Highly Accurate Numerical Schemes For The Spacce Fractional Ginzburg-Landau Equation

In this paper,we propose a fast and efficient numerical method:Jacobi poly-nomial method.We use it solve the Ginzburg-Landau equation with fractional Laplacian operator on space.The Jacobi polynomial method is developed and im-plemented in two steps:First,in space,we use the Jacobi polynomial method to discretize the one-and two-dimensional fractional Ginzburg-Landau equation.Us-ing the Jacobi-Gauss-Lobatto integral formula,the relationship between the Riesz fractional derivative and the Riemann-Liouville fractional derivative transforms the Ginzburg-Landau equation with fractional Laplace operator into an ordinary dif-ferential equation.Then applying the Jacobi-Gauss-Lobatto collocation point,the original equation with its initial value is converted to a system of ordinary differen-tial equations with the time variable.Second,in time,the Jacobi polynomial is used to approximate the equation.The Jacobi-Gauss-Radau integral formula is used to transform the ordinary differential equations obtained in the first step into algebraic equations,and then the approximate solution is solved by iterative method.Then use the basic lemma to prove the convergence of the Jacobi polynomial method that in the sense of~?and the weighted~2norm,the error between the exact solution and the numerical solution is exponentially convergent.Finally,we give several spe-cific numerical examples to confirm the correctness and efficiency of the method we proposed to solve the space fractional Ginzburg-Landau equation.