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Analysis of a moving collocation method for one-dimensional partial differential equations. (English) Zbl 1245.65138

The authors emphasize the similarities between the moving collocation method, the standard collocation method and the finite volume method. It is shown that the moving collocation method inherits the ease implementation and the high-order convergence rate from the traditional collocation method and the mass conservation from the finite volume method. For general linear two-point boundary value problems the authors obtain the convergence of the moving collocation method in the \(L^\infty\) norm. Several numerical results are presented to support the theoretical findings.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
34B05 Linear boundary value problems for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations

Software:

DASSL; MOVCOL4
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Full Text: DOI

References:

[1] Ascher U, Mattheij R M M, Russell R D. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Englewood Cliffs, NJ: Prentice-Hall, 1988 · Zbl 0671.65063
[2] Barbeiro S, Ferreira J A, Grigorieff R D. Supraconvergence of a finite difference scheme for solutions in H s(0, l). IMA J Numer Anal, 2005, 25: 797–811 · Zbl 1087.65070 · doi:10.1093/imanum/dri018
[3] Bialecki B, Ganesh M, Mustapha K. A Petrov-Galerkin method with quadrature for elliptic boundary value problems. IMA J Numer Anal, 2004, 24: 157–177 · Zbl 1057.65080 · doi:10.1093/imanum/24.1.157
[4] Bialecki B, Ganesh M, Mustapha K. A Crank-Nicolson Petrov-Galerkin method with quadrature for semi-linear parabolic problems. IMA J Numer Anal, 2005, 21: 918–937 · Zbl 1081.65092
[5] Budd C J, Carretero-González R, Russell R D. Precise computations of chemotactic collapse using moving mesh methods. J Comput Phys, 2005, 202: 463–487 · Zbl 1063.65096 · doi:10.1016/j.jcp.2004.07.010
[6] Budd C J, Galaktionov V A, Williams J F. Self-similar blow-up in higher-order semilinear parabolic equations. SIAM J Appl Math, 2004, 64: 1775–1809 · Zbl 1112.35095 · doi:10.1137/S003613990241552X
[7] Castillo P, Cockburn B, Schötzau D, et al. Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems. Math Comp, 2002, 71: 455–478 · Zbl 0997.65111
[8] Chen S H. A sufficient condition for blowup solutions of nonlinear heat equations. J Math Anal Appl, 2004, 293: 227–236 · Zbl 1058.35091 · doi:10.1016/j.jmaa.2003.12.038
[9] de Boor C, Swartz B. Collocation at Gaussian points. SIAM J Numer Anal, 1973, 10: 582–606 · Zbl 0232.65065 · doi:10.1137/0710052
[10] Evans J D, Galaktionov V A, Williams J F. Blow-up and global asymptotics of the limit unstable Cahn-Hilliard equation. SIAM J Math Anal, 2006, 38: 64–102 · Zbl 1110.35023 · doi:10.1137/S0036141004440289
[11] Ferziger J H, Perić M. Computational Methods for Fluid Dynamics. Berlin: Springer, 1996 · Zbl 0869.76003
[12] García-Archilia B, Mackenzie J A. Analysis of a supraconvergent cell vertex finite-volume method for one-dimensional convection-diffusion problems. IMA J Numer Anal, 1995, 15: 101–115 · Zbl 0815.65097 · doi:10.1093/imanum/15.1.101
[13] He Y, Sun W. Nonconforming spline collocation methods in irregular domain II: Error analysis. Numer Meth PDEs, to appear, DOI: 10.1002/num.2067
[14] Huang W, Ma J, Russell R D. A study of MMPDE moving mesh methods for the numerical simulation of blowup in reaction diffusion equations. J Comput Phys, 2008, 227: 6532–6552 · Zbl 1145.65080 · doi:10.1016/j.jcp.2008.03.024
[15] Huang W, Russell R D. A moving collocation method for the numerical solution of time dependent differential equations. Appl Numer Math, 1996, 20: 101–116 · Zbl 0859.65112 · doi:10.1016/0168-9274(95)00119-0
[16] Huang W, Russell R D. Adaptive Moving Mesh Methods. New York: Springer, 2011 · Zbl 1227.65090
[17] Lasis A, Süli E. hp-version discontinuous Galerkin finite element method for semilinear parabolic problems. SIAM J Numer Anal, 2007, 45: 1544–1569 · Zbl 1155.65073 · doi:10.1137/050642125
[18] Li R H, Chen Z Y, Wu W. Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. New York: Marcel Dekker Inc, 2000 · Zbl 0940.65125
[19] Ma J, Jiang Y. Moving collocation methods for time fractional differential equations and simulation of blowup. Sci China Math, 2011, 54: 611–622 · Zbl 1217.34010 · doi:10.1007/s11425-010-4133-1
[20] Morton K W. Numerical Solution of Convection-Diffusion Problems. London: Chapman & Hall, 1996
[21] Morton K W, Süli E. Finite volume methods and their analysis. IMA J Numer Anal, 1991, 11: 241–260 · Zbl 0729.65087 · doi:10.1093/imanum/11.2.241
[22] Petzold L R. A description of dassl: A differential/algebraic system solver. Technical Report SAND82-8637. Livermore: Sandia National Laboratories, 1982
[23] Russell R D, Williams J F, Xu X. Movcol4: A moving mesh code for fourth-order time-dependent partial differential equations. SIAM J Sci Comput, 2007, 29: 197–220 · Zbl 1138.65095 · doi:10.1137/050643167
[24] Shu C W. High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD. Int J Comput Fluid Dyn, 2003, 17: 107–118 · Zbl 1034.76044 · doi:10.1080/1061856031000104851
[25] Stakgold I. Green’s Functions and Boundary Value Problems. New York: John Wiley & Sons, 1979 · Zbl 0421.34027
[26] Sun W T, Ward M J, Russell R D. The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities. SIAM J Appl Dyn Syst, 2005, 4: 904–953 · Zbl 1145.35404 · doi:10.1137/040620990
[27] Tang T, Xu J. Adaptive Computations: Theory and Algorithms. Beijing: Science Press, 2007
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