Uncertain Data in Initial Boundary Value Problems: Impact on Short and Long Time Predictions

TL;DR

This paper analyzes how uncertainty in data affects solutions to initial boundary value problems over different time scales, highlighting the dominant sources of error for short and long-term predictions.

Contribution

It quantifies the relative influence of uncertainties in initial, boundary, and forcing data on solution accuracy over time, emphasizing the role of dissipative boundary conditions.

Findings

Uncertainty in initial data dominates short-term predictions.

For long-term predictions, forcing and boundary data uncertainties become more significant.

Errors from forcing functions grow linearly, while boundary data errors grow as the square root of time.

Abstract

We investigate the influence of uncertain data on solutions to initial boundary value problems. Uncertainty in the forcing function, initial conditions and boundary conditions are considered and we quantify their relative influence for short and long time calculations. It is shown that dissipative boundary conditions leading to energy bounds play a crucial role. For short time calculations, uncertainty in the initial data dominate. As time grows, the influence of initial data vanish exponentially fast. For longer time calculations, the uncertainty in the forcing function and boundary data dominate, as they grow in time. Errors due to the forcing function grows faster (linearly in time) than the ones due to the boundary data (grows as the square root of time). Roughly speaking, the results indicate that for short time calculations, the initial conditions are the most important, but for longer time calculations, focus should be on modelling efforts and boundary conditions. Our findings have impact on predictions where similar mathematical and numerical techniques are used for both short and long times as for example in regional weather and climate predictions.