arXiv subjects

Yushun Wang

Publications and source records attributed to Yushun Wang.

At least 19 recordsLinked to original sources

arXiv2026

High-order Lagrange multiplier schemes for general Hamiltonian PDEs

In this paper, we introduce a Lagrange multiplier approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. Unlike the widely used auxiliary variable methods, this novel approach does not require the nonlinear part of the energy to be bounded from below, thereby offering broader applicability. Moreover, this approach preserves the original energy exactly at both the continuous and discrete levels, as opposed to a modified energy preserved by the auxiliary variable methods. Rigorous proofs are provided for the energy conservation and numerical accuracy of all derived schemes. The trade-off for these advantages is the need to solve a nonlinear algebraic equation to determine the Lagrange multiplier. Nevertheless, numerical experiments show that the associated computational cost is generally not dominant, indicating that the new schemes retain computational efficiency comparable to the auxiliary variable-based schemes. Numerical results demonstrate the efficiency, accuracy, and structure-preserving properties of the proposed schemes.

math.NA↗

arXiv2025

Explicit and CPU/GPU parallel energy-preserving schemes for the Klein-Gordon-Schrödinger equations

A highly efficient energy-preserving scheme for univariate conservative or dissipative systems was recently proposed in [Comput. Methods Appl. Mech. Engrg. 425 (2024) 116938]. This scheme is based on a grid-point partitioned averaged vector field (AVF) method, allowing for pointwise decoupling and easy implementation of CPU parallel computing. In this article, we further extend this idea to multivariable coupled systems and propose a dual-partition AVF method that employs a dual partitioning strategy based on both variables and grid points. The resulting scheme is decoupled, energy-preserving, and exhibits greater flexibility. For the Klein-Gordon-Schrödinger equations, we apply the dual-partition AVF method and construct fully explicit energy-preserving schemes with pointwise decoupling, where the computational complexity per time step is $\mathcal{O}(N^d)$, with $d$ representing the problem dimension and $N$ representing the number of grid points in each direction. These schemes not only enable CPU parallelism but also support parallel computing on GPUs by adopting an update strategy based on a checkerboard grid pattern, significantly improving the efficiency of solving high-dimensional problems. Numerical experiments confirm the conservation properties and high efficiency of the proposed schemes.

math.NA↗

arXiv2024

Solution landscape of droplet on rough surfaces: wetting transition and directional transport

Droplets on rough surfaces can exhibit various stationary states that are crucial for designing hydrophobic materials and enabling directional liquid transport. Here we introduce a phase-field saddle dynamics method to construct the solution landscape of wetting transition and directional transport on pillared substrates. By applying this method, we reveal the full range of Cassie-Baxter and Wenzel states, along with the complete wetting transition paths. We further elucidate the mechanisms of directional droplet transport on both hydrophobic and hydrophilic surfaces, demonstrating how surface design can influence directional movement.

cond-mat.soft↗

arXiv2023

A novel high-order linearly implicit and energy-stable additive Runge-Kutta methods for gradient flow models

This paper introduces a novel paradigm for constructing linearly implicit and high-order unconditionally energy-stable schemes for general gradient flows, utilizing the scalar auxiliary variable (SAV) approach and the additive Runge-Kutta (ARK) methods. We provide a rigorous proof of energy stability, unique solvability, and convergence. The proposed schemes generalizes some recently developed high-order, energy-stable schemes and address their shortcomings. On the one other hand, the proposed schemes can incorporate existing SAV-RK type methods after judiciously selecting the Butcher tables of ARK methods \cite{sav_li,sav_nlsw}. The order of a SAV-RKPC method can thus be confirmed theoretically by the order conditions of the corresponding ARK method. Several new schemes are constructed based on our framework, which perform to be more stable than existing SAV-RK type methods. On the other hand, the proposed schemes do not limit to a specific form of the nonlinear part of the free energy and can achieve high order with fewer intermediate stages compared to the convex splitting ARK methods \cite{csrk}. Numerical experiments demonstrate stability and efficiency of proposed schemes.

