G. B. Wright, Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions, J. Comput. Phys., 366 (2018), pp. 170-190. · Zbl 1406.65099
[40] V. Shankar, G. B. Wright, R. M. Kirby, and A. L. Fogelson, A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction-diffusion equations on surfaces, J. Sci. Comput., 63 (2014), pp. 745-768. · Zbl 1319.65079
[41] E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal., 26 (1989), pp. 30-44. · Zbl 0667.65079
[42] H. Wendland, Scattered data approximation, Cambridge Monogr. Appl. Comput. Math. 17, Cambridge University Press, Cambridge, 2005. · Zbl 1075.65021
[43] D. L. Williamson, J. B. Drake, J. J. Hack, R. Jakob, and P. N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., 102 (1992), pp. 211-224. · Zbl 0756.76060
[44] G. B. Wright and B. Fornberg, Scattered node compact finite difference-type formulas generated from radial basis functions, J. Comput. Phys., 212 (2006), pp. 99-123. · Zbl 1089.65020
[45] G. B. Wright and B. Fornberg, Stable computations with flat radial basis functions using vector-valued rational approximations, J. Comput. Phys., 331 (2017), pp. 13756. · Zbl 1378.65045
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