A mixed formulation of proper generalized decomposition for solving the Allen-Cahn and Cahn-Hilliard equations

In this paper, we propose a new implementation strategy for solving the Allen-Cahn and Cahn-Hilliard equations with the proper generalized decomposition (PGD) method for parametric studies. As is common to all PGD methods, it is not necessary to do essentially repeating computations; instead, solutions for a range of parameters can be obtained in a single computation. The proposed implementation strategy includes a mixed formulation and a new data structure. The mixed formulation is applied with the staggered method which does not require the first variation of residual with the derivative of the bulk free energy and thereby avoid repeated matrix reassembling in the Newton-Raphson procedure. In addition, the proposed data structure stores pointwise values of the solution functions instead of the PGD representation for the residual of the Newton method, which is efficient for nonlinear time-dependent problems. Overall, the computational efficiency is significantly improved compared with most traditional PGD formulations and implementations. •Solved the Allen-Cahn and Cahn-Hilliard equations with the proper generalized decomposition method.•The mixed formulation is applied with staggered method.•Proposed a nodal-value data structure storing pointwise values of the solution functions instead of the commonly used method.