Section: New Results

Numerical schemes for Hamiltonian PDEs

On numerical Landau damping for splitting methods applied to the Vlasov-HMF model

In [14], we consider time discretizations of the Vlasov-HMF (Hamiltonian Mean-Field) equation based on splitting methods between the linear and non-linear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time stepsize.

Unconditional and optimal $H^2$-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions

In [17], The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the $d$-dimensional ($d=2,3$) nonlinear Klein-Gordon-Schrödinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a lifting technique are introduced to establish rigorously the optimal $H^2$-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at $O(h^2+{\tau }^2)$ with mesh-size $h$ and time step $\tau$ in the discrete $H^2$-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.

Modulation equations approach for solving vortex and radiation in nonlinear Schrödinger equation

In [16], we apply the modulation theory to study the vortex and radiation solution in the 2D nonlinear Schrödinger equation. The full modulation equations which describe the dynamics of the vortex and radiation separately are derived. A general algorithm is proposed to efficiently and accurately find vortices with prescribed values of energy and spin index. The modulation equations are solved by accurate numerical method. Numerical tests and simulations of radiation are given.

Unconditional $L^{\infty }$-convergence of two compact conservative finite difference schemes for the nonlinear Schrödinger equation in multi-dimensions

In [18], we are concerned with the unconditional and optimal $L^{\infty }$-error estimates of two fourth-order (in space) compact conservative finite difference time domain schemes for solving the nonlinear Schrödinger equation in two or three space dimensions. The fact of high space dimension and the approximation via compact finite difference discretization bring difficulties in the convergence analysis. The two proposed schemes preserve the total mass and energy in the discrete sense. To establish the optimal convergence results without any constraint on the time step, besides the standard energy method, the cut-off function technique as well as a lifting technique are introduced. On the contrast, previous works in the literature often require certain restriction on the time step. The convergence rate of the proposed schemes are proved to be of $O(h^4+{\tau }^2)$ with time step $\tau$ and mesh size $h$ in the discrete $L^{\infty }$-norm. The analysis method can be directly extended to other finite difference schemes for solving the nonlinear Schrödinger-type equations. Numerical results are reported to support our theoretical analysis, and investigate the effect of the nonlinear term and initial data on the blow-up solution.

Verification of $2D$x$2D$ and two-species Vlasov-Poisson solvers

Recently $1D$x$1D$ two-species Vlasov-Poisson simulations have been performed by the semi-Lagrangian method. Thanks to a classical first order dispersion analysis, we are able to check in [1] the validity of their simulations; the extension to second order is performed and shown to be relevant for explaining further details. In order to validate multi-dimensional effects, we propose a $2D$x$2D$ single species test problem that has true $2D$ effects coming from the sole second order dispersion analysis. Finally, we perform, in the same code, full $2D$x$2D$ non linear two-species simulations with mass ratio $\sqrt{0.01}$, and consider the mixing of semi-Lagrangian and Particle-in-Cell methods.

An exponential integrator for the drift-kinetic model

In [11], we propose an exponential integrator for the drift-kinetic equations in polar geometry. This approach removes the CFL condition from the linear part of the system (which is often the most stringent requirement in practice) and treats the remainder explicitly using Arakawa's finite difference scheme. The present approach is mass conservative, up to machine precision, and significantly reduces the computational effort per time step. In addition, we demonstrate the efficiency of our method by performing numerical simulations in the context of the ion temperature gradient instability. In particular, we find that our numerical method can take time steps comparable to what has been reported in the literature for the (predominantly used) splitting approach. In addition, the proposed numerical method has significant advantages with respect to conservation of energy and efficient higher order methods can be obtained easily. We demonstrate this by investigating the performance of a fourth order implementation.

Convergence of a normalized gradient algorithm for computing ground states

In [15], we consider the approximation of the ground state of the one-dimensional cubic nonlinear Schrödinger equation by a normalized gradient algorithm combined with linearly implicit time integrator, and finite difference space approximation. We show that this method, also called imaginary time evolution method in the physics literature, is con-vergent, and we provide error estimates: the algorithm converges exponentially towards a modified solitons that is a space discretization of the exact soliton, with error estimates depending on the discretization parameters.