Figure 3.
The logarithmic plot of the L 2 norm over the domain Omega of the error between the numerical phase field phi n and a reference solution phi reference at time t n, plotted against the nondimensional parameter u zero times tau divided by epsilon, where u zero is a characteristic velocity, tau is the time step, and epsilon is an interface thickness parameter. Three time integration methods are compared: Implicit Euler, Crank-Nicolson, and the Generalized alpha method. The slopes marked 1, 2, and 3 indicate first order, second order, and third order convergence, respectively. Implicit Euler exhibits first order accuracy, Crank-Nicolson shows second order accuracy over a range of parameters, and the Generalized alpha method achieves higher, near third order accuracy for small values of u zero tau over epsilon, before all methods converge similarly at larger values.Convergence of the -norm of the discretization error in the phase field at . The time-step size is normalized with respect to the interface-displacement time