A set of compact finite-difference approximations to first and second derivatives, related to the extended Numerov method of Chawla on nonuniform grids

Allen B.T. (1966). A new method of solving second-order differential equations when the first derivative is present. Comput J 8: 392–394 Ascher U.M., Mattheij R.M.M. and Russell R.D. (1995). Numerical solution of boundary value problems for ordinary differential equations. SIAM, Philadelphia Bieniasz L.K. (2004). Improving the accuracy of the spatial discretization in finite-difference electrochemical kinetic simulations, by means of the extended Numerov method. J Comput Chem 25: 1075–1083 Bieniasz L.K. (2007). A fourth-order accurate, three-point compact approximation of the boundary gradient, for electrochemical kinetic simulations by the extended Numerov method. Electrochim Acta 52: 2203–2209 Bieniasz L.K. (2007). Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 16. Patch-adaptive strategy combined with the extended Numerov spatial discretisation. Electrochim Acta 52: 3929–3940 Bieniasz, L. K.: Experiments with a local adaptive grid h-refinement for the finite-difference solution of BVPs in singularly perturbed second-order ODEs. Appl Math Comput (in press) Britz D. (2005). Digital simulation in electrochemistry, 3rd edn. Springer, Berlin Chawla M.M. (1978). A fourth-order tridiagonal finite difference method for general nonlinear two-point boundary value problems with mixed boundary conditions. J Inst Math Appl 21: 83–93 Chawla M.M. (1978). A fourth-order tridiagonal finite difference method for general two-point boundary value problems with nonlinear boundary conditions. J Inst Math Appl 22: 89–97 Collatz L. (1960). The numerical treatment of differential equations, 3rd edn. Springer, Berlin Eigenberger G. and Butt J.B. (1976). A modified Crank–Nicolson technique with non-equidistant space steps. Chem Eng Sci 31: 681–691 Evans D.J. and Mohanty R.K. (2002). Alternating group explicit method for the numerical solution of non-linear singular two-point boundary value problems using a fourth-order finite difference method. Int J Comput Math 79: 1121–1133 Fletcher C.A.J. (1982). Burgers’ equation: a model for all reasons. In: Noye, J. (eds) Numerical solution of partial differential equations, pp 139–225. North-Holland, Amsterdam Fornberg B. (1988). Generation of finite difference formulas on arbitrarily spaced grids. Math Comput 51: 699–706 Fox L. (1957). The numerical solution of two-point boundary problems in ordinary differential equations. Clarendon Press, Oxford Hirsh, R. S.: Higher-order approximations in fluid mechanics – compact to spectral. Von Karman Institute for Fluid Dynamics Lecture Series 1983–04 (1983) Jain M.K., Iyengar S.R.K. and Subramanyam G.S. (1984). Variable mesh methods for the numerical solution of two-point singular perturbation problems. Comput Meth Appl Mech Engng 42: 273–286 Jain M.K., Jain R.K. and Mohanty R.K. (1990). High-order difference methods for system of 1D nonlinear parabolic partial differential equations. Int J Comput Math 37: 105–112 Jain M.K., Jain R.K. and Mohanty R.K. (1990). A fourth-order difference method for the one-dimensional general quasilinear parabolic partial differential equation. Numer Meth PDEs 6: 311–319 Keller H.B. (1968). Numerical methods for two-point boundary-value problems. Blaisdell, Waltham Kreiss H.-O., Manteuffel T.A., Swartz B., Wendroff B. and White A.B. (1986). Supra-convergent schemes on irregular grids. Math Comput 47: 537–554 Manteuffel T.A. and White A.B. Jr. (1986). The numerical solution of second-order boundary value problems on nonuniform meshes. Math Comput 47: 511–535 Mohanty R.K., Jain M.K. and Kumar D. (2000). Single cell finite difference approximations of O (kh2 + h4) for du/dx for one space dimensional nonlinear parabolic equation. Numer Meth PDEs 16: 408–415 Mohanty R.K., Evans D.J. and Dey S. (2001). Three point discretization of order four and six for (du/dx) of the solution of nonlinear singular two point boundary value problem. Int J Comput Math 78: 123–139 Mohanty R.K. (2005). A family of variable mesh methods for the estimates of (du/dr) and solution of nonlinear two point boundary value problems with singularity. J Comput Appl Math 182: 173–187 Mohanty R.K. and Evans D.J. (2005). Highly accurate two parameter CAGE parallel algorithms for nonlinear singular two point boundary value problems. Int J Comput Math 82: 433–444 Mohanty R.K. and Khosla N. (2005). A third-order accurate variable-mesh TAGE iterative method for the numerical solution of two-point nonlinear singular boundary value problems. Int J Comput Math 82: 1261–1273 Noumerov B.V. (1924). A method of extrapolation of perturbations. Mon Notices Roy Astron Soc 84: 592–601 Ramos J.I. (1987). Hermitian operator methods for reaction-diffusion equations. Numer Meth PDEs 3: 241–287 Roos H.-G., Stynes M. and Tobiska L. (1996). Numerical methods for singularly perturbed differential equations, convection-diffusion and flow problems. Springer, Berlin Stepleman R.S. (1976). Tridiagonal fourth-order approximations to general two-point nonlinear boundary value problems with mixed boundary conditions. Math Comput 30: 92–103 Thomas L.H. (1949). Elliptic problems in linear difference equations over a network. Watson Scientific Computing Laboratory Report. Columbia University, New York Vetter K.J. (1961). Elektrochemische Kinetik. Springer, Berlin