High order numerical algorithms based on biquadratic spline collocation for two-dimensional parabolic partial differential equations

References

• C. Allouch, P. Sablonnière, and D. Sbibih, A collocation method for the numerical solution of a two dimensional integral equation using a quadratic spline quasi-interpolant, Numer. Algorithms 62 (2013), pp. 445–468. doi: 10.1007/s11075-012-9598-2
   Web of Science ®Google Scholar
• D. Archer, An O(h4) cubic spline collocation method for quasilinear parabolic equations, SIAM J. Numer. Anal. 14(4) (1977), pp. 620–637. doi: 10.1137/0714042
   Google Scholar
• B. Bialecki and R.I. Fernandes, An orthogonal spline collocation alternating direction implicit Crank–Nicolson method for linear parabolic problems on rectangles, SIAM J. Numer. Anal. 36(5) (1999), pp. 1414–1434. doi: 10.1137/S0036142997310387
   Web of Science ®Google Scholar
• B. Bialecki and R.I. Fernandes, An alternating direction implicit backward differentiation orthogonal spline collocation method for linear variable coefficient parabolic equations, SIAM J. Numer. Anal. 47(5) (2009), pp. 3429–3450. doi: 10.1137/080739112
   Web of Science ®Google Scholar
• B. Bialecki, G. Fairweather, A. Karageorghis, and Q.N. Nguyen, Optimal superconvergent one step quadratic spline collocation methods, BIT Numer. Math. 48 (2008), pp. 449–472. doi: 10.1007/s10543-008-0188-6
   Web of Science ®Google Scholar
• A. Bourlioux, A.T. Layton, and M.L. Minion, High-order multi-implicit spectral deferred correction methods for problems of reactive flow, J. Comput. Phys. 189 (2003), pp. 651–675. doi: 10.1016/S0021-9991(03)00251-1
   Web of Science ®Google Scholar
• C.C. Christara, Quadratic spline collocation methods for elliptic partial differential equations, BIT Numer. Math. 34(1) (1994), pp. 33–61. doi: 10.1007/BF01935015
   Google Scholar
• C.C. Christara and K.S. Ng, Optimal quadratic and cubic spline collocation on nonuniform partitions, Computing 76(3–4) (2006), pp. 227–257. doi: 10.1007/s00607-005-0140-4
   Web of Science ®Google Scholar
• C.C. Christara, T. Chen, and D.M. Dang, Quadratic spline collocation for one-dimensional linear parabolic partial differential equations, Numer. Algorithms 53(4) (2010), pp. 511–553. doi: 10.1007/s11075-009-9317-9
   Web of Science ®Google Scholar
• A. Christlieb, B. Ong, and J.M. Qiu, Spectral deferred correction methods with high order Runge–Rutta schemes in prediction and correction steps, Comm. Appl. Math. Comput. Sci. 4 (2009), pp. 27–56. doi: 10.2140/camcos.2009.4.27
   Google Scholar
• A. Christlieb, B. Ong, and J.M. Qiu, Integral deferred correction methods constructed with high order Runge–Kutta integrators, Math. Comput. 79 (2010), pp. 761–783. doi: 10.1090/S0025-5718-09-02276-5
   Web of Science ®Google Scholar
• A. Christlieb, M. Morton, B. Ong, and J.M. Qiu, Semi-implicit integral deferred correction constructed with additive Runge–Kutta methods, Comm. Math. Sci. 9 (2011), pp. 879–902. doi: 10.4310/CMS.2011.v9.n3.a10
   Web of Science ®Google Scholar
• J. Douglas and T. Dupont, Collocation methods for parabolic equations in a single-space variable, Lect. Notes Math. 385 (1974), pp. 1–147. doi: 10.1007/BFb0057338
   Google Scholar
• A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT Numer. Math. 40(2) (2000), pp. 241–266. doi: 10.1023/A:1022338906936
   Web of Science ®Google Scholar
• G. Fairweather, A. Karageorghis, and J. Maack, Compact optimal quadratic spline collocation methods for the Helmholtz equation, J. Comput. Phys. 230 (2011), pp. 2880–2895. doi: 10.1016/j.jcp.2010.12.041
