1. The Impact of Velocity Update Frequency on Time Accuracy for Mantle Convection Particle Methods.
- Author
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Trim, S. J., Butler, S. L., and Spiteri, R. J.
- Subjects
FLOW velocity ,PARTICLE tracks (Nuclear physics) ,VELOCITY ,ADVECTION - 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 explicit 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. Plain Language Summary: Computation of the flow velocity is an expensive process for mantle convection codes. For high‐velocity flows, tracer particle methods can be used to model the transport of mantle material while minimizing model errors such as artificial diffusion. A major element of tracer particle methods is the computation of particle trajectories, which is commonly done using Runge–Kutta (RK) methods. A popular choice is the classical fourth‐order accurate RK method, which is comprised of four stages per model time step. Strictly speaking, flow velocities are to be computed for each RK stage. However, flow velocities can be computed less often in order to reduce computation time. In this study, we examine the impact of how often the flow velocities are computed during time integration on accuracy and efficiency. We find that velocities can be computed less frequently to efficiently compute solutions to modest error tolerances. However, computing the velocities for every time integration stage is the most efficient way to find solutions with tight error tolerances. Specifically, highly accurate results can be achieved with less computation time using the classical fourth‐order explicit RK method with velocity updates at all four stages. Key Points: Computation of the velocity field is expensive for mantle convection codesComputing the velocity for a subset of Runge–Kutta stages can give acceptable results for modest error tolerances at reduced costThe reduced cost from not updating the velocity at each time integration stage is not worth the accuracy lost for tight error tolerances [ABSTRACT FROM AUTHOR]
- Published
- 2024
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