| 09 Dec 2025
Automatic tuning of iterative pseudo-transient solvers for modelling the deformation of heterogeneous media
Abstract. Geodynamic modeling has become a crucial tool for investigating the dynamics of Earth deformation across various scales. Such simulations often involve solving mechanical problems with significant material heterogeneities (e.g., strong viscosity contrasts) under nearly incompressible conditions. Recent advancements have enabled the development of iterative solvers based on Dynamic Relaxation or Pseudo-Transient schemes, which require minimal global communication and exhibit quasi-linear scaling on GPU and supercomputing architectures. These solvers incorporate automatic tuning of iterative parameters, including pseudo-time steps and damping coefficients, based on spectral estimates of the discrete operators, ensuring both robust and rapid convergence. We demonstrate the effectiveness of this approach on problems discretized using finite-difference and face-centered finite volume methods, including heterogeneous incompressible Stokes flows. Moreover, the relative algorithmic simplicity of DR-based methods allows for straightforward extensions to compressible flow, multiphase flow, and nonlinear constitutive laws, opening promising avenues for large-scale, high-resolution simulations of geoscientific problems.
Competing interests: one of the authors is also editor
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In their paper, T. Duretz and colleagues suggest improvements to existing strategies of numerical geodynamic modeling through exploring the application of the direct relaxation (DR) method to various geodynamic problems. The key contributions compared to previous works are in the introduction of automated iteration parameter selection in the pseudo-transient method (pseudo-time stepping and damping) and solving incompressible Stokes flow equations using this method combined with Powell-Hestenes iterations. The iteration parameters are determined based on the eigenvalues of the discrete problem. The paper also addresses challenges associated with large viscosity contrasts and the enforcement of incompressibility, showing a systematic analysis. Finite Difference (FD) and Face-Centered Finite Volume (FCFV) discretization methods are employed, with the FCFV approach yielding smooth solutions across viscosity discontinuities. The study is complemented by numerical examples and an accompanying code repository, providing useful resources for further development of practical applications within and beyond the geodynamic community. The paper is well written, but it can still be improved in a few places with more explanation.
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