arXiv subjects

Chaoyu Quan

Publications and source records attributed to Chaoyu Quan.

At least 19 recordsLinked to original sources

arXiv2026

Energy Dissipation Analysis of Implicit-Explicit Linear Multistep Methods for Gradient Flows Using General Multipliers

A unified framework is proposed to establish the energy dissipation of implicit-explicit linear multistep methods (IMEX-LMMs) for gradient flows, based on general multipliers that are linear combinations of first-order differences of numerical solutions. A generalized Dahlquist's theory is developed to establish the energy dissipation of IMEX-LMMs. It is shown that given an IMEX-LMM, to find a multiplier ensuring the energy dissipation is relaxed to solve a linear programming that can be easily solved. Within this framework, two specific multipliers are discovered to establish the energy dissipation of the sixth-order IMEX backward differentiation formula (IMEX-BDF6) method and a seventh-order IMEX weighted and shifted BDF method, and a new eighth-order energy-dissipative IMEX-LMM is provided. To the best of our knowledge, these are the first energy-dissipation results for the IMEX-BDF6 method and the IMEX-LMMs of order higher than six. In addition, this framework can be used directly to establish the $L^2$- or $H^1$-stability of general LMMs for linear parabolic problems. Numerical experiments illustrate the temporal accuracy and energy dissipation of these methods.

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arXiv2026

Energy Dissipation Analysis of Implicit-Explicit Linear Multistep Methods for Gradient Flows Using a Simple Multiplier

This paper proposes a theoretical framework for establishing the energy dissipation of general implicit-explicit linear multistep methods (IMEX-LMMs) for gradient flows, by constructing a dissipative modified energy consisting of the original energy and a non-negative quadratic modification. We first test IMEX-LMMs with a simple multiplier, the first-order time difference of numerical solutions. Then, it is shown that the associated non-negative quadratic modification can be constructed if and only if two generating polynomials (corresponding to the LMM) are positive on $[-1,1]$. Based on this, the modified energy is proved to decay over time under a mild time-step restriction depending on the lower bounds of the associated generating polynomials. As a consequence, the energy dissipation of the well-known backward differentiation formula methods up to fifth order can be obtained straightforwardly. Furthermore, we construct for the first time (to the best of our knowledge) a sixth-order energy-dissipative IMEX-LMM and also prove the sixth-order barrier of energy-dissipative IMEX-LMMs when testing the simple multiplier. Some numerical experiments are conducted to verify our theoretical results.

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arXiv2026

A semi-generating function approach to the stability of implicit-explicit multistep methods for nonlinear parabolic equations

The rigorous stability analysis of high-order implicit-explicit linear multistep (IELM) methods for nonlinear parabolic equations by using discrete energy arguments is a long standing open issue due to their non-A-stability property. A novel semi-generating function approach combined with a global discrete energy analysis is suggested for the stability and convergence of general IELM methods in solving nonlinear parabolic equations. Inspired from the Grenander-Szeg\H{o} theorem for Toeplitz matrices, the semi-generating function approach is used to handle the three groups of discrete coefficients via three complex polynomials on the unit circle. A unified theoretical framework is then presented to establish the unconditional stability of IELM methods if the minimum eigenvalue of composite convolution kernels for the implicit part is properly large and the spectral norm bound of composite convolution kernels for the explicit part is properly small. An indicator, called implicit-explicit controllability intensity, is then introduced to evaluate the degree of controllability of the implicit part over the explicit part. Some of the existing IELM methods, up to fifth-order time accuracy, are revisited and compared by computing the associated implicit-explicit controllability intensities such that one can choose an IELM method or proper parameter to maintain the unconditional stability for a specific nonlinear parabolic model. We also propose a new parameterized class of IELM methods, up to the ninth-order time accuracy, which satisfy the a priori settings of our theory and have a large value of the implicit-explicit controllability intensity by choosing a proper parameter so that they would be well suited for a wide class of nonlinear parabolic problems.