math.NA↗

arXiv2023

Energy stable and maximum bound principle preserving schemes for the Allen-Cahn equation based on the Saul'yev methods

The energy dissipation law and maximum bound principle are significant characteristics of the Allen-Chan equation. To preserve discrete counterpart of these properties, the linear part of the target system is usually discretized implicitly, resulting in a large linear or nonlinear system of equations. The Fast Fourier Transform (FFT) algorithm is commonly used to solve the resulting linear or nonlinear systems with computational costs of $\mathcal{O}(M^d log M)$ at each time step, where $M$ is the number of spatial grid points in each direction, and $d$ is the dimension of the problem. Combining the Saul'yev methods and the stabilized technique, we propose and analyze novel first- and second-order numerical schemes for the Allen-Cahn equation in this paper. In contrast to the traditional methods, the proposed methods can be solved by components, requiring only $\mathcal{O}(M^d)$ computational costs per time step. Additionally, they preserve the maximum bound principle and original energy dissipation law at the discrete level. We also propose rigorous analysis of their consistency and convergence. Numerical experiments are conducted to confirm the theoretical analysis and demonstrate the efficiency of the proposed methods.

math.NA↗

arXiv2022

Unconditional convergence of conservative spectral Galerkin methods for the coupled fractional nonlinear Klein-Gordon-Schrödinger equations

In this work, two novel classes of structure-preserving spectral Galerkin methods are proposed which based on the Crank-Nicolson scheme and the exponential scalar auxiliary variable method respectively, for solving the coupled fractional nonlinear Klein-Gordon-Schrödinger equation. The paper focuses on the theoretical analyses and computational efficiency of the proposed schemes, the Crank-Nicoloson scheme is proved to be unconditionally convergent and has the maximum-norm boundness of numerical solutions. The exponential scalar auxiliary variable scheme is linearly implicit and decoupled, but lack of the maximum-norm boundness, also, the energy structure has been modified. Subsequently, the efficient implementations of the proposed schemes are introduced in detail. Both the theoretical analyses and the numerical comparisons show that the proposed spectral Galerkin methods have high efficiency in long-time computations.

math.NA↗

arXiv2022

Linearly implicit energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator and convergence analysis

In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. A second-order scheme is first proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence in the $H^1$ norm without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to overcome the difficulty posed by the unavailability of a priori $L^\infty$ estimates of numerical solutions. Based on the integrating factor Runge-Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linear and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.

math.NA↗

arXiv2020

Two novel classes of arbitrary high-order structure-preserving algorithms for canonical Hamiltonian systems

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.

math.NA↗

arXiv2020

Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs

In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such novel strategy is based on the newly developed exponential scalar variable (ESAV) approach that can remove the bounded-from-blew restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products, which make it more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize the symplectic Runge-Kutta method for both solution variables and the auxiliary variable, where the values of internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms are of constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes.

math.NA↗

arXiv2020

An explicit and practically invariants-preserving method for conservative systems

An explicit numerical strategy that practically preserves invariants is derived for conservative systems by combining an explicit high-order Runge-Kutta (RK) scheme with a simple modification of the standard projection approach, which is named the explicit invariants-preserving (EIP) method. The proposed approach is shown to have the same order as the underlying RK method, while the error of invariants is analyzed in the order of $\mathcal{O}\left(h^{2(p+1)}\right),$ where $h$ is the time step and $p$ represents the order of the method. When $p$ is appropriately large, the EIP method is practically invariants-conserving because the error of invariants can reach the machine accuracy. The method is illustrated for the cases of single and multiple invariants, with regard to both ODEs and high-dimensional PDEs. Extensive numerical experiments are presented to verify our theoretical results and demonstrate the superior behaviors of the proposed method in a long time numerical simulation. Numerical results suggest that the fourth-order EIP method preserves much better the qualitative properties of the flow than the standard fourth-order RK method and it is more efficient in practice than the fully implicit integrators.

math.NA↗

arXiv2020

Mass- and energy-preserving exponential Runge-Kutta methods for the nonlinear Schrödinger equation

