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Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions

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Abstract

It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results.

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Notes

  1. This is unfortunately generally not true in the nonlinear case, as the energy might increase after one iteration of the explicit Euler method if no dissipation is added to the numerical solution.

  2. A more rigorous study is clearly needed and will be included in future investigations.

  3. The conference presentation [99] by Tolstykh in 2000 seems to be the earliest reference to RBF-FD methods.

  4. For a discussion on the accuracy of infinitely smooth kernels as well as the role of the shape parameter \(\varepsilon \) see for instance [5, 28, 75, 92] and references therein.

  5. In our implementation we are using the MATLAB function integral for their computation so that strictly speaking, none of our integration is exact. This MATLAB function uses global adaptive quadrature and certain (default) error tolerances.

References

  1. Abgrall, R., Bacigaluppi, P., Tokareva, S.: High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics. Comput. Math. Appl. 78(2), 274–297 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Abgrall, R., Nordström, J., Öffner, P., Tokareva, S.: Analysis of the SBP-SAT stabilization for finite element methods part i: Linear problems. J. Sci. Comput. 85(2), 1–29 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Abgrall, R., Nordström, J., Öffner, P., Tokareva, S.: Analysis of the SBP-SAT stabilization for finite element methods part ii: Entropy stability. arXiv:1912.08390 (2020). Accepted in Communications on Applied Mathematics and Computation

  4. Buhmann, M.D.: Radial basis functions. Acta Numer. 9, 1–38 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations, vol. 12. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  6. Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 1998, 1–49 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Springer, Berlin (2006)

    MATH  Google Scholar 

  8. Canuto, C., Quarteroni, A.: Error estimates for spectral and pseudospectral approximations of hyperbolic equations. SIAM J. Numer. Anal. 19(3), 629–642 (1982)

    MathSciNet  MATH  Google Scholar 

  9. Chen, T., Shu, C.W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Cheng, J., Shu, C.W.: Positivity-preserving Lagrangian scheme for multi-material compressible flow. J. Comput. Phys. 257, 143–168 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Cockburn, B., Hou, S., Shu, C.W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Cockburn, B., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  14. Cockburn, B., Shu, C.W.: The Runge–Kutta local projection \({P}^1\)-discontinuous-Galerkin finite element method for scalar conservation laws. ESAIM Math. Model. Numer. Anal. 25(3), 337–361 (1991)

    MATH  Google Scholar 

  15. Cockburn, B., Shu, C.W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, vol. 3. Springer, Berlin (2005)

    MATH  Google Scholar 

  17. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Courier Corporation (2007)

  18. Dick, J., Kuo, F.Y., Sloan, I.H.: High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Don, W.S., Gao, Z., Li, P., Wen, X.: Hybrid compact-WENO finite difference scheme with conjugate Fourier shock detection algorithm for hyperbolic conservation laws. SIAM J. Sci. Comput. 38(2), A691–A711 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive Theory of Functions of Several Variables, pp. 85–100. Springer (1977)

  21. Engels, H.: Numerical Quadrature and Cubature. Academic Press, Berlin (1980)

    MATH  Google Scholar 

  22. Fasshauer, G.E.: Solving partial differential equations by collocation with radial basis functions. In: Proceedings of Chamonix, Vol. 1997, pp. 1–8. Vanderbilt University Press Nashville, TN (1996)

  23. Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB, vol. 6. World Scientific, Singapore (2007)

    MATH  Google Scholar 

  24. Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014)

    MathSciNet  MATH  Google Scholar 

  25. 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)

    MathSciNet  MATH  Google Scholar 

  26. 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)

    MathSciNet  MATH  Google Scholar 

  27. Fornberg, B., Driscoll, T.A., Wright, G., Charles, R.: Observations on the behavior of radial basis function approximations near boundaries. Comput. Math. Appl. 43(3–5), 473–490 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions With Applications to the Geosciences. SIAM, Philadelphia (2015)

    MATH  Google Scholar 

  29. Fornberg, B., Zuev, J.: The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Comput. Math. Appl. 54(3), 379–398 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Funaro, D., Gottlieb, D.: A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations. Math. Comput. 51(184), 599–613 (1988)

    MathSciNet  MATH  Google Scholar 

  31. Funaro, D., Gottlieb, D.: Convergence results for pseudospectral approximations of hyperbolic systems by a penalty-type boundary treatment. Math. Comput. 57(196), 585–596 (1991)

