Abstract
We propose, analyze and numerically validate a new energy dissipative scheme for the Ginzburg–Landau equation by using the invariant energy quadratization approach. First, the Ginzburg–Landau equation is transformed into an equivalent formulation which possesses the quadratic energy dissipation law. After the space-discretization of the Fourier pseudo-spectral method, the semi-discrete system is proved to be energy dissipative. Using diagonally implicit Runge–Kutta scheme, the semi-discrete system is integrated in the time direction. Then the presented full-discrete scheme preserves the energy dissipation, which is beneficial to the numerical stability in long-time simulations. Several numerical experiments are provided to illustrate the effectiveness of the proposed scheme and verify the theoretical analysis.
Highlights
The Ginzburg–Landau equation (GLE) was first proposed by Ginzburg and Landau in 1950 and applied to model the electrodynamics, quantum mechanics and thermodynamic properties of superconductors
As a state transformation equation, the GLE is closely related to many other state transformation equations, such as Allen–Cahn equation and Chafee–Infante equation
The GLE plays an important role in the study of state transformation and unstable wave theory
Summary
The Ginzburg–Landau equation (GLE) was first proposed by Ginzburg and Landau in 1950 and applied to model the electrodynamics, quantum mechanics and thermodynamic properties of superconductors. The main challenge issue is to construct a high order energy dissipative scheme for the GLE. The invariant energy quadratization (IEQ) approach [17] is a new method developed in recent years by introducing a Lagrange multiplier and has been successfully applied to a variety of gradient flow models [18,19,20,21,22,23,24]. Gong [28] presented a novel class of arbitrarily high-order and unconditionally energy-stable algorithms for gradient flow models by combining the IEQ approach and a specific class of Runge–Kutta (RK) methods. Since the solution to the Ginzburg–Landau equation shows instability very soon and this high order implicit scheme can unconditionally keep Ginzburg–Landau equation energy dissipation, we adopt this method to solve the Ginzburg–Landau equation.
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