   Web of Science ®Google Scholar
• C.E. Greenwell-Yanik and G. Fairweather, Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables, SIAM J. Numer. Anal. 23 (1986), pp. 282–296. doi: 10.1137/0723020
   Web of Science ®Google Scholar
• E.N. Houstis, C.C. Christara, and J.R. Rice, Quadratic-spline collocation methods for two-point boundary value problems, Int. J. Numer. Methods Eng. 26(4) (1988), pp. 935–952. doi: 10.1002/nme.1620260412
   Web of Science ®Google Scholar
• J.F. Huang, J. Jia, and M. Minion, Accelerating the convergence of spectral deferred correction methods, J. Comput. Phys. 214 (2006), pp. 633–656. doi: 10.1016/j.jcp.2005.10.004
   Web of Science ®Google Scholar
• A. Kaya, Finite difference approximations of multidimensional unsteady convection–diffusion–reaction equations, J. Comput. Phys. 285 (2015), pp. 331–349. doi: 10.1016/j.jcp.2015.01.024
   Web of Science ®Google Scholar
• A. Kaya and A. Sendur, Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid, J. Comput. Phys. 300 (2015), pp. 574–591. doi: 10.1016/j.jcp.2015.08.007
   Web of Science ®Google Scholar
• A.T. Layton, On the choice of correctors for semi-implicit Picard deferred correction methods, Appl. Numer. Math. 58 (2008), pp. 845–858. doi: 10.1016/j.apnum.2007.03.003
   Web of Science ®Google Scholar
• A.T. Layton and M.L. Minion, Implications of the choice of quadrature nodes for picard integral deferred corrections methods for ordinary differential equations, BIT Numer. Math. 45 (2005), pp. 341–373. doi: 10.1007/s10543-005-0016-1
   Web of Science ®Google Scholar
• A.T. Layton and M.L. Minion, Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods, Comm. Appl. Math. Comput. Sci. 2(1) (2007), pp. 1629–1643. doi: 10.2140/camcos.2007.2.1
   Google Scholar
• P.A. Markowich and C.A. Ringhofer, Collocation methods for boundary value problems on ‘long’ intervals, Math. Comput. 40(161) (1983), pp. 123–150.
   Web of Science ®Google Scholar
• M.L. Minion, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci. 1(3) (2003), pp. 471–500. doi: 10.4310/CMS.2003.v1.n3.a6
   Google Scholar
• R. Pallav and A. Pedas, Quadratic spline collocation for the smoothed weakly singular Fredholm integral equations, Numer. Funct. Anal. Optim. 30(9-10) (2009), pp. 1048–1064. doi: 10.1080/01630560903408705
   Web of Science ®Google Scholar
• R.D. Russell and L.F. Shampine, A collocation method for boundary value problems, Numer. Math. 19(1) (1972), pp. 1–28. doi: 10.1007/BF01395926
   Web of Science ®Google Scholar
• T.W.H. Sheu, S.K. Wang, and R.K. Lin, An implicit scheme for solving the convection–diffusion–reaction equation in two dimensions, J. Comput. Phys. 164 (2000), pp. 123–142. doi: 10.1006/jcph.2000.6588
   Web of Science ®Google Scholar
• R. Speck, D. Ruprecht, M. Emmett, M. Minion, M. Bolten, and R. Krause, A multi-level spectral deferred correction method, BIT Numer. Math. 55(3) (2015), pp. 843–867. doi: 10.1007/s10543-014-0517-x
   Web of Science ®Google Scholar
• T. Tang, H.H. Xie, and X.B. Yin, High-order convergence of spectral deferred correction methods on general quadrature nodes, J. Sci. Comput. 56 (2013), pp. 1–13. doi: 10.1007/s10915-012-9657-9
   Web of Science ®Google Scholar
• R. Wang, P. Keast, and P. Muir, A high-order global spatially adaptive collocation method for 1-D parabolic PDEs, Appl. Numer. Math. 50 (2004), pp. 239–260. doi: 10.1016/j.apnum.2003.12.023
   Web of Science ®Google Scholar