A volumetric integral radial basis function method for time-dependent partial differential equations. I. Formulation.

*(English)*Zbl 1159.76363Summary: A strictly conservative volume integral formulation of the time dependent conservation equations in terms of meshless radial basis functions (RBFs) is presented. Rotational and translational transformations are considered that simplify the partial differential equations (PDEs) to be solved. As a result, the solutions that are represented at a finite sample of knots, \(x \in \Omega\partial \Omega \subset \mathbb R^d\) are permitted to move as the system of equations evolves in time. Knots are inserted, deleted, or rearranged in such a manner to conserve the extensive physical quantities of mass, momentum components, and total energy.

Our study consists of the following parts:

(A) Local rotational and Galilean translational transformations can be obtained to reduce the conservation equations into steady-state forms for the inviscid Euler equations or Navier — Stokes equations.

(B) The entire set of PDEs are transformed into the method of lines approach yielding a set of coupled ordinary differential equations whose homogeneous solution is exact in time.

(C) The spatial components are approximated by expansions of meshless RBFs; each individual RBF is volumetrically integrated at one of the sampling knots \(x_{i}\), yielding a collocation formulation of the method of lines structure of the ODEs

.(D) Because the volume integrated RBFs increase more rapidly away from the data center than the commonly used RBFs, we use a higher order preconditioner to counter-act the ill-conditioning problem. Domain decomposition is used over each piecewise continuous subdomain.

Our study consists of the following parts:

(A) Local rotational and Galilean translational transformations can be obtained to reduce the conservation equations into steady-state forms for the inviscid Euler equations or Navier — Stokes equations.

(B) The entire set of PDEs are transformed into the method of lines approach yielding a set of coupled ordinary differential equations whose homogeneous solution is exact in time.

(C) The spatial components are approximated by expansions of meshless RBFs; each individual RBF is volumetrically integrated at one of the sampling knots \(x_{i}\), yielding a collocation formulation of the method of lines structure of the ODEs

.(D) Because the volume integrated RBFs increase more rapidly away from the data center than the commonly used RBFs, we use a higher order preconditioner to counter-act the ill-conditioning problem. Domain decomposition is used over each piecewise continuous subdomain.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76N15 | Gas dynamics (general theory) |

##### Keywords:

Galilean transformations; momentum component transformations; shock waves; volumetric formulation##### Software:

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\textit{E. J. Kansa} et al., Eng. Anal. Bound. Elem. 28, No. 10, 1191--1206 (2004; Zbl 1159.76363)

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##### References:

[1] | Baxter, B., The asymptotic cardinal function of the multiquadratic φ(r)=(r2+c2)1/2 as c→∞, Comput math appl, 24, 12, 1-6, (1992) · Zbl 0764.41016 |

[2] | Baxter, B., Preconditioned conjugate gradients, radial basis functions, and Toeplitz matrices, Comput math appl, 43, 3-5, 305-318, (2002) · Zbl 1002.65018 |

[3] | Beatson, R.K.; Cherrie, J.B.; Mouat, C.T., Fast Fitting of radial basis functions: methods based on preconditioned GMRES iteration, Adv comput math, 11, 253-270, (1999) · Zbl 0940.65011 |

[4] | Beatson, R.K.; Cherrie, J.B.; Newsam, G.N., Fast evaluation of radial basis functions: methods for generalized multiquadrics in \(R\^{}\{n\}\), SIAM J sci comput, 23, 5, 1549-1571, (2002) · Zbl 1009.65007 |

[5] | Beatson, R.K.; Light, W.A., Fast evaluation of radial basis function: methods for two-dimensional polyharmonic splines, IMA J numer anal, 17, 343-372, (1997) · Zbl 0929.65004 |

[6] | Beatson, R.K.; Light, W.A.; Billings, S., Fast solution of the radial basis function interpolation methods: domain decomposition methods, SIAM J sci comput, 22, 5, 1717-1740, (2000) · Zbl 0982.65015 |

[7] | Beatson, R.K.; Newsam, G.N., Fast evaluation of radial basis functions. I, Computers math appl, 24, 12, 7-19, (1992) · Zbl 0765.65021 |

[8] | Bird, R.B.; Stewart, W.E.; Lightfoot, E.N., Transport phenomena, (1960), Wiley New York |

[9] | Buhmann, M.D.; Micchelli, C.A., Multiquadric interpolation improved, Comput math appl, 24, 12, 21-25, (1992) · Zbl 0764.41001 |

[10] | Chapman, C.J., High speed flow, Cambridge texts in applied mathematics, (2000), Cambridge University Press Cambridge |

[11] | Chen, J.-S.; Wu, C.T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin mesh-free methods, Int J numer meth engng, 50, 435-466, (2001) · Zbl 1011.74081 |

