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Supra-convergent schemes on irregular grids. (English) Zbl 0619.65055

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

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65D25 Numerical differentiation
34A30 Linear ordinary differential equations and systems
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