arXiv subjects

Chaoyu Quan

Publications and source records attributed to Chaoyu Quan.

At least 37 records · Page 2Linked to original sources

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 $0.475329$.

math.NA↗

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.

math.NA↗

arXiv2022

Stability and convergence of Strang splitting. Part I: Scalar Allen-Cahn equation

We consider a class of second-order Strang splitting methods for Allen-Cahn equations with polynomial or logarithmic nonlinearities. For the polynomial case both the linear and the nonlinear propagators are computed explicitly. We show that this type of Strang splitting scheme is unconditionally stable regardless of the time step. Moreover we establish strict energy dissipation for a judiciously modified energy which coincides with the classical energy up to $\mathcal O(τ)$ where $τ$ is the time step. For the logarithmic potential case, since the continuous-time nonlinear propagator no longer enjoys explicit analytic treatments, we employ a second order in time two-stage implicit Runge--Kutta (RK) nonlinear propagator together with an efficient Newton iterative solver. We prove a maximum principle which ensures phase separation and establish energy dissipation law under mild restrictions on the time step. These appear to be the first rigorous results on the energy dissipation of Strang-type splitting methods for Allen-Cahn equations.

math.NA↗

arXiv2022

A decreasing upper bound of energy for time-fractional phase-field equations

In this article, we study the energy dissipation property of time-fractional Allen-Cahn equation. We propose a decreasing upper bound of energy that decreases with respect to time and coincides with the original energy at $t = 0$ and as $t$ tends to $\infty$. This upper bound can also be viewed as a nonlocal-in-time modified energy, the summation of the original energy and an accumulation term due to the memory effect of time fractional derivative. In particular, this indicates that the original energy indeed decays w.r.t. time in a small neighborhood at $t=0$. We illustrate the theory mainly with the time-fractional Allen-Cahn equation, but it could be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation. On the discrete level, the first-order L1 and second-order L2 schemes for time-fractional Allen-Cahn equation have similar decreasing modified energies, so that the stability can be established. Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.

math.NA↗

arXiv2021

An F-modulated stability framework for multistep methods

We introduce a new $\mathbf F$-modulated energy stability framework for general linear multistep methods. We showcase the theory for the two dimensional molecular beam epitaxy model with no slope selection which is a prototypical gradient flow with Lipschitz-bounded nonlinearity. We employ a class of representative BDF$k$, $2\le k \le 5$ discretization schemes with explicit $k^{\mathrm{th}}$-order extrapolation of the nonlinear term. We prove the uniform-in-time boundedness of high Sobolev norms of the numerical solution. The upper bound is unconditional, i.e. regardless of the size of the time step. We develop a new algebraic theory and calibrate nearly optimal and \emph{explicit} maximal time step constraints which guarantee monotonic $\mathbf F$-modulated energy dissipation.

math.NA↗

arXiv2021

The BDF3/EP3 scheme for MBE with no slope selection is stable

We consider the classical molecular beam epitaxy (MBE) model with logarithmic type potential known as no-slope-selection. We employ a third order backward differentiation (BDF3) in time with implicit treatment of the surface diffusion term. The nonlinear term is approximated by a third order explicit extrapolation (EP3) formula. We exhibit mild time step constraints under which the modified energy dissipation law holds. We break the second Dahlquist barrier and develop a new theoretical framework to prove unconditional uniform energy boundedness with no size restrictions on the time step. This is the first unconditional result for third order BDF methods applied to the MBE models without introducing any stabilization terms or fictitious variables. A novel theoretical framework is also established for the error analysis of high order methods.

math.NA↗

arXiv2021

Energy Stable L2 Schemes for Time-Fractional Phase-Field Equations

In this article, the energy stability of two high-order L2 schemes for time-fractional phase-field equations is established. We propose a reformulation of the L2 operator and also some new properties on it. We prove the energy boundedness (by initial energy) of an L2 scalar auxiliary variable scheme for any phase-field equation and the fractional energy law of an implicit-explicit L2 Adams--Bashforth scheme for the Allen--Cahn equation. The stability analysis is based on a new Cholesky decomposition proposed recently by some of us.

math.NA↗

arXiv2021

Negative time splitting is stable

For high order (than two) in time operator-splitting methods applied to dissipative systems, a folklore issue is the appearance of negative-time/backward-in-time linear evolution operators such as backward heat operators interwoven with nonlinear evolutions. The stability of such methods has remained an ensuing difficult open problem. In this work we consider a fourth order operator splitting discretization for the Allen-Cahn equation which is a prototypical high order splitting method with negative time-stepping, i.e. backward in time integration for the linear parabolic part. We introduce a new theoretical framework and prove uniform energy stability and higher Sobolev stability. This is the first strong stability result for negative time stepping operator-splitting methods.

math.NA↗

arXiv2021

On the energy stability of Strang-splitting for Cahn-Hilliard

We consider a Strang-type second order operator-splitting discretization for the Cahn-Hilliard equation. We introduce a new theoretical framework and prove uniform energy stability of the numerical solution and persistence of all higher Sobolev norms. This is the first strong stability result for second order operator-splitting methods for the Cahn-Hilliard equation. In particular we settle several long-standing open issues in the work of Cheng, Kurganov, Qu and Tang \cite{Tang15}.

