A new implicit formulation of the enthalpy method using flag updates

In the computation of problems involving phase changes, numerical approaches formulated on enthalpy offer numerous advantages to ‘front-tracking’ methods where the moving boundary between phases is explicitly tracked. However, due to the piecewise definition of enthalpy, such formulations effectively insert additional nonlinearity into the governing equations, thus adding increased complexity to implicit time-evolution schemes. In this paper, we develop and present a new ‘flag-update’ enthalpy method that crucially results in a linear set of equations at each time step. The equations can then be formulated as a sparse linear system, and subsequently solved using a more efficient inversion process. In a detailed error analysis, and via benchmarking on the classic Stefan problem in 1D and 2D, we show that the flag-update scheme is significantly faster than traditional implicit (Gauss–Seidel SOR) methods. However, speedup does not persist in 3D due to the significant memory and storage manipulations required. This study highlights the need to develop rigorous numerical analysis thresholds on such schemes.