Jain, M. K.; Jain, R. K.; Mohanty, R. K. A fourth order difference method for the one-dimensional general quasilinear parabolic partial differential equation. (English) Zbl 0715.65067 Numer. Methods Partial Differ. Equations 6, No. 4, 311-319 (1990). The fourth order compact difference scheme for the equation \(u_{xx}=f(u,u_ t,u_ x,x,t)\) is constructed, the order of truncation error being \(O(k^ 2+kh^ 2+h^ 4)\) where k and h are the mesh sizes in t and x directions, respectively. The scheme is shown to be unconditionally stable and to give oscillation-free solutions independently on cell Reynolds number for the convection-diffusion equation with constant coefficients. Numerical examples are presented. Reviewer: A.I.Tolstykh Cited in 1 ReviewCited in 26 Documents MSC: 65M06 Finite difference 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 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations Keywords:unconditional stability; error estimate; fourth order compact difference scheme; oscillation-free solutions; cell Reynolds number; convection- diffusion equation; Numerical examples PDFBibTeX XMLCite \textit{M. K. Jain} et al., Numer. Methods Partial Differ. Equations 6, No. 4, 311--319 (1990; Zbl 0715.65067) Full Text: DOI References: [1] ”The construction of finite difference analogs of some finite element schemes,” in Ed., Mathematical Aspects of Finite Element in Partial Differential Equations, Academic, New York, 1974, pp. 279-312. · doi:10.1016/B978-0-12-208350-1.50014-5 [2] Hirsh, J. Comput. Phys. 19 pp 90– (1975) [3] Ciment, J. Comput. Phys. 28 pp 135– (1978) [4] Christie, J. Comput. Phys. 59 pp 353– (1985) [5] Stoyan, ZAMM 59 pp 361– (1979) [6] Khosla, Comp. and Fluids 2 pp 207– (1974) [7] Spalding, Int. J. Numer. Meth. Engrg. 4 pp 551– (1972) [8] Roscoe, J. Inst. Maths. Applic. 16 pp 291– (1975) [9] Il’in, Math. Notes Acad. Sci. USSR 6 pp 596– (1969) [10] Jain, Appl. Math. Modelling 7 pp 57– (1983) [11] Christie, Int. J. Numer. Meth. in Engrg. 10 pp 1389– (1976) [12] Christie, Int. J. Numer. Meth. Engrg. 12 pp 1764– (1978) [13] and , eds., Numerical Analysis of Singular Perturbation Problems, Academic, New York, 1979. [14] and , eds., Numerical Modelling in Diffusion-Convection, Pentech, London, 1982. [15] Rubin, Comp. and Fluids 3 pp 1– (1975) [16] Ramos, Comput. Meth. Appl. Mech. Engrg. 64 pp 195– (1987) [17] Ramos, Comput. Meth. Appl. Mech. Engrg. 64 pp 221– (1987) [18] An Extrapolated Crank-Nicolson Difference Scheme for Quasilinear Parabolic Equations, Nonlinear Partial Differential Equations, Academic, 1967, pp. 193-201. [19] Douglas, J. Soc. Indust. Appl. Math. 11 pp 195– (1963) [20] ”Towards time-stepping algorithms for convective diffusion,” and , eds., Numerical Analysis of Singular Perturbation Problems, Academic, New York, 1979, pp. 199-216. [21] Chawla, J. Inst. Math. Applic. 21 pp 83– (1978) [22] Numerical Solution of Differential Equations, 2nd ed., Wiley-Eastern, New Delhi, 1987, pp. 357-359. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.