Numerical efficiency of explicit time integrators for phase-field models

TL;DR

The paper tackles the computational efficiency of explicit time integrators for phase-field models that couple phase fields to concentration under obstacle potentials. By comparing FEuler, SSP, and STS schemes across reproducible sharp-interface benchmarks, it demonstrates that STS-based explicit time integration can achieve speedups of $4$ to $114$ times over forward Euler while maintaining accuracy. It also shows that energy stability is not the dominant factor in approaching the sharp-interface limit, and that practical tolerances and interface width primarily govern errors. The findings advocate using STS schemes for phase-field simulations, including complex 3D scenarios with pores and particles, to achieve substantial speedups without sacrificing fidelity.

Abstract

Phase-field simulations are a practical but also expensive tool to calculate microstructural evolution. This work aims to compare explicit time integrators for a broad class of phase-field models involving coupling between the phase-field and concentration. Particular integrators are adapted to constraints on the phase-field as well as storage scheme implications. Reproducible benchmarks are defined with a focus on having exact sharp interface solutions, allowing for identification of dominant error terms. Speedups of 4 to 114 over the classic forward Euler integrator are achievable while still using a fully explicit scheme without appreciable accuracy loss. Application examples include final stage sintering with pores slowing down grain growth as they move and merge over time.

Figures (7)

• Figure 1: Geometry for the double triple junction.
• Figure 2: Results for the embedded phase case, showing non-uniform convergence to the sharp interface limit. The equilibrium grand potential $\psi$ is shown exemplary as an inset; it clearly divides the two bulk phases with homogeneous values inside. The energy is observed to decrease monotonically.
• Figure 3: Results for the double triple junction case, showing convergence to the sharp interface but also non-monotonic decrease of energy; the equilibrium shape in the inset is that of the predicted vesica piscis shape. At the practical interface resolution, the analytical angle is still approximated well within the entire range of tested interface energy ratios. The circular marks in \ref{['fig:tjangle-time']} indicate STS2 results for $W=3$ and the triangles for the FEuler integrator.
• Figure 4: Results for the embedded grain geometry. The exact solution is closely approximated by all simulations. The convergence plot shows that for $a_\phi=1.0e-2$ the convergence stalls starting between $\Delta x = \{0.25, 0.125\}$, whereas consistent convergence is obtained for $a_\phi=1.0e-4$. Comparison to a reference solution shows the expected temporal orders. Comparing to the exact solution shows the STS integrators as the most efficient ones.
• Figure 5: Results for the Stefan problem. The analytical solution is closely approximated and is being converged to with grid refinement. The choice of integrator seems to have little effect on the error and the STS integrators are observed to be the most efficient ones.