Open Access
Issue
ESAIM: M2AN
Volume 53, Number 3, May-June 2019
Page(s) 925 - 958
DOI https://doi.org/10.1051/m2an/2019011
Published online 21 June 2019
  1. T. Aboiyar, E.H. Georgoulis and A. Iske, High order weno finite volume schemes using polyharmonic spline reconstruction. In: Proceedings of the International Conference on Numerical Analysis and Approximation Theory NAAT2006. Dept. of Mathematics. University of Leicester, Cluj-Napoca (Romania) (2006). [Google Scholar]
  2. T. Aboiyar, E.H. Georgoulis and A. Iske, Adaptive ader methods using kernel-based polyharmonic spline weno reconstruction. SIAM J. Sci. Comput. 32 (2010) 3251–3277. [Google Scholar]
  3. C. Bigoni and J.S. Hesthaven, Adaptive weno methods based on radial basis function reconstruction. J. Sci. Comput. 72 (2017) 986–1020. [Google Scholar]
  4. J.P. Boyd, Error saturation in gaussian radial basis functions on a finite interval, J. Comput. Appl. Math. 234 (2010) 1435–1441. [Google Scholar]
  5. P. Chandrashekar, Kinetic energy preserving and entropy stable finite volume schemes for compressible euler and navier-stokes equations. Commun. Comput. Phys. 14 (2013) 1252–1286. [Google Scholar]
  6. D. Derigs, A.R. Winters, G.J. Gassner and S. Walch, A novel averaging technique for discrete entropy-stable dissipation operators for ideal MHD. J. Comput. Phys. 330 (2017) 624–632. [Google Scholar]
  7. T.A. Driscoll and B. Fornberg, Interpolation in the limit of increasingly flat radial basis functions. Comput. Math. Appl. 43 (2002) 413–422. [Google Scholar]
  8. J. Duchon, Splines Minimizing Rotation-invariant Semi-norms in Sobolev Spaces. Springer, Berlin (1977) 85–100. [Google Scholar]
  9. G.E. Fasshauer and J.G. Zhang, On choosing ``optimal’’ shape parameters for RBF approximation. Numer. Algorithms 45 (2007) 345–368. [Google Scholar]
  10. U.S. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230 (2011) 5587–5609. [Google Scholar]
  11. U.S. Fjordholm, S. Mishra and E. Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50 (2012) 544–573. [Google Scholar]
  12. U.S. Fjordholm, S. Mishra and E. Tadmor, Eno reconstruction and eno interpolation are stable. Found. Comput. Math. 13 (2013) 139–159. [CrossRef] [Google Scholar]
  13. U.S. Fjordholm and D. Ray, A sign preserving weno reconstruction method. J. Sci. Comput. 68 (2016) 42–63. [Google Scholar]
  14. B. Fornberg, E. Larsson and N. Flyer, Stable computations with gaussian radial basis functions in 2-D. Technical report, Department of Information Technology, Uppsala University (2009). [Google Scholar]
  15. B. Fornberg and C. Piret, A stable algorithm for flat radial basis functions on a sphere. SIAM J. Sci. Comput. 30 (2007) 60–80. [Google Scholar]
  16. B. Fornberg and G. Wright, Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 48 (2004) 853–867. [Google Scholar]
  17. S. Gottlieb, D.I. Ketcheson and C.-W. Shu, High order strong stability preserving time discretizations. J. Sci. Comput. 38 (2009) 251–289. [Google Scholar]
  18. A. Harten, B. Engquist, S. Osher and S.R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys. 71 (1987) 231–303. [Google Scholar]
  19. A. Harten, J.M. Hyman, P.D. Lax and B. Keyfitz, On finitedifference approximations and entropy conditions for shocks. Commun. Pure Appl. Math. 29 (1976) 297–322. [Google Scholar]
  20. J.S. Hesthaven, Numerical Methods for Conservation Laws: From Analysis to Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PN (2018). [Google Scholar]
  21. S.N. Kružkov, First order quasilinear equations in several independent variables. Math. USSR-Sbornik 10 (1970) 217. [CrossRef] [Google Scholar]
  22. E. Larsson and B. Fornberg, Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput. Math. Appl. 49 (2005) 103–130. [Google Scholar]
  23. A.-Y. le Roux, A numerical conception of entropy for quasi-linear equations. Math. Comput. 31 (1977) 848–872. [Google Scholar]
  24. P.G. Lefloch, J.-M. Mercier and C. Rohde, Fully discrete, entropy conservative schemes of arbitraryorder. . SIAM J. Numer. Anal. 40 (2002) 1968–1992. [CrossRef] [MathSciNet] [Google Scholar]
  25. R.J. LeVeque, Numerical Methods for Conservation Laws. Springer Science & Business Media, Berlin (1992). [CrossRef] [Google Scholar]
  26. M.L. Merriam, An Entropy-based Approach to Nonlinear Stability. Stanford University, Stanford, CA, USA (1989). [Google Scholar]
  27. C.A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2 (1986) 11–22. [Google Scholar]
  28. M.S. Mock, Systems of conservation laws of mixed type. J. Differ. Equ. 37 (1980) 70–88. [Google Scholar]
  29. G. Mühlbach, A recurrence formula for generalized divided differences and some applications. J. Approx. Theory 9 (1973) 165–172. [Google Scholar]
  30. G. Mühlbach, The general neville-aitken-algorithm and some applications. Numer. Math. 31 (1978) 97–110. [Google Scholar]
  31. F.J. Narcowich and J.D. Ward, Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices. J. Approx. Theory 69 (1992) 84–109. [Google Scholar]
  32. D. Ray, P. Chandrashekar, U.S. Fjordholm and S. Mishra, Entropy stable scheme on two-dimensional unstructured grids for euler equations. Commun. Comput. Phys. 19 (2016) 1111–1140. [Google Scholar]
  33. S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math. 11 (1999) 193–210. [Google Scholar]
  34. R. Schaback, Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3 (1995) 251–264. [Google Scholar]
  35. R. Schaback, Native hilbert spaces for radial basis functions I. New Deve. Approx. Theory 132 (1998) 255–282. [Google Scholar]
  36. R. Schaback, Multivariate interpolation by polynomials and radial basis functions. Constr. Approx. 21 (2005) 293–317. [Google Scholar]
  37. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439–471. [Google Scholar]
  38. E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49 (1987) 91–103. [CrossRef] [MathSciNet] [Google Scholar]
  39. B. Van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to godunov’s method. J. Comput. Phys. 32 (1979) 101–136. [Google Scholar]
  40. H. Wendland, Scattered Data Approximation. Cambridge University Press, Cambridge (2004). [Google Scholar]
  41. P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54 (1984) 115–173. [Google Scholar]
  42. G.B. Wright and B. Fornberg, Stable computations with flat radial basis functions using vector-valued rational approximations. J. Comput. Phys. 331 (2017) 137–156. [Google Scholar]

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