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\begin{document}

\title{The impact of velocity update frequency on time accuracy for mantle
convection particle methods}



\author[1]{Sean James Trim}%
\author[1]{Samuel L. Butler}%
\author[1]{Raymond J Spiteri}%
\affil[1]{University of Saskatchewan}%


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  \date{May 22, 2024}


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\begin{abstract}
Computing the velocity field is an expensive process for mantle
convection codes. This has implications for particle methods used to
model the advection of quantities such as temperature or composition. A
common choice for the numerical treatment of particle trajectories is
classical fourth-order Runge--Kutta (ERK4) integration, which involves a
velocity computation at each of its four stages. To reduce the cost per
time step, it is possible to evaluate the velocity for a subset of the
four time integration stages. We explore two such alternative schemes,
in which velocities are only computed for: a) stage 1 on odd-numbered
time steps and stages 2--4 for even-numbered time steps, and b) stage 1
for all time steps. A theoretical analysis of stability and accuracy is
presented for all schemes. It was found that the alternative schemes are
first-order accurate with stability regions different from that of ERK4.
The efficiency and accuracy of the alternate schemes were compared
against ERK4 in four test problems covering isothermal, thermal, and
thermochemical flows. Exact solutions were used as reference solutions
when available. In agreement with theory, the alternate schemes were
observed to be first-order accurate for all test problems. Accordingly,
they may be used to efficiently compute solutions to within modest error
tolerances. For small error tolerances, however, ERK4 was the most
efficient.%
\end{abstract}%



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