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.


76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
76M99 Basic methods in fluid mechanics
65Z05 Applications to the sciences


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