[12] | Chen, J.-S.; Yoon, S.; Wu, C.T., Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods, Int J numer meth engng, 53, 2587-2615, (2002) · Zbl 1098.74732 |

[13] | Chen, W.; He, J., A study on radial basis function and quasi-Monte Carlo methods, Int J nonlinear sci numer simul, 1, 4, 343-352, (2000) |

[14] | Chen, W., New RBF collocation schemes and kernel RBFs with applications, Lecture notes comput sci engng, 26, 75-86, (2002) · Zbl 1016.65094 |

[15] | Chen, W., Symmetric boundary knot method, Engng anal bound elem, 26, 6, 489-494, (2002) · Zbl 1006.65500 |

[16] | Chen W. Distance function wavelets. Part III. Exotic transforms and series. Research Report of Simula Research Laboratory. CoRR Preprint; June 2002. |

[17] | Chern, I.-L.; Glimm, J.; McBryan, O.; Plohr, B.; Yaniv, S., Front tracking for gas dynamics, J comput phys, 62, 83-110, (1986) · Zbl 0577.76068 |

[18] | Chui, C.K.; Stöckler, J.; Ward, J.W., Analytic wavelets generated by radial functions, Adv comput math, 5, 95-123, (1996) · Zbl 0855.65145 |

[19] | Deconinck, H.; Hirsch, C.; Peutman, J., Characteristic decomposition methods for the multidimensional Euler equations, Lecture notes in physics, vol. 274, (1986), Springer New York |

[20] | Fornberg, B.; Driscoll, T.A., Interpolation in the limit of increasingly flat radial basis functions, Comput math appl, 43, 3-5, 413-421, (2002) · Zbl 1006.65013 |

[21] | Fornberg, B.; Wright, G., Stable computation of multiquadric interpolants for all values of the shape parameter, Bit, (2004), in press · Zbl 1072.41001 |

[22] | Fornberg B, Wright G, Larsson E. Some observations regarding interpolants in the limit of flat radial basis functions. Comput Math Appl 2004;47:37-55. · Zbl 1048.41017 |

[23] | Franke, R., Scattered data interpolation: tests of some methods, Math comput, 98, 181-200, (1982) · Zbl 0476.65005 |

[24] | Glimm, J.; Grove, J.W.; Li, X.-L.; Shyue, K.-M.; Zhang, Q.; Zeng, Y., Three dimensional front tracking, SIAM J sci comput, 19, 703-727, (1998) · Zbl 0912.65075 |

[25] | Hon, Y.C.; Wu, Z., Additive Schwarz domain decomposition with radial basis approximation, Int J appl math, 4, 81-98, (2002) · Zbl 1051.65121 |

[26] | Kansa, E.J., Local, point-wise rotational transformations of the conservation equations into stream-wise coordinates, Comput math appl, 43, 3-5, 501-511, (2002) · Zbl 1073.76625 |

[27] | Kansa, E.J.; ion, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations, Comput math appl, 39, 7/8, 123-137, (2000) · Zbl 0955.65086 |

[28] | Laney, C.B., Computational gas dynamics, (1998), Cambridge University Press Cambridge, UK |

[29] | Larsson, E.; Fornberg, B., A numerical study of radial basis functions based solution methods for elliptic pdes, Comput math appl, 46, 891-902, (2003) · Zbl 1049.65136 |

[30] | Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions. II, Math comput, 54, 211-230, (1990) · Zbl 0859.41004 |

[31] | Mai-Duy, N.; Tran-Cong, T., Numerical solution of differential equations using multiquadric radial basis function networks, Neural networks, 14, 185-199, (2001) |

[32] | Mai-Duy, N.; Tran-Cong, T., Numerical solution of navier – stokes equations using multiquadric radial basis function networks, Int J numer meth fluids, 37, 65-86, (2001) · Zbl 1047.76101 |

[33] | Mouat CT. Fast algorithms and preconditioning techniques for fitting radial basis functions. PhD Thesis, Mathematics Department, University of Canterbury, Christchurch, New Zealand; 2001. |

[34] | Schaback, R., On the efficiency of interpolation by radial basis functions, (), 309-318 · Zbl 0937.65013 |

[35] | Smith, B.F.; Bjørstad, P.E.; Gropp, W.D., Domain decomposition: parallel multilevel methods for elliptic partial differential equations, (1996), Cambridge University Press Cambridge, UK · Zbl 0857.65126 |

[36] | Thompson, P.A., Compressible fluid dynamics, (1972), McGraw-Hill New York · Zbl 0251.76001 |

[37] | Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer Berlin · Zbl 0923.76004 |

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