Truncation error analysis for the piecewise parabolic method

Accuracy is the cornerstone of numerical schemes for obtaining computational results that reliably represent physical phenomena. This work provides a theoretical truncation error analysis of the well‐established piecewise parabolic method (PPM) for unforced linear advection problems with non‐constant velocity on uniform grids. Despite its many higher order accurate features, only first‐order accuracy is found in the case of deformational flows. Independent of the spatio‐temporal form of the velocity, the leading‐order error term is always of purely temporal nature, even if certain elements of a Lagrangian approach are identified in the scheme that link space and time together. These findings are confirmed in a set of numerical experiments for multiple velocity types. Thereby, a strong connection between the resulting accuracy of the advected quantity and the imposed accuracy of the velocity is ascertained. The root cause of the leading‐order error term is identified and a minimal set of modifications is presented by which the temporal accuracy of PPM can be improved for arbitrary flows.