×

zbMATH — the first resource for mathematics

A strictly conservative spatial approximation scheme for the governing engineering and physics equations over irregular regions and inhomogeneously scattered nodes. (English) Zbl 0761.65096
From the author’s summary: This paper reports the progress made in multiquadrics as a spatial approximation scheme for systems of governing equations of engineering and physics by minimizing the spatial truncation errors without excessive refinement. We develope a strictly conservative interpolation scheme over irregular regions from which the partial derivatives are obtained. In addition, we develope a non-iterative scheme to be used with domain decomposition to ensure derivative continuity over continuous regions. Jump discontinuities for shock and material interfaces are likewise treated by appropriate modification of the algorithm.

MSC:
65Z05 Applications to the sciences
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
Software:
pchip
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fritsch, F.N.; Carlson, R.E., Monotone piecewise cubic interpolation, SIAM J. numer. anal., 17, 238-246, (1980) · Zbl 0423.65011
[2] Carlson, R.E.; Fritsch, F.N., Monotone piecewise bicubic interpolation, SIAM J. numer. anal., 22, 386-400, (1982) · Zbl 0571.65005
[3] Franke, R., Scattered data interpolation: test of some methods, Math. comput., 38, 181-200, (1982) · Zbl 0476.65005
[4] Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, J. geophys. res., 176, 1905-1915, (1971)
[5] Hardy, R.L., Theory and applications of the multiquadric-biharmonic method: 20 years of discovery 1968-1988, Comp. math. with applic., 19, 8/9, 163-208, (1990) · Zbl 0692.65003
[6] Hardy, R.L.; Nelson, S.A., A multiquadric-biharmonic representation and approximation of disturbing potentials, Geophy. res. lett., 13, 18-21, (1986)
[7] Micchelli, C.A., Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. approx., 2, 11-22, (1986) · Zbl 0625.41005
[8] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions, J. approx. theory and its applic., 4, 77-89, (1988) · Zbl 0703.41008
[9] Buhmann, M.D., Convergence of univariate quasi-interpolation using multiquadrics, IMA J. num. anal., 8, 365-383, (1988) · Zbl 0659.41003
[10] Buhmann, M.D.; Micchelli, C.A., Completely monotonic functions for cardinal interpolation, (), 1-4 · Zbl 0705.41003
[11] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions I, J. approx. theory and its applications, 4, 77-89, (1988) · Zbl 0703.41008
[12] Stead, S., Estimation of gradients from scattered data, Rocky mountain J. math., 14, 265-279, (1984) · Zbl 0558.65009
[13] Kansa, E.J., Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics, I. surface approximations and derivative estimates, Comp. math. with applic., 19, 8/9, 127-145, (1990) · Zbl 0692.76003
[14] Dyn, N.; Levin, D., Iterative solution of systems originating from integral equations and surface interpolation, SIAM J. numer. anal., 20, 377-390, (1983) · Zbl 0517.65096
[15] Dyn, N.; Levin, D.; Rippa, S., Numerical procedures for surface Fitting of scattered data by radial functions, SIAM J. sci. stat. comput., 7, 639-659, (1986) · Zbl 0631.65008
[16] 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, Comp. math. with applic., 19, 8/9, 147-161, (1990) · Zbl 0850.76048
[17] Gradshteyn, I.S.; Ryzhik, I.M., Table of integrals, series, and products, (1988), Academic Press NY, (Corrected and Enlarged Edition by A. Jeffrey) · Zbl 0918.65002
[18] Bjoerstad, P.E.; Widlund, O.B., Iterative methods for the solution of elleptic problems on regions partitioned into substructures, SIAM J. numer. anal., 23, 1097-1120, (1986) · Zbl 0615.65113
[19] Dryja, M.; Proskurowski, W., Capacitance matrix method using strips with alternating Neumann and Dirichlet boundary conditions, Appl. numer. math., 1, 285-298, (1985) · Zbl 0619.65094
[20] Gustafson, J.L.; Montry, G.R.; Benner, R.E., Development of parallel methods for a 1024-processor hypercube, SIAM J. sci. stat. comput., 9, 609-638, (1988) · Zbl 0652.65091
[21] Glimm, J.; Lindquist, B.; McBryan, O.; Tryggvason, G., Sharp and diffuse fronts in oil reservoirs. front tracking and capillarity, Report DOE/ER/03077-262, NYU, (1985)
[22] Chern, L.; Glimm, J.; McBryan, O.; Ploir, B.; Yanin, S., Front tracking for gas dynamics, Report DOE/ER/03077-223, NYU, (1984)
[23] Osher, S.; Sethian, J.A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. comp. phys., 79, 12-49, (1988) · Zbl 0659.65132
[24] Clark, R.A., Compressible Lagrangian hydrodynamics without Lagrangian cells, (), 281-294 · Zbl 0582.76080
[25] Trease, H., Three-dimensional free Lagrangian hydro-dynamics, (), 145-157
[26] Clark, R.A., Evolution of HOBO, Comput. phys. commun., 48, 61-64, (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.