Kreiss, H.-O.; Manteuffel, T. A.; Swartz, B.; Wendroff, B.; White, A. B. jun. Supra-convergent schemes on irregular grids. (English) Zbl 0619.65055 Math. Comput. 47, 537-554 (1986). For the ordinary differential equation \(D^ ky=F\) with initial conditions \(D^ py=b_ p\) at \(x=0\) \((p=0,1,...,k-1)\) the finite difference solution is not always accurate to \(O(h^ 2)\) for odd k or for nonuniform grids with maximum interval h. But when the source function f is approximated by a class of functions F the solution error remains \(O(h^ 2)\). Such enhancement of the truncation error is termed supra-convergence. For any k, there is a [k/2]-parameter family of symmetric averages of the values of the kth derivative at the points of the stencil which, when set equal to the kth difference quotient, yields second-order convergence. It is further extended to stable compact schemes for equations with lower order terms under general boundary conditions. A counterexample with Numerov’s scheme is discussed to show that a given difference scheme which possesses a higher order truncation error when the mesh is uniform, does not always attain the same higher-order convergence when the mesh is not uniform. Super-convergence can yield various orders of enhancement over the ordinary errors but it is not so far noted that supra-convergence for compact schemes yields more than one higher order of convergence than the truncation error for polynomial based schemes. Reviewer: V.Subba Rao Cited in 59 Documents MSC: 65L05 Numerical methods for initial value problems involving ordinary differential equations 65D25 Numerical differentiation 34A30 Linear ordinary differential equations and systems Keywords:compact difference schemes; divided differences; superconvergence; finite difference solution; nonuniform grids; truncation error; supra- convergence; counterexample; Numerov’s scheme PDFBibTeX XMLCite \textit{H. O. Kreiss} et al., Math. Comput. 47, 537--554 (1986; Zbl 0619.65055) Full Text: DOI