Arbitrarily high-order accurate and energy-stable schemes for solving the conservative Allen-Cahn equation

In this paper, three high-order accurate and unconditionally energy-stable methods are proposed for solving the conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier. One is developed based on an energy linearization Runge-Kutta (EL-RK) method which combines an energy linearization technique with a specific class of RK schemes, the other two are based on the Hamiltonian boundary value method (HBVM) including a Gauss collocation method, which is the particular instance of HBVM, and a general class of cases. The system is first discretized in time by these methods in which the property of unconditional energy stability is proved. Then the Fourier pseudo-spectral method is employed in space along with the proofs of mass conservation. To show the stability and validity of the obtained schemes, a number of 2D and 3D numerical simulations are presented for accurately calculating geometric features of the system. In addition, our numerical results are compared with other known structure-preserving methods in terms of numerical accuracy and conservation properties.