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arXiv2026

A generalized matrix-valued Allen--Cahn model and its numerical solution

This paper introduces a generalized matrix-valued Allen--Cahn model, where the unknown matrix-valued field belongs to $\mathbb{R}^{m_1\times m_2}$ with dimension $m_1\geq m_2$. By taking different values of $m_1$ and $m_2$, this model covers the classical scalar-valued, vector-valued, and square-matrix-valued Allen--Cahn equations. At the continuous level, the proposed model is proven to admit a unique solution satisfying the maximum bound principle (MBP) and the energy dissipation law. At the discrete level, a class of arbitrarily high-order exponential time differencing Runge-Kutta (ETDRK) schemes is investigated that preserve the MBP unconditionally. Moreover, we prove that the first- and second-order ETDRK schemes satisfy the discrete energy dissipation unconditionally, while third- and higher-order schemes preserve the discrete energy dissipation under suitable time-step constraints. The proof of sharp convergence order in time is provided. Numerical experiments are carried out to confirm our theoretical results.

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arXiv2025

Unconditional energy dissipation of Strang splitting for the matrix-valued Allen-Cahn equation

The energy dissipation property of the Strang splitting method was first demonstrated for the matrix-valued Allen-Cahn (MAC) equation under restrictive time-step constraints [J. Comput. Phys. 454, 110985, 2022]. In this work, we eliminate this limitation through a refined stability analysis framework, rigorously proving that the Strang splitting method preserves the energy dissipation law unconditionally for arbitrary time steps. The refined proof hinges on a precise estimation of the double-well potential term in the modified energy functional. Leveraging this unconditional energy dissipation property, we rigorously establish that the Strang splitting method achieves global-in-time $H^1$-stability, preserves determinant boundedness, and maintains second-order temporal convergence for the matrix-valued Allen-Cahn equation. To validate these theoretical findings, we conduct numerical experiments confirming the method's energy stability and determinant bound preservation for the MAC equation.

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arXiv2025

Stability and convergence of multi-product expansion splitting methods with negative weights for semilinear parabolic equations

The operator splitting method has been widely used to solve differential equations by splitting the equation into more manageable parts. In this work, we resolves a long-standing problem -- how to establish the stability of multi-product expansion (MPE) splitting methods with negative weights. The difficulty occurs because negative weights in high-order MPE method cause the sum of the absolute values of weights larger than one, making standard stability proofs fail. In particular, we take the semilinear parabolic equation as a typical model and establish the stability of arbitrarily high-order MPE splitting methods with positive time steps but possibly negative weights. Rigorous convergence analysis is subsequently obtained from the stability result. Several numerical experiments validate the stability and accuracy of various high-order MPE splitting methods, highlighting their efficiency and robustness.

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arXiv2025

Maximum bound preservation of exponential integrators for Allen-Cahn equations

We develop and analyze a class of arbitrarily high-order, maximum bound preserving time-stepping schemes for solving Allen-Cahn equations. These schemes are constructed within the iterative framework of exponential integrators, combined with carefully chosen numerical quadrature rules, including the Gauss-Legendre quadrature rule and the left Gauss-Radau quadrature rule. Notably, the proposed schemes are rigorously proven to unconditionally preserve the maximum bound without requiring any additional postprocessing techniques, while simultaneously achieving arbitrarily high-order temporal accuracy. A thorough error analysis in the $L^2$ norm is provided. Numerical experiments validate the theoretical results, demonstrate the effectiveness of the proposed methods, and highlight that an inappropriate choice of quadrature rules may violate the maximum bound principle, leading to incorrect dynamics.

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arXiv2025

Stability and Convergence of Strang Splitting Method for the Allen-Cahn Equation with Homogeneous Neumann Boundary Condition

The Strang splitting method has been widely used to solve nonlinear reaction-diffusion equations, with most theoretical convergence analysis assuming periodic boundary conditions. However, such analysis presents additional challenges for the case of homogeneous Neumann boundary condition. In this work the Strang splitting method with variable time steps is investigated for solving the Allen--Cahn equation with homogeneous Neumann boundary conditions. Uniform $H^k$-norm stability is established under the assumption that the initial condition $u^0$ belongs to the Sobolev space $H^k(Ω)$ with integer $k\ge 0$, using the Gagliardo--Nirenberg interpolation inequality and the Sobolev embedding inequality. Furthermore, rigorous convergence analysis is provided in the $H^k$-norm for initial conditions $u^0 \in H^{k+6}(Ω)$, based on the uniform stability. Several numerical experiments are conducted to verify the theoretical results, demonstrating the effectiveness of the proposed method.