    MathSciNet  MATH  Google Scholar 

  32. Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), A1233–A1253 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Gelb, A., Hou, X., Li, Q.: Numerical analysis for conservation laws using \(\ell _1\) minimization. J. Sci. Comput. 81(3), 1240–1265 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Glaubitz, J.: Shock capturing by Bernstein polynomials for scalar conservation laws. Appl. Math. Comput. 363, 124593 (2019). https://doi.org/10.1016/j.amc.2019.124593

    Article  MathSciNet  MATH  Google Scholar 

  35. Glaubitz, J.: Constructing positive interpolatory cubature formulas. arXiv preprint arXiv:2009.11981 (2020). Submitted

  36. Glaubitz, J.: jglaubitz/weakrbf (2020). 10.5281/zenodo.4310328. https://doi.org/10.5281/zenodo.4310328

  37. Glaubitz, J.: Shock Capturing and High-Order Methods for Hyperbolic Conservation Laws. Logos Verlag, Berlin (2020). https://doi.org/10.30819/5084

    Book  MATH  Google Scholar 

  38. Glaubitz, J.: Stable high-order cubature formulas for experimental data. arXiv preprint arXiv:2009.03452 (2020). Submitted

  39. Glaubitz, J.: Stable high order quadrature rules for scattered data and general weight functions. SIAM J. Numer. Anal. 58(4), 2144–2164 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Glaubitz, J., Gelb, A.: High order edge sensors with \(\ell ^1\) regularization for enhanced discontinuous Galerkin methods. SIAM J. Sci. Comput. 41(2), A1304–A1330 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Glaubitz, J., Nogueira, A., Almeida, J., Cantão, R., Silva, C.: Smooth and compactly supported viscous sub-cell shock capturing for discontinuous Galerkin methods. J. Sci. Comput. 79, 249–272 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Glaubitz, J., Öffner, P.: Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points. Appl. Numer. Math. 151, 98–118 (2020). https://doi.org/10.1016/j.apnum.2019.12.020

    Article  MathSciNet  MATH  Google Scholar 

  43. Glaubitz, J., Öffner, P., Ranocha, H., Sonar, T.: Artificial viscosity for correction procedure via reconstruction using summation-by-parts operators. In: XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications, pp. 363–375. Springer (2016)

  44. Glaubitz, J., Öffner, P., Sonar, T.: Application of modal filtering to a spectral difference method. Math. Comput. 87(309), 175–207 (2018)

    MathSciNet  MATH  Google Scholar 

  45. Gottlieb, D., Hesthaven, J.S.: Spectral methods for hyperbolic problems. J. Comput. Appl. Math. 128(1), 83–131 (2001)

    MathSciNet  MATH  Google Scholar 

  46. Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Aomputation 67(221), 73–85 (1998)

    MathSciNet  MATH  Google Scholar 

  47. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    MathSciNet  MATH  Google Scholar 

  48. Gustafsson, B.: High Order Difference Methods for Time Dependent PDE, vol. 38. Springer, Berlin (2007)

    MATH  Google Scholar 

  49. Gustafsson, B., Kreiss, H.O., Oliger, J.: Time Dependent Problems and Difference Methods, vol. 24. Wiley, New York (1995)

    MATH  Google Scholar 

  50. Haber, S.: Numerical evaluation of multiple integrals. SIAM Rev. 12(4), 481–526 (1970)

    MathSciNet  MATH  Google Scholar 

  51. Hesthaven, J., Kirby, R.: Filtering in Legendre spectral methods. Math. Comput. 77(263), 1425–1452 (2008)

    MathSciNet  MATH  Google Scholar 

  52. Hesthaven, J.S.: Spectral penalty methods. Appl. Numer. Math. 33(1–4), 23–41 (2000)

    MathSciNet  MATH  Google Scholar 

  53. Hesthaven, J.S., Mönkeberg, F.: Entropy stable essentially nonoscillatory methods based on RBF reconstruction. ESAIM Math. Model. Numer. Anal. 53(3), 925–958 (2019)

    MathSciNet  MATH  Google Scholar 

  54. Hesthaven, J.S., Mönkeberg, F.: Two-dimensional RBF-ENO method on unstructured grids. J. Sci. Comput. 82(3), 1–24 (2020)

    MathSciNet  MATH  Google Scholar 

  55. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007)

    MATH  Google Scholar 

  56. Hon, Y., Mao, X.: An efficient numerical scheme for Burgers’ equation. Appl. Math. Comput. 95(1), 37–50 (1998)

    MathSciNet  MATH  Google Scholar 

  57. Huybrechs, D.: Stable high-order quadrature rules with equidistant points. J. Comput. Appl. Math. 231(2), 933–947 (2009)