math.NA↗

arXiv2021

The operator-splitting method for Cahn-Hilliard is stable

We prove energy stability of a standard operator-splitting method for the Cahn-Hilliard equation. We establish uniform bound of Sobolev norms of the numerical solution and convergence of the splitting approximation. This is the first unconditional energy stability result for the operator-splitting method for the Cahn-Hilliard equation. Our analysis can be extended to many other models.

math.NA↗

arXiv2021

On a sinc-type MBE model

We introduce a new sinc-type molecular beam epitaxy model which is derived from a cosine-type energy functional. The landscape of the new functional is remarkably similar to the classical MBE model with double well potential but has the additional advantage that all its derivatives are uniformly bounded. We consider first order IMEX and second order BDF2 discretization schemes. For both cases we quantify explicit time step constraints for the energy dissipation which is in good accord with the practical numerical simulations. Furthermore we introduce a new theoretical framework and prove unconditional uniform energy boundedness with no size restrictions on the time step. This is the first unconditional (i.e. independent of the time step size) result for semi-implicit methods applied to the phase field models without introducing any artificial stabilization terms or fictitious variables.

math.NA↗

arXiv2021

Energy-dissipation for time-fractional phase-field equations

We consider a class of time-fractional phase field models including the Allen-Cahn and Cahn-Hilliard equations. We establish several weighted positivity results for functionals driven by the Caputo time-fractional derivative. Several novel criterions are examined for showing the positive-definiteness of the associated kernel functions. We deduce strict energy-dissipation for a number of non-local energy functionals, thereby proving fractional energy dissipation laws.

math.AP↗

arXiv2021

Sharp convergence to steady states of Allen-Cahn

In our recent work we found a surprising breakdown of symmetry conservation: using standard numerical discretization with very high precision the computed numerical solutions corresponding to very nice initial data may converge to completely incorrect steady states due to the gradual accumulation of machine round-off error. We solved this issue by introducing a new Fourier filter technique for solutions with certain band gap properties. To further investigate the attracting basin of steady states we classify in this work all possible bounded nontrivial steady states for the Allen-Cahn equation. We characterize sharp dependence of nontrivial steady states on the diffusion coefficient and prove strict monotonicity of the associated energy. In particular, we establish a certain self-replicating property amongst the hierarchy of steady states and give a full classification of their energies and profiles. We develop a new modulation theory and prove sharp convergence to the steady state with explicit rates and profiles.

math.AP↗

arXiv2021

On a parabolic sine-Gordon model

We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional stabilization term. We rigorously prove the energy stability of the numerical schemes under nearly sharp and quite mild time step constraints. We demonstrate the striking similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.

math.AP↗

arXiv2021

On symmetry breaking of Allen-Cahn

We consider numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational solutions may converge to completely incorrect steady states. This happens for very smooth initial data and state-of-the-art algorithms. We analyze this phenomenon and showcase the resolution of this problem by a new symmetry-preserving filter technique. We develop a new theoretical framework and rigorously prove the convergence to steady states for the filtered solutions.

math.AP↗

arXiv2021

Numerical Energy Dissipation for Time-Fractional Phase-Field Equations

The numerical integration of phase-field equations is a delicate task which needs to recover at the discrete level intrinsic properties of the solution such as energy dissipation and maximum principle. Although the theory of energy dissipation for classical phase field models is well established, the corresponding theory for time-fractional phase-field models is still incomplete. In this article, we study certain nonlocal-in-time energies using the first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete fractional energy law and a discrete weighted energy law. The extension for a $(2-α)$-order L1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate that the energy bound is preserved for the L1 schemes with nonuniform time steps. Several numerical experiments are carried to verify our theoretical analysis.

math.NA↗

arXiv2020

How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations

There exists a well defined energy for classical phase-field equations under which the dissipation law is satisfied, i.e., the energy is non-increasing with respect to time. However, it is not clear how to extend the energy definition to time-fractional phase-field equations so that the corresponding dissipation law is still satisfied. In this work, we will try to settle this problem for phase-field equations with Caputo time-fractional derivative, by defining a nonlocal energy as an averaging of the classical energy with a time-dependent weight function. As the governing equation exhibits both nonlocal and nonlinear behavior, the dissipation analysis is challenging. To deal with this, we propose a new theorem on judging the positive definiteness of a symmetric function, that is derived from a special Cholesky decomposition. Then, the nonlocal energy is proved to be dissipative under a simple restriction of the weight function. Within the same framework, the time fractional derivative of classical energy for time-fractional phase-field models can be proved to be always nonpositive.

math.AP↗

arXiv2018

A Domain Decomposition Method for the Poisson-Boltzmann Solvation Models

In this paper, a domain decomposition method for the Poisson-Boltzmann (PB) solvation model that is widely used in computational chemistry is proposed. This method, called ddLPB for short, solves the linear Poisson-Boltzmann (LPB) equation defined in $\mathbb R^3$ using the van der Waals cavity as the solute cavity. The Schwarz domain decomposition method is used to formulate local problems by decomposing the cavity into overlapping balls and only solving a set of coupled sub-equations in balls. A series of numerical experiments is presented to test the robustness and the efficiency of this method including the comparisons with some existing methods. We observe exponential convergence of the solvation energy with respect to the number of degrees of freedom which allows this method to reach the required level of accuracy when coupling with quantum mechanical descriptions of the solute.

math.NA↗