Vortices in a stochastic parabolic Ginzburg-Landau equation

Abstract

Abstract. We consider the variant of a stochastic parabolic Ginzburg-Landau equation that allows for the formation of point defects of the solution. The noise in the equation is multiplicative of the gradient type. We show that the family of the Jacobians associated to the solution is tight on a suitable space of measures. Our main result is the characterization of the limit points of this family. They are concentrated on finite sums of delta measures with integer weights. The point defects of the solution coincide with the points at which the delta measures are centered.

A Fourth Order Energy Dissipative Scheme for a Traffic Flow Model

“…In this section, we carry out two numerical experiments by applying the implicit high order energy dissipative scheme. The stability and the convergence of the new scheme (15) are verified by these numerical experiments. Meanwhile, we compare the proposed scheme (15) with the AVF method in energy errors and accuracy.…”

“…The stability and the convergence of the new scheme (15) are verified by these numerical experiments. Meanwhile, we compare the proposed scheme (15) with the AVF method in energy errors and accuracy. The numerical iteration formula of the AVF method is…”

“…Figure 2b shows that they both can keep the energy reducing steadily. According to Figure 2c, comparatively speaking, the energy errors obtained by the proposed scheme (15) is even smaller than that by the scheme (18 Setting M 1 = 0.5 and ε = 0.001, we list the convergence rates of the proposed new scheme (15) in Tables 1 and 2. For the time accuracy test, we select the numerical solution with ∆t = 0.0001 and N = 50 as the reference solution.…”

“…First, we set M 1 = 0.3, the number of grid points N = 200 and the time step ∆t = 0.001. Letting ε = 0, 0.00025, 0.0005, 0.00075, 0.001 respectively, the energy variations and energy errors of the proposed scheme (15) are presented in Figure 1. Based on Figure 1a, we find the proposed scheme (15) can preserve the discrete energy conservation exactly when = 0, and the energy dissipations of the GLE can be observed when > 0.…”

“…Meanwhile, there are many effective ways to solve the GLE, such as the discontinuous Galerkin method [5], the semi-implicit method [6] and the trial equation method [7]. In addition, there also exist relevant studies on the fractional GLE [8][9][10][11] and the stochastic GLE [12][13][14][15][16]. However, most of previous researches can only achieve second order accuracy in the time direction.…”