    MathSciNet  MATH  Google Scholar 

  58. Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper 4079, 2007 (2007)

    Google Scholar 

  59. Iske, A.: Radial basis functions: basics, advanced topics and meshfree methods for transport problems. Rend. Sem. Mat. Univ. Pol. Torino 61(3), 247–285 (2003)

    MathSciNet  MATH  Google Scholar 

  60. Iske, A.: Scattered data approximation by positive definite kernel functions. Rend. Sem. Mat. Univ. Pol. Torino 69(3), 217–246 (2011)

    MathSciNet  MATH  Google Scholar 

  61. Iske, A.: Ten good reasons for using polyharmonic spline reconstruction in particle fluid flow simulations. In: Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, pp. 193–199. Springer (2020)

  62. Iske, A., Sonar, T.: On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions. Numerische Mathematik 74(2), 177–201 (1996)

    MathSciNet  MATH  Google Scholar 

  63. Jameson, A., Vincent, P.E., Castonguay, P.: On the non-linear stability of flux reconstruction schemes. J. Sci. Comput. 50(2), 434–445 (2012)

    MathSciNet  MATH  Google Scholar 

  64. Jiang, G.S., Shu, C.W.: On a cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62(206), 531–538 (1994)

    MathSciNet  MATH  Google Scholar 

  65. Kansa, E., Hon, Y.: Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput. Math. Appl. 39(7–8), 123–138 (2000)

    MathSciNet  MATH  Google Scholar 

  66. Kansa, E.J.: Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-II Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8–9), 147–161 (1990)

    MathSciNet  MATH  Google Scholar 

  67. Ketcheson, D.I.: Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30(4), 2113–2136 (2008)

    MathSciNet  MATH  Google Scholar 

  68. Klöckner, A., Warburton, T., Hesthaven, J.S.: Viscous shock capturing in a time-explicit discontinuous Galerkin method. Math. Model. Nat. Phenomena 6(3), 57–83 (2011)

    MathSciNet  MATH  Google Scholar 

  69. Kreiss, H.O., Lorenz, J.: Initial-Boundary Value Problems and the Navier–Stokes Equations, vol. 47. SIAM (1989)

  70. Krivodonova, L., Xin, J., Remacle, J.F., Chevaugeon, N., Flaherty, J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3–4), 323–338 (2004)

    MathSciNet  MATH  Google Scholar 

  71. Larsson, E., Fornberg, B.: A numerical study of some radial basis function based solution methods for elliptic PDES. Comput. Math. Appl. 46(5–6), 891–902 (2003)

    MathSciNet  MATH  Google Scholar 

  72. Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia (1973)

    MATH  Google Scholar 

  73. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, vol. 31. Cambridge University Press (2002)

  74. Levy, D., Tadmor, E.: From semidiscrete to fully discrete: Stability of Runge–Kutta schemes by the energy method. SIAM Rev. 40(1), 40–73 (1998)

    MathSciNet  MATH  Google Scholar 

  75. Madych, W.: Miscellaneous error bounds for multiquadric and related interpolators. Comput. Math. Appl. 24(12), 121–138 (1992)

    MathSciNet  MATH  Google Scholar 

  76. Martel, J.M., Platte, R.B.: Stability of radial basis function methods for convection problems on the circle and sphere. J. Sci. Comput. 69(2), 487–505 (2016)

    MathSciNet  MATH  Google Scholar 

  77. Metropolis, N., Ulam, S.: The Monte Carlo method. J. Am. Stat. Assoc. 44(247), 335–341 (1949)

    MATH  Google Scholar 

  78. Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992)

    MATH  Google Scholar 

  79. Öffner, P., Glaubitz, J., Ranocha, H.: Stability of correction procedure via reconstruction with summation-by-parts operators for Burgers’ equation using a polynomial chaos approach. ESAIM Math. Model. Numer. Anal. 52(6), 2215–2245 (2018)

    MathSciNet  MATH  Google Scholar 

  80. Öffner, P., Glaubitz, J., Ranocha, H.: Analysis of artificial dissipation of explicit and implicit time-integration methods. Int. J. Numer. Anal. Model. 17(3), 332–349 (2020)

