اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدHigh-order linearly implicit structure-preserving exponential integrators for the nonlinear Schrodinger equation
78
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
A novel class of high-order linearly implicit energy-preserving exponential integrators are proposed for the nonlinear Schrodinger equation. We firstly done that the original equation is reformulated into a new form with a modified quadratic energy by the scalar auxiliary variable approach. The spatial derivatives of the system are then approximated with the standard Fourier pseudo-spectral method. Subsequently, we apply the extrapolation technique to the nonlinear term of the semi-discretized system and a linearized system is obtained. Based on the Lawson transformation, the linearized system is rewritten as an equivalent one and we further apply the symplectic Runge-Kutta method to the resulting system to gain a fully discrete scheme. We show that the proposed scheme can produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution and is extremely efficient in the sense that only linear equations with constant coefficients need to be solved at every time step. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other high-order structure-preserving method.
قيم البحث
اقرأ أيضاً
In this paper, a family of arbitrarily high-order structure-preserving exponential Runge-Kutta methods are developed for the nonlinear Schrodinger equation by combining the scalar auxiliary variable approach with the exponential Runge-Kutta method. B
y introducing an auxiliary variable, we first transform the original model into an equivalent system which admits both mass and modified energy conservation laws. Then applying the Lawson method and the symplectic Runge-Kutta method in time, we derive a class of mass- and energy-preserving time-discrete schemes which are arbitrarily high-order in time. Numerical experiments are addressed to demonstrate the accuracy and effectiveness of the newly proposed schemes.
This article presents and analyses an exponential integrator for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. We first prove that the strong order of the numerical approx
imation is $1/2$ if the nonlinear term in the system is globally Lipschitz-continuous. Then, we use this fact to prove that the exponential integrator has convergence order $1/2$ in probability and almost sure order $1/2$, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the exponential integrator as well as a modified version of it.
In this paper, we present a linearly implicit energy-preserving scheme for the Camassa-Holm equation by using the multiple scalar auxiliary variables approach, which is first developed to construct efficient and robust energy stable schemes for gradi
ent systems. The Camassa-Holm equation is first reformulated into an equivalent system by utilizing the multiple scalar auxiliary variables approach, which inherits a modified energy. Then, the system is discretized in space aided by the standard Fourier pseudo-spectral method and a semi-discrete system is obtained, which is proven to preserve a semi-discrete modified energy. Subsequently, the linearized Crank-Nicolson method is applied for the resulting semi-discrete system to arrive at a fully discrete scheme. The main feature of the new scheme is to form a linear system with a constant coefficient matrix at each time step and produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution. Several numerical results are addressed to confirm accuracy and efficiency of the proposed scheme.
In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, our key ideas mainly follow the
extrapolation/prediction-correction technique and symplectic Runge-Kutta (RK) methods in time combined with the standard Fourier pseudo-spectral method in space. We show that it is uniquely solvable, unconditionally stable and can exactly preserve the momentum of the system. Subsequently, based on the energy quadratization approach and the analogous linearized idea used in the construction of the linear momentum-preserving scheme, the energy-preserving scheme is presented and it is proven to preserve both the discrete mass and quadratic energy. Numerical results are addressed to demonstrate the accuracy and efficiency of the schemes.
In this paper, we analyse a new exponential-type integrator for the nonlinear cubic Schrodinger equation on the $d$ dimensional torus $mathbb T^d$. The scheme has recently also been derived in a wider context of decorated trees in [Y. Bruned and K. S
chratz, arXiv:2005.01649]. It is explicit and efficient to implement. Here, we present an alternative derivation, and we give a rigorous error analysis. In particular, we prove second-order convergence in $H^gamma(mathbb T^d)$ for initial data in $H^{gamma+2}(mathbb T^d)$ for any $gamma > d/2$. This improves the previous work in [Knoller, A. Ostermann, and K. Schratz, SIAM J. Numer. Anal. 57 (2019), 1967-1986]. The design of the scheme is based on a new method to approximate the nonlinear frequency interaction. This allows us to deal with the complex resonance structure in arbitrary dimensions. Numerical experiments that are in line with the theoretical result complement this work.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
