INVARIANTS-PRESERVING DU FORT-FRANKEL SCHEMES AND THEIR ANALYSES FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH WAVE OPERATOR

Du Fort-Frankel finite difference method(FDM)was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953.It is an explicit and unconditionally von Neumann stable scheme.However,there has been no research work on numerical solutions of nonlinear Schrödinger equations with wave operator by using Du Fort-Frankel-type finite difference methods(FDMs).In this study,a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional(1D)and two-dimensional(2D)nonlinear Schrödinger equations with wave operator.By using the discrete energy method,it is shown that their solutions possess the discrete energy and mass conservative laws,and conditionally converge to exact solutions with an order of for ofο(T^(2)+h_(x)^(2)+(T/h_(x))^(2))1D problem and an order ofο(T^(2)+h_(x)^(2)+h_(Y)^(2)+(T/h_(X))^(2)+(T/h_(y))^(2))for 2D problem in H1-norm.Here,τdenotes time-step size,while,hx and hy represent spatial meshsizes in x-and y-directions,respectively.Then,by introducing a stabilized term,a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised.They not only preserve the discrete energies and masses,but also own much better stability than original schemes.Finally,numerical results demonstrate the theoretical analyses.

Energy law preserving continuous finite element schemes for a gas metal arc welding system

In this paper a modifed continuous energy law was explored to investigate transport behavior in a gas metal arc welding(GMAW)system.The energy law equality at a discrete level for the GMAW system was derived by using the finite element scheme.The mass conservation and current density continuous equation with the penalty scheme was applied 10 improve the stability.According to the phase-field model coupled with the energy law preserving method,the GMAW model was discretized and a metal transfer process with a pulse current was simulated.It was found that the numerical solution agrees well with the data of the metal transfer process obtained by high-speed photography.Compared with the numerical solution of the volume of fuid model,which was widely studied in the GMAW system based on the finite element method Euler scheme,the energy law preserving method can provide better accuracy in predicting the shape evolution of the droplet and with a greater computing efficiency.