    MathSciNet  Google Scholar 

  81. Osher, S.: Riemann solvers, the entropy condition, and difference. SIAM J. Numer. Anal. 21(2), 217–235 (1984)

    MathSciNet  MATH  Google Scholar 

  82. Platte, R.B., Driscoll, T.A.: Computing eigenmodes of elliptic operators using radial basis functions. Comput. Math. Appl. 48(3–4), 561–576 (2004)

    MathSciNet  MATH  Google Scholar 

  83. Platte, R.B., Driscoll, T.A.: Polynomials and potential theory for Gaussian radial basis function interpolation. SIAM J. Numer. Anal. 43(2), 750–766 (2005)

    MathSciNet  MATH  Google Scholar 

  84. Platte, R.B., Driscoll, T.A.: Eigenvalue stability of radial basis function discretizations for time-dependent problems. Comput. Math. Appl. 51(8), 1251–1268 (2006)

    MathSciNet  MATH  Google Scholar 

  85. Powell, M.J.: The theory of radial basis function approximation in 1990. Advances in numerical analysis, pp. 105–210 (1992)

  86. Randall, J.L.: Numerical Methods for Conservation Laws. Lectures in Mathematics ETH Zürich (1992)

  87. Ranocha, H., Glaubitz, J., Öffner, P., Sonar, T.: Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators. Appl. Numer. Math. 128, 1–23 (2018)

    MathSciNet  MATH  Google Scholar 

  88. Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299–328 (2016)

    MathSciNet  MATH  Google Scholar 

  89. Sarra, S.A., Heryudono, A.R., Wang, C.: A numerical study of a technique for shifting eigenvalues of radial basis function differentiation matrices. Tech Report, MU-MTH-TR-2011-1 (2011)

  90. Scarnati, T., Gelb, A., Platte, R.B.: Using \(\ell _1\) regularization to improve numerical partial differential equation solvers. J. Sci. Comput. 75(1), 225–252 (2018)

    MathSciNet  MATH  Google Scholar 

  91. Schaback, R.: Creating surfaces from scattered data using radial basis functions. In: Mathematical Methods for Curves and Surfaces, pp. 477–496. University Press (1995)

  92. Schaback, R.: Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3(3), 251–264 (1995)

    MathSciNet  MATH  Google Scholar 

  93. Schaback, R.: Multivariate interpolation and approximation by translates of a basis function. Ser. Approx. Decompos. 6, 491–514 (1995)

    MathSciNet  MATH  Google Scholar 

  94. Schaback, R.: Multivariate interpolation by polynomials and radial basis functions. Constr. Approx. 21(3), 293–317 (2005)

    MathSciNet  MATH  Google Scholar 

  95. Shu, C.W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9(6), 1073–1084 (1988)

    MathSciNet  MATH  Google Scholar 

  96. Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, New York (1971)

    MATH  Google Scholar 

  97. Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)

    MathSciNet  MATH  Google Scholar 

  98. Tadmor, E.: Shock capturing by the spectral viscosity method. Comput. Methods Appl. Mech. Eng. 80(1–3), 197–208 (1990)

    MathSciNet  MATH  Google Scholar 

  99. Tolstykh, A.I.: On using RBF-based differencing formulas for unstructured and mixed structured-unstructured grid calculations. In: Proceedings of the 16th IMACS World Congress, vol. 228, pp. 4606–4624. Lausanne (2000)

  100. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2013)

    Google Scholar 

  101. Trefethen, L.N.: Cubature, approximation, and isotropy in the hypercube. SIAM Rev. 59(3), 469–491 (2017)

    MathSciNet  MATH  Google Scholar 

  102. Vincent, P.E., Castonguay, P., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47(1), 50–72 (2011)

    MathSciNet  MATH  Google Scholar 

  103. Wendland, H.: Scattered Data Approximation, vol. 17. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  104. Wilson, M.W.: Discrete least squares and quadrature formulas. Math. Comput. 24(110), 271–282 (1970)

    MathSciNet  MATH  Google Scholar 

  105. Wilson, M.W.: Necessary and sufficient conditions for equidistant quadrature formula. SIAM J. Numer. Anal. 7(1), 134–141 (1970)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Simon-Christian Klein for helpful advice. This work is partially supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) #GL 927/1-1 (Glaubitz), AFOSR #F9550-18-1-0316 (Glaubitz and Gelb), NSF-DMS #1502640, NSF-DMS #1912685, and ONR #N00014-20-1-2595 (Gelb).

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    Jan Glaubitz & Anne Gelb

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Glaubitz, J., Gelb, A. Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions. J Sci Comput 87, 40 (2021). https://doi.org/10.1007/s10915-021-01453-8

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