The modified equation approach to the stability and accuracy analysis of finite-difference methods

doi:10.1016/0021-9991(74)90011-4

Journal of Computational Physics

Volume 14, Issue 2, February 1974, Pages 159-179

https://doi.org/10.1016/0021-9991(74)90011-4 Get rights and content

Abstract

The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by the algebraic manipulations described herein. The connection between “heuristic” stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.

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The modified equation approach to the stability and accuracy analysis of finite-difference methods

Abstract

J. Computational Phys.

J. Computational Phys.

J. Computational Phys.

J. Computational Phys.

J. Computational Phys.

J. Computational Phys.

Difference Methods for Initial-Value Problems

The numerical solution of partial differential equations governing convection

AGARDograph No. 146