Figure 2.
The illustration presents three panels illustrating different time integration approaches for a phase field variable phi as a function of the spatial coordinate x, with values ranging from minus 1 to 1. In the left panel, the multi stage method shows several intermediate profiles, labeled phi n to the power 1, phi n to the power 2, and phi n to the power 3, which equals phi n, progressing from the previous profile phi n minus 1. The red horizontal segments indicate incremental shifts in x during each internal stage. In the middle panel, the multi step method compares profiles phi n minus 3, phi n minus 2, phi n minus 1, and the current phi n, highlighting how information from multiple earlier steps contributes to the update. In the right panel, the two step method shows a simpler update using only phi n minus 1 and phi n, with a single larger shift along the x axis.Illustration of the required area of mesh refinement (indicated in red) for a multi-stage time-integration method (left), a multi-step method (center) and a two-step method (right)