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Register a new userArbitrary high-order linear structure-preserving schemes for the regularized long-wave equation
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Added by
Chaolong Jiang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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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 variable 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.
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.
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.
We study a family of structure-preserving deterministic numerical schemes for Lindblad equations, and carry out detailed error analysis and absolute stability analysis. Both error and absolute stability analysis are validated by numerical examples.
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.
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