Semi-Implicit Spectral Deferred Correction Method Based on the Invariant Energy Quadratization Approach for Phase Field Problems

This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.

Novel High-Order Mass-and Energy-Conservative Runge-Kutta Integrators for the Regularized Logarithmic Schrodinger Equation

We develop a class of conservative integrators for the regularized logarithmic Schrodinger equation(RLogSE)using the quadratization technique and symplectic Runge-Kutta schemes.To preserve the highly nonlinear energy functional,the regularized equation is first transformed into an equivalent system that admits two quadratic invariants by adopting the invariant energy quadratization approach.The reformulation is then discretized using the Fourier pseudo-spectral method in the space direction,and integrated in the time direction by a class of diagonally implicit Runge-Kutta schemes that conserve both quadratic invariants to round-off errors.For comparison purposes,a class of multi-symplectic integrators are developed for RLogSE to conserve the multi-symplectic conservation law and global mass conservation law in the discrete level.Numerical experiments illustrate the convergence,efficiency,and conservative properties of the proposed methods.