## 1. Introduction

[2] Modern computational geodynamics heavily relies on parallel algorithms to speed up calculations. Such a tendency is continuously growing over time as the available parallel resources increase, in particular with the development of multi-core architectures and Graphical Processing Unit computations [*Schmidt et al.*, 2010]. One of the most widely used approaches in parallel geodynamic codes is spatial decomposition [*Bunge and Baumgardner*, 1995; *Zhong et al.*, 2000; *Schmalzl and Hansen*, 2000; *Kageyama and Sato*, 2004; *Choblet et al.*, 2007; *Zhong et al.*, 2008; *Tackley*, 2008; *Hütigg and Stemmer*, 2008; *Aleksandrov and Samuel*, 2011], where the physical computational space is subdivided into smaller domains that are attributed to one processor or to a set of processors. Each sub-domain carries out its own calculation in parallel and exchanges information periodically with other sub-domains. Such approaches are efficient as long as the size of the sub-domains is large enough so that computational time remains larger than communication time. However, when the size of the sub-domains becomes too small, the speedup stagnates, which puts bounds on the maximum performances of the algorithm (Figure 1).

[3] Other approaches consist in solving at each time step the global set of discretized governing equations using parallel direct libraries or parallel iterative solvers [e.g., *Katz et al.*, 2007; *Braun et al.*, 2008; *Tosi et al.*, 2010; *Suckale et al.*, 2010; *Samuel and Evonuk*, 2010; *Thieulot*, 2011]. Similar to the domain-decomposition approach, such a parallelization only concerns the spatial domain and the performances of parallel solvers also saturate and sometimes decrease when the number of processors becomes too large.

[4] I present here an alternative approach named *parareal* [*Lions et al.*, 2001], which is based on time domain decomposition. This method has been successfully applied to solve ordinary differential equations and time-dependent systems of partial differential equations in various scientific areas, including molecular dynamic simulations [*Baffico et al.*, 2002], wave propagation [*Mercerat et al.*, 2009] and finite Prandtl number fluid dynamics [*Fisher et al.*, 2003; *Trindade and Pereira*, 2004; *Liu and Hu*, 2008; *Samaddar et al.*, 2010]. However, to my knowledge, the *parareal* algorithm has not been applied in geodynamic studies where motions relevant to the Earth and other planetary mantles are that of a convective fluid at infinite Prandtl number. In that case, the time dependence of the mass and momentum equations is only implicit, due to thermal and/or viscous couplings with the explicitly time-dependent energy equation. This requires a number of modifications to the original algorithm.

[5] This *parareal* approach can be combined to spatial domain decomposition or to any other parallel algorithm, allowing an additional increase in speedup with increasing the number of processors.

[6] The main objective of this study is to adapt the original version of the *parareal* algorithm to the governing equations for solid state convection, to test its robustness, and to evaluate its parallel performances theoretically and experimentally, using test cases representative of typical geodynamic scenarios.

[7] The paper is organized as follows: section 2 introduces the set of governing equations to be solved with the *parareal* approach. Section 3 describes the algorithm. Section 4 presents the theoretical performances of the *parareal* algorithm, which are compared with those measured experimentally in the context of two geodynamic scenarios, presented in section 5, preceding the discussion.