Due to the increasing availability of high-performance computing over the past few decades, numerical models have become an important tool for research in geodynamics.
Several generations of mantle convection software have been developed, but due to their differing methods and increasing complexity it is important to evaluate the accuracy of each new model generation to ensure published geodynamic research is reliable and reproducible.
Here we explore the accuracy of the open-source, finite-element codes ASPECT and CitcomS as a function of mesh spacing using low to moderate-Rayleigh-number models in steady-state thermal convection.
ASPECT (Advanced Solver for Problems in Earth's ConvecTion) is a new-generation mantle convection code that enables modeling global mantle convection with realistic parameters and complicated physical processes using adaptive mesh refinement

While there have been significant efforts to develop software capable of modeling mantle convection in a 3-D spherical shell

There have been a number of studies comparing ASPECT results with other codes using Cartesian geometry

In this work we report a comparison of steady-state thermal convection at low to moderate Rayleigh number using both CitcomS and ASPECT.
A number of previous studies have focused on the low-Rayleigh-number calculations (

The conservation of mass, momentum, and energy equations for an incompressible Boussinesq fluid in their nondimensional forms are given by

Parameters used in

The Rayleigh number and appropriate boundary conditions can describe this problem if all material properties and gravity are held constant.
The Rayleigh number is given by

For the ASPECT calculations, we use version 2.2.0

The cases that we consider use temperature-dependent, nondimensional viscosity expressed as

Values of the

We compare the results from ASPECT and CitcomS on a variety of meshes and we report the top and bottom Nusselt number, mean temperature, and rms velocity.
The Nusselt number,

ASPECT uses quadratic velocity and temperature elements by default and has the capability to refine the mesh based on a variety of measured properties of the solution.
CitcomS, by comparison, uses linear velocity and temperature elements with a mesh spacing that remains fixed throughout the calculation.
The authors of

In order to more accurately reproduce the CitcomS results, the rheology (Eq.

For CitcomS we used the default parameter setting in the CitcomS-3.3.1 version from CIG with the following exceptions:

CitcomS requires the user to specify the coarsest mesh and number of multigrid levels with the formula for each direction being

The mesh structure used for CitcomS calculations. The terms

Degrees of freedom (DoFs) for each resolution of each code. Note that

Results from the advection-in-annulus benchmark in ASPECT.
This shows how mesh refinement influences the heat flux out of the system depending on whether entropy viscosity

The default ASPECT temperature solver is the entropy viscosity (EV) method

Isotherms from Cases

In this section we focus on the Rayleigh number

To create a tetragonal pattern, a degree 3 and order 2 spherical harmonic perturbation is used.
The magnitude of this perturbation for both the cosine and sine terms is

To assess how each code handles temperature-dependent rheology, we selected three cases:

Results for Case

The results from

Plots of rms velocity and average temperature with respect to depth for all cases tested. Dashed dark grey lines are CitcomS data, and solid black lines are ASPECT data. All cases are in excellent agreement across both codes.

We compare the convergence of the solutions from CitcomS and ASPECT for the

The rms velocity, average temperature, and Nusselt number at the top and bottom of the model for Case

The plots of rms velocity, mean temperature, and Nusselt numbers at the outer and inner shell boundaries on different meshes for each code (Fig.

The rms velocity plotted against Nusselt numbers at both the top (dashed lines) and bottom (solid lines) of the model for Case

We also show that in addition to the small differences in the steady-state global quantities between the two codes, the time series evolutions of the global diagnostics follow nearly identical paths. Figure

The rms velocity, average temperature, and Nusselt number at the top and bottom of the model for Case

For the

Results for Case

Case

The rms velocity, average temperature, and Nusselt number at the top and bottom of the model for Case

Solutions for Case

Results for Case

The rms velocity, average temperature, and Nusselt number at the top and bottom of the model for Case

The rms velocity, average temperature, and Nusselt number at the top and bottom of the model for Case

The rms velocity, average temperature, and Nusselt number at the top and bottom of the model for Case

Reported values for this case show more difference between the two codes, even at the highest mesh refinements tested (Table

The

The results from all three

As with the tetragonal-planform cases, we compare the convergence of the solutions from CitcomS and ASPECT for the

Results for Case

Plots of rms velocity, mean temperature, and top and bottom Nusselt numbers on different meshes are again produced, with overall convergence increasing with mesh refinement (Fig.

Data points trend smoothly towards convergence for ASPECT, with a slight outlier at

Results for Case

Results for Case

Case

The ASPECT results show well-behaved convergence for all parameters calculated (Fig.

Case

Convergence for both codes is good for all parameters tested (Fig.

We note excellent agreement in the rms velocity, mean temperature, and top and bottom Nusselt number between the two codes on the most refined meshes.
If we take the difference between the top and bottom Nusselt numbers as a measure of the accuracy of the solution, which should be zero for incompressible flow at steady state, the

For the higher-Rayleigh-number

In general, globally averaged diagnostics from both codes at the highest mesh resolutions tested agree to within 0.6 %, and the Richardson extrapolation of the results from increasing mesh resolution agrees to within 1.0 %.
Often the Richardson extrapolation agrees to within 0.5 %.
ASPECT generates better-resolved solutions on coarser meshes than CitcomS, as would be expected because it uses higher-order elements.
ASPECT also has several methods of improving the performance of these calculations.
Adaptive refinement, both dynamically or statically through refining the boundary layers, can resolve features while reducing computational effort.
This was not used in this study to facilitate similarity between the two codes.
ASPECT also has a newer geometric multigrid solver

Many studies use CitcomS results computed on a

Using the streamline upwind Petrov–Galerkin (SUPG) energy solver for ASPECT we find extremely good agreement between CitcomS and ASPECT results for these low to intermediate Rayleigh number calculations.
Both CitcomS and ASPECT use the SUPG algorithm to solve the energy equation.
For the selected cases

All software used to generate these results is freely available.
ASPECT is publicly available on GitHub at

SDK, GTE, and SL were responsible for the initial conceptualization of this study. Software updates for use of SUPG in ASPECT were developed by TH and RG. ASPECT simulations were designed by SL. Models calculated using ASPECT were performed by GTE and SL, and models calculated using CitcomS were performed by SDK. GTE performed the data curation and formal analysis and prepared the manuscript with contributions from all co-authors.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the Computational Infrastructure for Geodynamics (

Shangxin Liu was partially supported by EarthScope and GeoPRISMS programs of the National Science Foundation through the Mid-Atlantic Geophysical Integrative Collaboration (MAGIC) project (grant no. EAR-1250988). Timo Heister was partially supported by the National Science Foundation (award nos. DMS-2028346, OAC-2015848, and EAR-1925575) and by the Computational Infrastructure for Geodynamics through the National Science Foundation (award nos. EAR-0949446 and EAR-1550901). Rene Gassmöller was partially supported by the National Science Foundation (award nos. EAR-1925677 and EAR-2054605) and by the Computational Infrastructure for Geodynamics through the National Science Foundation (award nos. EAR-0949446 and EAR-1550901).

This paper was edited by Ludovic Räss and reviewed by Christian Hüttig and one anonymous referee.