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arXiv2025

Convergence analysis of a solver for the linear Poisson--Boltzmann model

This work investigates the convergence of a domain decomposition method for the Poisson-Boltzmann model that can be formulated as an interior-exterior transmission problem. To study its convergence, we introduce an interior-exterior constant providing an upper bound of the $L^2$ norm of any harmonic function in the interior, and establish a spectral equivalence for related Dirichlet-to-Neumann operators to estimate the spectrum of interior-exterior iteration operator. This analysis is nontrivial due to the unboundedness of the exterior subdomain, which distinguishes it from the classical analysis of the Schwarz alternating method with nonoverlapping bounded subdomains. It is proved that for the linear Poisson-Boltzmann solvent model in reality, the convergence of interior-exterior iteration is ensured when the relaxation parameter lies between 0 and 2. This convergence result interprets the good performance of ddLPB method developed in [SIAM Journal on Scientific Computing, 41 (2019), pp. B320-B350] where the relaxation parameter is set to 1. Numerical simulations are conducted to verify our convergence analysis and to investigate the optimal relaxation parameter for the interior-exterior iteration.

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arXiv2024

Maximum bound principle preserving and energy decreasing exponential time differencing schemes for the matrix-valued Allen-Cahn equation

This work delves into the exponential time differencing (ETD) schemes for the matrix-valued Allen-Cahn equation. In fact, the maximum bound principle (MBP) for the first- and second-order ETD schemes is presented in a prior publication [SIAM Review, 63(2), 2021], assuming a symmetric initial matrix field. Noteworthy is our novel contribution, demonstrating that the first- and second-order ETD schemes for the matrix-valued Allen-Cahn equation -- both being linear schemes -- unconditionally preserve the MBP, even in instances of nonsymmetric initial conditions. Additionally, we prove that these two ETD schemes preserve the energy dissipation law unconditionally for the matrix-valued Allen-Cahn equation. Some numerical examples are presented to verify our theoretical results and to simulate the evolution of corresponding matrix fields.

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arXiv2024

A Relaxed Lagrange Multiplier Approach for Phase Field Models

This paper introduces a novel relaxed Lagrange multiplier (RLM) method for designing efficient and energy-stable numerical schemes for phase-field models. The proposed approach reformulates the original model by introducing a time-dependent Lagrange multiplier (r(t)), whose evolution is governed by an ordinary differential equation involving a relaxation parameter (\alpha>0). This relaxation technique slows down the evolution of the multiplier, thereby improving the consistency between the modified and original systems after temporal discretization, while avoiding the nonlinear algebraic equations arising in the original Lagrange multiplier (LM) method. We construct first- and second-order temporal discretizations based on the RLM approach and rigorously prove their unconditional stability with respect to a modified energy. Furthermore, we establish the boundedness of the numerical Lagrange multiplier and show that it converges to 1 as the relaxation parameter tends to zero. We also demonstrate that the modified energy converges to the original energy as the time-step size (\tau) tends to zero. Extensive numerical experiments confirm the theoretical results, demonstrating second-order accuracy, unconditional energy stability, and significantly improved computational efficiency, with approximately half the computational cost of comparable scalar auxiliary variable (SAV) and LM methods. The RLM method resolves the consistency issue associated with the SAV approach without requiring the nonlinear free energy to be bounded from below, thereby providing a robust and highly efficient alternative for simulating phase-field models.

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arXiv2024

Maximum bound principle and original energy dissipation of arbitrarily high-order rescaled exponential time differencing Runge-Kutta schemes for Allen--Cahn equations

The energy dissipation law and the maximum bound principle are two critical physical properties of the Allen--Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law, most apply to a modified form of energy. In this work, we demonstrate that, when the nonlinear term of the Allen--Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge--Kutta (ETDRK) schemes preserve the original energy dissipation property, under a mild step-size constraint. Additionally, we guarantee the Lipschitz condition on the nonlinear term by applying a rescaling post-processing technique, which ensures that the numerical solution unconditionally satisfies the maximum bound principle. Consequently, our proposed schemes maintain both the original energy dissipation law and the maximum bound principle and can achieve arbitrarily high-order accuracy. We also establish an optimal error estimate for the proposed schemes. Some numerical experiments are carried out to verify our theoretical results.

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arXiv2023

$H^1$-norm stability and convergence of an L2-type method on nonuniform meshes for subdiffusion equation

This work establishes $H^1$-norm stability and convergence for an L2 method on general nonuniform meshes when applied to the subdiffusion equation. Under mild constraints on the time step ratio $ρ_k$, such as $0.4573328\leq ρ_k\leq 3.5615528$ for $k\geq 2$, the positive semidefiniteness of a crucial bilinear form associated with the L2 fractional-derivative operator is proved. This result enables us to derive long time $H^1$-stability of L2 schemes. These positive semidefiniteness and $H^1$-stability properties hold for standard graded meshes with grading parameter $1 5/α-1$. To the best of our knowledge, this study is the first work on $H^1$-norm stability and convergence of L2 methods on general nonuniform meshes for the subdiffusion equation.

