A comparison of linear and non-linear theories for modelling solid Earth dynamics

To date, most computational work in solid Earth geophysics has been based on linearised continuum mechanics. This is justified so long as deformation from the reference state remains sufficiently small. The dependence on linearisation also reflects computational limitations of the past: most tractable problems relied on geometric symmetries along with linearity to reduce the calculation to the solution of decoupled systems of ordinary differential equations.

Increases in computational power have allowed for increasingly routine applications of fully numerical techniques such as finite-difference, finite-element, and finite-volume methods. This has allowed geophysical problems to be solved in increasingly realistic Earth models. Although for the most part, the equations being solved are the same as linearised ones used previously, keeping nonlinear terms significantly increases the complexity of solution schemes. Within the context of fully numerical methods, non-linear problems are solved using iterative schemes that involve repeated solution of the corresponding linearised equations. This implies that solving non-linear equations should only be appreciably more expensive if non-linear effects are physically important.

Within this presentation, we compare the use of linearised and non-linear equations of motion, focusing on quasi-static elastic and viscoelastic loading problems of relevance to studies of glacial isostatic adjustment. This is achieved using the open-source finite-element package FeniCSx which facilitates rapid development and testing. Starting from simple representative examples, we quantify the errors associated with linearisation along with the added cost of solving non-linear problems.