# The numerical solution of second-order boundary value problems on nonuniform meshes

@article{Manteuffel1986TheNS, title={The numerical solution of second-order boundary value problems on nonuniform meshes}, author={Thomas A. Manteuffel and Andrew B. White}, journal={Mathematics of Computation}, year={1986}, volume={47}, pages={511-535} }

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to…

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## References

SHOWING 1-10 OF 33 REFERENCES

FINITE DIFFERENCE COLLOCATION METHODS FOR NONLINEAR TWO POINT BOUNDARY VALUE PROBLEMS

- Mathematics
- 1979

A general class of finite difference methods for solving nonlinear two point boundary value problems is considered. These methods can also be interpreted as collocation methods. A convergence…

A Variable Mesh Finite Difference Method for Solving a Class of Parabolic Differential Equations in One Space Variable

- Mathematics
- 1978

A variable mesh finite difference scheme for a class of parabolic differential equations which exhibit shock-like structures is developed. It is shown that a properly chosen variable mesh will yield…

On the efficient numerical solution of systems of second order boundary value problems

- Mathematics
- 1986

An explicit reduction is given of the centered Euler scheme for a first order two-dimensional system producing an accurate difference approximation for a second order scalar equation. All of the…

An Adaptive Finite Element Method for Initial-Boundary Value Problems for Partial Differential Equations

- Computer Science
- 1982

A finite element method is developed to solve initial-boundary value problems for vector systems of partial differential equations in one space dimension and time that utomatically adapts the computational mesh as the solution progresses in time and permits an accurate solution to be calculated with fewer mesh points than would be necessary with a uniform mesh.

The Construction and Comparison of Finite Difference Analogs of Some Finite Element Schemes

- Mathematics
- 1974

A new method for solving two-point boundary value problems using optimal node distribution

- Mathematics
- 1972

Supra-convergent schemes on irregular grids

- Mathematics
- 1986

As Tikhonov and Samarskil showed for k = 2, it is not essential that k th-order compact difference schemes be centered at the arithmetic mean of the stencil's points to yield second-order convergence…

On Selection of Equidistributing Meshes for Two-Point Boundary-Value Problems

- Mathematics
- 1979

A general theory is developed for calculating equidistributing meshes $\{ t_i \} $ for difference methods for boundary-value problems of the form \[ u' = f(u,t),\qquad b(u(0),u(1)) = 0. \] It is…

Some Stability Inequalities for Compact Finite Difference Schemes

- Mathematics
- 1988

For finite difference schemes of compact form on nonuniform grids approximating m-th order two-point boundary value problems stability inequalities are proved which use a norm analogous to the…