The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: A study of the boundary error.

*(English)*Zbl 0839.65098The paper deals with the loss of accuracy due to the imposition of time-dependent boundary conditions, dictated by the physics of the problem. It is shown that this conventional boundary condition imposition leads to a numerical scheme that is only first-order accurate in the neighborhood of the boundary, leading to a global accuracy of second order only.

The authors analyze and pinpoint the reasons for the deterioration of the accuracy and provide a simple recipe for restoring the accuracy in the case of linear, constant coefficient hyperbolic systems of partial differential equations.

Two methods for this restoring of the accuracy are analyzed. For nonlinear hyperbolic equations the studied method is effective only for Runge-Kutta schemes of third-order accuracy or less. Numerical studies are presented to verify the efficiency of each approach.

The authors analyze and pinpoint the reasons for the deterioration of the accuracy and provide a simple recipe for restoring the accuracy in the case of linear, constant coefficient hyperbolic systems of partial differential equations.

Two methods for this restoring of the accuracy are analyzed. For nonlinear hyperbolic equations the studied method is effective only for Runge-Kutta schemes of third-order accuracy or less. Numerical studies are presented to verify the efficiency of each approach.

Reviewer: K.Najzar (Praha)

##### MSC:

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35L60 | First-order nonlinear hyperbolic equations |