Conference presentation given at 20th OpenFOAM Workshop 2025, Vienna, Austria. Abstract of presentation: The development of finite-volume solid mechanics has relied on second-order discretisation, with the majority of solvers adopting semi-implicit segregated solution procedures similar to those used in fluid dynamics. These solvers have proven their robustness and efficiency; however, multiphysics applications, such as electro-mechanical coupling, might benefit from solvers capable of exceeding second-order accuracy. Demirdzic’s research on fourth-order finite volume discretisation highlighted its potential to eliminate shear locking, but the discretisation was not easily generalised to unstructured grids and the segregated solution procedure was shown to be inefficient. Demirdzic suggested that a least squares procedure might provide a more practical approach for implementing high-order discretization. Building on this, Castrilo utilised concepts from high-order (> 2) fluid solvers to develop implicit high-order structural solvers based on a cell-centred grid arrangement. Both authors used in-house codes (which are not publicly available), which has led to a lack of comparative studies on the efficiency and accuracy of high-order discretisations versus second-order ones. This research seeks to fill this gap by implementing a cell-centred high-order discretisation within the OpenFOAM framework and an open Python framework, and evaluating its performance against a second-order discretisation. Given the challenges of larger computational stencils, denser matrices, and greater memory requirements associated with high-order discretisation, this study aims to assess its viability and effectiveness for solid mechanics applications. Conference slides included.