zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. I: Surface approximations and partial derivative estimates. (English) Zbl 0692.76003
Summary: We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptical partial differential equations. The parabolic problem uses an implicit time- marching scheme whereas the hyperbolic problems uses an explicit time- marching scheme. We show that MQ is also exceptionally accurate and efficient.

MSC:
76A99Foundations, constitutive equations, rheology
76M99Basic methods in fluid mechanics
65Z05Applications of numerical analysis to physics
Software:
pchip
WorldCat.org
Full Text: DOI
References:
[1] Fritsch, F. N.; Carlson, R. E.: Monotone piecewise cubic interpolation. SIAM jl numer. Analysis 17, 238-246 (1980) · Zbl 0423.65011
[2] Carlson, R. E.; Fritsch, F. N.: Monotone piecewise bicubic interpolation. SIAM J numer. Analysis 22, 386-400 (1982) · Zbl 0571.65005
[3] Hyman, J. M.: Accurate monotonicity preserving cubic interpolation. SIAM jl sci. Stat. comput. 4, 645-654 (1983) · Zbl 0533.65004
[4] R. L. Dougherty, A. S. Edelman and J. M. Hyman, Positivity, monotonicity, or convexity-preserving cubic and quintic Hermite interpolation. LA-UR-85-2877 (submitted to SIAM Jl numer. Analysis). · Zbl 0693.41004
[5] Berger, M.; Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. comput. Phys. 53, 484-512 (1984) · Zbl 0536.65071
[6] Van Leer, B.: Toward the ultimate conservation scheme, III. J. comput. Phys. 23, 276-299 (1975)
[7] Franke, R.: Scattered data interpolation: test of some methods. Math. comput. 38, 181-200 (1982) · Zbl 0476.65005
[8] Foley, J. A.: Smooth multivariate interpolation to scattered data. Ph.d. dissertation (1979)
[9] Foley, T. A.; Nielson, G. M.: Multivariate interpolation to scattered data using delta iteration. Approximation theory III, 419-424 (1980) · Zbl 0505.41003
[10] Duchon, J.: Fonctions--spline du type plaque mince en dimension 2. Report #231 (1975)
[11] Duchon, J.: Fonctions--spline a energie invariante par rotation. Report #27 (1976)
[12] Hardy, R. L.: Multiquadric equations of topography and other irregular surfaces. J. geophys. Res. 176, 1905-1915 (1971)
[13] Hardy, R. L.: Research results in the application of multiquadric equations to surveying and mapping problems. Survg mapp. 35, 321-332 (1975)
[14] Tarwater, A. E.: A parameter study of Hardy’s multiquadric method for scattered data interpolation. Ucrl-54670 (Sept. 1985)
[15] Micchelli, C. A.: Interpolation of scattering data: distance matrices and conditionally positive definite functions. Constr. approx. 2, 11-22 (1986) · Zbl 0625.41005
[16] Korn, G. A.; Korn, T. M.: Mathematical handbook for scientists and engineers. (1968) · Zbl 0177.29301
[17] Lancaster, P.; Salkauskas, K.: Curve and surface Fitting: an introduction. (1986) · Zbl 0649.65012
[18] Stead, S.: Estimation of gradients from scattered data. Rocky mount. J. math. 14, 265-279 (1984) · Zbl 0558.65009
[19] Kansa, E. J.: Application of Hardy’s multiquadric interpolation to hydrodynamics. Proc. 1986 multiconf. Computer simulation 4, 111-117 (Jan. 1986)
[20] Foley, T. A.: Interpolation and approximation of 3-D and 4-D scattered data. Computers math. Applic. 13, 711-740 (1987) · Zbl 0635.65007
[21] Franke, R.: Recent advances in the approximation of surfaces from scattered data. (Dec. 1986)
[22] Schiro, R.; Williams, G.: An adaptive application of multiquadric interpolants for numerically modeling large numbers of irregularly spaced hydrographic data. Survg mapp. 44, 365-381 (1984)
[23] Damon, A.: Extensions of smoothing spline methods using generalized cross validation. Technical note COMA 3/86 (1986)
[24] Foley, T. A.: Scattered data interpolation and approximation with error bounds, vol. 3. Comput. aided. Geometric des. 3, 163-177 (1986) · Zbl 0619.65005
[25] Hardy, R. L.; Nelson, S. A.: A multiquadric-biharmonic representation and approximation of disturbing potentials. Geophys. res. Lett. 13, 18-21 (1986)
[26] Hardy, R. L.: Surface Fitting with biharmonic and harmonic models. 135-147 (May 1982)
[27] W. R. Madych and S. A. Nelson, Multivariate interpolation: a variational theory, J. Approx. Theory Applic. (in press). · Zbl 0703.41008
[28] Blake, F. G.: Spherical wave propagation in solid media. J. acoust. Soc. am. 24, 211-215 (1952)
[29] Dukowicz, J. K.; Kodis, J. W.: Accurate conservative remapping (REZONING) for arbitrary Lagrangian-Eulerian computations. SIAM jl sci. Stat. comput. 8, 305-320 (1987) · Zbl 0644.76085
[30] R. E. Carlson, Private communication, LLNL (1986).
[31] Dyn, N.; Levin, D.: Iterative solution of systems originating from integral equations and surface interpolation. SIAM jl numer. Analysis 20, 377-390 (1983) · Zbl 0517.65096
[32] Dyn, N.; Levin, D.; Rippa, S.: Numerical procedures for surface Fitting of scattered data by radial functions. SIAM jl sci. Stat. comput. 7, 639-659 (1986) · Zbl 0631.65008