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arXiv2022

Roundoff error problem in L2-type methods for time-fractional problems

Roundoff error problems have occurred frequently in interpolation methods of time-fractional equations, which can lead to undesirable results such as the failure of optimal convergence. These problems are essentially caused by catastrophic cancellations. Currently, a feasible way to avoid these cancellations is using the Gauss--Kronrod quadrature to approximate the integral formulas of coefficients rather than computing the explicit formulas directly for example in the L2-type methods. This nevertheless increases computational cost and arises additional integration errors. In this work, a new framework to handle catastrophic cancellations is proposed, in particular, in the computation of the coefficients for standard and fast L2-type methods on general nonuniform meshes. We propose a concept of $δ$-cancellation and then some threshold conditions ensuring that $δ$-cancellations will not happen. If the threshold conditions are not satisfied, a Taylor-expansion technique is proposed to avoid $δ$-cancellation. Numerical experiments show that our proposed method performs as accurate as the Gauss--Kronrod quadrature method and meanwhile much more efficient. This enables us to complete long time simulations with hundreds of thousands of time steps in short time.

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arXiv2022

Linear scaling computation of forces for the domain-decomposition linear Poisson--Boltzmann method

The Linearized Poisson--Boltzmann (LPB) equation is a popular and widely accepted model for accounting solvent effects in computational (bio-) chemistry. In the present article we derive the analytical forces of the domain-decomposition-based ddLPB-method with vdW or SAS surface. We present an efficient strategy to compute the forces and its implementation, allowing linear scaling of the method with respect to the number of atoms using the fast multipole method (FMM). Numerical tests illustrates the accuracy of the computation of the analytical forces and compares efficiency with other available methods.

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arXiv2022

Long time $H^1$-stability of fast L2-1$_σ$ method on general nonuniform meshes for subdiffusion equations

In this work, the global-in-time $H^1$-stability of a fast L2-1$_σ$ method on general nonuniform meshes is studied for subdiffusion equations, where the convolution kernel in the Caputo fractional derivative is approximated by sum of exponentials. Under some mild restrictions on time stepsize, a bilinear form associated with the fast L2-1$_σ$ formula is proved to be positive semidefinite for all time. As a consequence, the uniform global-in-time $H^1$-stability of the fast L2-1$_σ$ schemes can be derived for both linear and semilinear subdiffusion equations, in the sense that the $H^1$-norm is uniformly bounded as the time tends to infinity. To the best of our knowledge, this appears to be the first work for the global-in-time $H^1$-stability of fast L2-1$_σ$ scheme on general nonuniform meshes for subdiffusion equations. Moreover, the sharp finite time $H^1$-error estimate for the fast L2-1$_σ$ schemes is reproved based on more delicate analysis of coefficients where the restriction on time step ratios is relaxed comparing to existing works.

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arXiv2022

Global-in-time $H^1$-stability of L2-1$_σ$ method on general nonuniform meshes for subdiffusion equation

In this work the L2-1$_σ$ method on general nonuniform meshes is studied for the subdiffusion equation. When the time step ratio is no less than $0.475329$, a bilinear form associated with the L2-1$_σ$ fractional-derivative operator is proved to be positive semidefinite and a new global-in-time $H^1$-stability of L2-1$_σ$ schemes is then derived under simple assumptions on the initial condition and the source term. In addition, the sharp $L^2$-norm convergence is proved under the constraint that the time step ratio is no less than $0.475329$.

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arXiv2022

Stability and convergence of Strang splitting. Part II: tensorial Allen-Cahn equations

We consider the second-order in time Strang-splitting approximation for vector-valued and matrix-valued Allen-Cahn equations. Both the linear propagator and the nonlinear propagator are computed explicitly. For the vector-valued case, we prove the maximum principle and unconditional energy dissipation for a judiciously modified energy functional. The modified energy functional is close to the classical energy up to $\mathcal O(τ)$ where $τ$ is the splitting step. For the matrix-valued case, we prove a sharp maximum principle in the matrix Frobenius norm. We show modified energy dissipation under very mild splitting step constraints. We exhibit several numerical examples to show the efficiency of the method as well as the sharpness of the results.

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