In this paper, a family of arbitrarily high-order structure-preserving exponential Runge-Kutta methods are developed for the nonlinear Schrödinger equation by combining the scalar auxiliary variable approach with the exponential Runge-Kutta method. By 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.

math.NA↗

arXiv2020

Efficient energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions

We present two novel classes of fully discrete energy-preserving algorithms for the sine-Gordon equation subject to Neumann boundary conditions. The cosine pseudo-spectral method is first used to develop structure-preserving spatial discretizations under two different meshes, which result two finite-dimensional Hamiltonian ODE systems. Then we combine the prediction-correction Crank-Nicolson scheme with the projection approach to arrive at fully discrete energy-preserving methods. Alternatively, we introduce a supplementary variable to transform the initial model into a relaxation system, which allows us to construct structure-preserving algorithms more easily. We then discretize the relaxation system directly by using the cosine pseudo-spectral method in space and the prediction-correction Crank-Nicolson scheme in time to derive a new class of energy-preserving schemes. The proposed methods can be solved effectively by the discrete Cosine transform. Some benchmark examples and numerical comparisons are presented to demonstrate the accuracy, efficiency and superiority of the proposed schemes.

math.NA↗

arXiv2020

Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations

A novel class of explicit high-order energy-preserving methods are proposed for general Hamiltonian partial differential equations with non-canonical structure matrix. When the energy is not quadratic, it is firstly done that the original system is reformulated into an equivalent form with a modified quadratic energy conservation law by the energy quadratization approach. Then the resulting system that satisfies the quadratic energy conservation law is discretized in time by combining explicit high-order Runge-Kutta methods with orthogonal projection techniques. The proposed schemes are shown to share the order of explicit Runge-Kutta method and thus can reach the desired high-order accuracy. Moreover, the methods are energy-preserving and explicit because the projection step can be solved explicitly. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other structure-preserving methods.

math.NA↗

arXiv2020

Arbitrarily high-order structure-preserving schemes for the Gross-Pitaevskii equation with angular momentum rotation in three dimensions

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.

math.NA↗

arXiv2020

A linearized and conservative Fourier pseudo-spectral method for the damped nonlinear Schrödinger equation in three dimensions

In this paper, we propose a linearized Fourier pseudo-spectral method, which preserves the total mass and energy conservation laws, for the damped nonlinear Schrödinger equation in three dimensions. With the aid of the semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, an optimal $L^2$-error estimate for the proposed method without any restriction on the grid ratio is established by analyzing the real and imaginary parts of the error function. Numerical results are addressed to confirm our theoretical analysis.

math.NA↗

arXiv2020

A linearly implicit structure-preserving scheme for the Camassa-Holm equation based on multiple scalar auxiliary variables approach

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 gradient 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.

math.NA↗

arXiv2020

A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation

In this paper, we generalize the exponential energy-preserving integrator proposed in the recent paper [SIAM J. Sci. Comput. 38(2016) A1876-A1895] for conservative systems, which now becomes linearly implicit by further utilizing the idea of the scalar auxiliary variable approach. Comparing with the original exponential energy-preserving integrator which usually leads to a nonlinear algebraic system, our new method only involve a linear system with constant coefficient matrix. Taking the nonlinear Klein-Gordon equation for example, we derive the concrete energy-preserving scheme and demonstrate its high efficiency through numerical experiments.

math.NA↗

arXiv2019

Structure-preserving algorithms for multi-dimensional fractional Klein-Gordon-Schrödinger equation

This paper aims to construct structure-preserving numerical schemes for multi-dimensional space fractional Klein-Gordon-Schrödinger equation, which are based on the newly developed partitioned averaged vector field methods. First, we derive an equivalent equation, and reformulate the equation as a canonical Hamiltonian system by virtue of the variational derivative of the functional with fractional Laplacian. Then, we develop a semi-discrete conservative scheme via using the Fourier pseudo-spectral method to discrete the equation in space direction. Further applying the partitioned averaged vector field methods on the temporal direction gives a class of fully-discrete schemes that can preserve the mass and energy exactly. Numerical examples are provided to confirm our theoretical analysis results at last.

math.NA↗