اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدArbitrarily high-order conservative schemes for the generalized Korteweg-de Vries equation
89
0
0.0
(
0
)
تأليف
Kai Yang
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This paper proposes a new class of arbitarily high-order conservative numerical schemes for the generalized Korteweg-de Vries (KdV) equation. This approach is based on the scalar auxiliary variable (SAV) method. The equation is reformulated into an equivalent system by introducing a scalar auxiliary variable, and the energy is reformulated into a sum of two quadratic terms. Therefore, the quadratic preserving Runge-Kutta method will preserve both the mass and the reformulated energy in the discrete time flow. With the Fourier pseudo-spectral spatial discretization, the scheme conserves the first and third invariant quantities (momentum and energy) exactly in the fully discrete sense. The discrete mass possesses the precision of the spectral accuracy.
قيم البحث
اقرأ أيضاً
In this paper, we develop a new class of high-order energy-preserving schemes for the Korteweg-de Vries equation based on the quadratic auxiliary variable technique, which can conserve the original energy of the system. By introducing a quadratic aux
iliary variable, the original system is reformulated into an equivalent form with a modified quadratic energy, where the way of the introduced variable naturally produces a quadratic invariant of the new system. A class of Runge-Kutta methods satisfying the symplectic condition is applied to discretize the reformulated model in time, arriving at arbitrarily high-order schemes, which naturally conserve the modified quadratic energy and the produced quadratic invariant. Under the consistent initial condition, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In order to match the high order precision of time, the Fourier pseudo-spectral method is employed for spatial discretization, resulting in fully discrete energy-preserving schemes. To solve the proposed nonlinear schemes effectively, we present a very efficient practically-structure-preserving iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed schemes. This new class of numerical strategies is rather general so that they can be readily generalized for any conservative systems with a polynomial energy.
This paper proposes a new class of mass or energy conservative numerical schemes for the generalized Benjamin-Ono (BO) equation on the whole real line with arbitrarily high-order accuracy in time. The spatial discretization is achieved by the pseudo-
spectral method with the rational basis functions, which can be implemented by the Fast Fourier transform (FFT) with the computational cost $mathcal{O}( Nlog(N))$. By reformulating the spatial discretized system into the different equivalent forms, either the spatial semi-discretized mass or energy can be preserved exactly under the continuous time flow. Combined with the symplectic Runge-Kutta, with or without the scalar auxiliary variable reformulation, the fully discrete energy or mass conservative scheme can be constructed with arbitrarily high-order temporal accuracy, respectively. Our numerical results show the conservation of the proposed schemes, and also the superior accuracy and stability to the non-conservative (Leap-frog) scheme.
We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the general gradient flow model into an
equivalent gradient flow model with a quadratic free energy and a modified mobility. Given solutions up to $t_n=n Delta t$ with $Delta t$ the time step size, we linearize the EQ-reformulated gradient flow model in $(t_n, t_{n+1}]$ by extrapolation. Then we employ an algebraically stable Runge-Kutta method to discretize the linearized model in $(t_n, t_{n+1}]$. Then we use the Fourier pseudo-spectral method for the spatial discretization to match the order of accuracy in time. The resulting fully discrete scheme is linear, unconditionally energy stable, uniquely solvable, and may reach arbitrarily high order. Furthermore, we present a family of linear schemes based on prediction-correction methods to complement the new linear schemes. Some benchmark numerical examples are given to demonstrate the accuracy and efficiency of the schemes.
In this paper, we design a novel linearized and momentum-preserving Fourier pseudo-spectral scheme to solve the Rosenau-Korteweg de Vries equation. With the aid of a new semi-norm equivalence between the Fourier pseudo-spectral method and the finite
difference method, a prior bound of the numerical solution in discrete $L^{infty}$-norm is obtained from the discrete momentum conservation law. Subsequently, based on the energy method and the bound of the numerical solution, we show that, without any restriction on the mesh ratio, the scheme is convergent with order $O(N^{-s}+tau^2)$ in discrete $L^infty$-norm, where $N$ is the number of collocation points used in the spectral method and $tau$ is the time step. Numerical results are addressed to confirm our theoretical analysis.
In this paper, we design a novel class of arbitrarily high-order structure-preserving numerical schemes for the time-dependent Gross-Pitaevskii equation with angular momentum rotation in three dimensions. Based on the idea of the scalar auxiliary var
iable approach which is proposed in the recent papers [J. Comput. Phys., 416 (2018) 353-407 and SIAM Rev., 61(2019) 474-506] for developing energy stable schemes for gradient flow systems, we firstly reformulate the Gross-Pitaevskii equation into an equivalent system with a modified energy conservation law. The reformulated system is then discretized by the Gauss collocation method in time and the standard Fourier pseudo-spectral method in space, respectively. We show that the proposed schemes can preserve the discrete mass and modified energy exactly. Numerical results are addressed to verify the efficiency and high-order accuracy of the proposed schemes.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
