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Mantle convection and long-term lithosphere dynamics in the Earth and other planets can be treated as the slow deformation of a highly viscous fluid, and as such can be described using the compressible Navier–Stokes equations. Since on Earth-sized planets the influence of compressibility is not a dominant effect, density deviations from a reference profile are at most on the order of a few percent and using the full governing equations poses numerical challenges, most modelling studies have simplified the governing equations. Common approximations assume a temporally constant, but depth-dependent reference profile for the density (the anelastic liquid approximation), or drop compressibility altogether and use a constant reference density (the Boussinesq approximation). In most previous studies of mantle convection and crustal dynamics, one can assume that the error introduced by these approximations was small compared to the errors that resulted from poorly constrained material behaviour and limited numerical accuracy. However, as model parametrizations have become more realistic, and model resolution has improved, this may no longer be the case and the error due to using simplified conservation equations might no longer be negligible: while such approximations may be reasonable for models of mantle plumes or slabs traversing the whole mantle, they may be unsatisfactory for layered materials experiencing phase transitions or materials undergoing significant heating or cooling. For example, at boundary layers or close to dynamically changing density gradients, the error arising from the use of the aforementioned compressibility approximations can be the dominant error source, and common approximations may fail to capture the physical behaviour of interest. In this paper, we discuss new formulations of the continuity equation that include dynamic density variations due to temperature, pressure and composition without using a reference profile for the density. We quantify the improvement in accuracy relative to existing formulations in a number of benchmark models and evaluate for which practical applications these effects are important. Finally, we consider numerical aspects of the new formulations. We implement and test these formulations in the freely available community software aspect , and use this code for our numerical experiments. [ABSTRACT FROM AUTHOR]

SUMMARY Numerical simulation of the processes in the Earth's mantle is a key piece in understanding its dynamics, composition, history and interaction with the lithosphere and the Earth's core. However, doing so presents many practical difficulties related to the numerical methods that can accurately represent these processes at relevant scales. This paper presents an overview of the state of the art in algorithms for high-Rayleigh number flows such as those in the Earth's mantle, and discusses their implementation in the Open Source code Aspect (Advanced Solver for Problems in Earth's ConvecTion). Specifically, we show how an interconnected set of methods for adaptive mesh refinement (AMR), higher order spatial and temporal discretizations, advection stabilization and efficient linear solvers can provide high accuracy at a numerical cost unachievable with traditional methods, and how these methods can be designed in a way so that they scale to large numbers of processors on compute clusters. Aspect relies on the numerical software packages deal.II and Trilinos, enabling us to focus on high level code and keeping our implementation compact. We present results from validation tests using widely used benchmarks for our code, as well as scaling results from parallel runs. [ABSTRACT FROM AUTHOR]