Arbitrarily high-order accurate and energy-stable schemes for solving the conservative Allen-Cahn equation. (English) Zbl 1533.65194
Summary: 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.
{© 2022 Wiley Periodicals LLC.}
{© 2022 Wiley Periodicals LLC.}
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65D32 | Numerical quadrature and cubature formulas |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
conservative equation; energy linearization approach; energy-stable; Hamiltonian boundary value method; massReferences:
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