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Jian-Guo Liu

Publications and source records attributed to Jian-Guo Liu.

At least 73 records · Page 4Linked to original sources
arXiv2021

Stripe solitons and lump solutions to a generalized (3 + 1)-dimensional B-type Kadomtsev-Petviashvili equation with variable coefficients in fluid dynamics

Under investigation is a generalized (3 + 1)-dimensional B-type Kadomtsev-Petviashvili equation with variable coefficients in fluid dynamics. Based on the Hirota's bilinear form and the positive quadratic function, abundant lump solutions are obtained.The interaction solutions between lump solutions and other solitons are also presented. Their dynamical behaviors are graphically shown with different choices of the free parameters.

nlin.SI↗
arXiv2021

Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective

In the mean field integrate-and-fire model, the dynamics of a typical neuron within a large network is modeled as a diffusion-jump stochastic process whose jump takes place once the voltage reaches a threshold. In this work, the main goal is to establish the convergence relationship between the regularized process and the original one where in the regularized process, the jump mechanism is replaced by a Poisson dynamic, and jump intensity within the classically forbidden domain goes to infinity as the regularization parameter vanishes. On the macroscopic level, the Fokker-Planck equation for the process with random discharges (i.e. Poisson jumps) are defined on the whole space, while the equation for the limit process is on the half space. However, with the iteration scheme, the difficulty due to the domain differences has been greatly mitigated and the convergence for the stochastic process and the firing rates can be established. Moreover, we find a polynomial-order convergence for the distribution by a re-normalization argument in probability theory. Finally, by numerical experiments, we quantitatively explore the rate and the asymptotic behavior of the convergence for both linear and nonlinear models.

math.PR↗
arXiv2021

Projection method for droplet dynamics on groove-textured surface with merging and splitting

The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After introducing an infinite dimensional manifold with an admissible tangent space on the boundary of the manifold, by Onsager's principle for an obstacle problem, we derive the associated parabolic variational inequalities. These variational inequalities can be used to simulate the contact line dynamics with unavoidable merging and splitting of droplets due to the impermeable obstacle. To efficiently solve the parabolic variational inequality, we propose an unconditional stable explicit boundary updating scheme coupled with a projection method. The explicit boundary updating efficiently decouples the computation of the motion by mean curvature of the capillary surface and the moving contact lines. Meanwhile, the projection step efficiently splits the difficulties brought by the obstacle and the motion by mean curvature of the capillary surface. Furthermore, we prove the unconditional stability of the scheme and present an accuracy check. The convergence of the proposed scheme is also proved using a nonlinear Trotter-Kato's product formula under the pinning contact line assumption. After incorporating the phase transition information at splitting points, several challenging examples including splitting and merging of droplets are demonstrated.

math.NA↗
arXiv2020

Analysis of a fourth order exponential PDE arising from a crystal surface jump process with Metropolis-type transition rates

We analytically and numerically study a fourth order PDE modeling rough crystal surface diffusion on the macroscopic level. We discuss existence of solutions globally in time and long time dynamics for the PDE model. The PDE, originally derived by the second author, is the continuum limit of a microscopic model of the surface dynamics, given by a Markov jump process with Metropolis type transition rates. We outline the convergence argument, which depends on a simplifying assumption on the local equilibrium measure that is valid in the high temperature regime. We provide numerical evidence for the convergence of the microscopic model to the PDE in this regime.

math.AP↗
arXiv2020

Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface

We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the capillary surface. Using a single graph representation, we interpret this geometric motion as a gradient flow on a Hilbert manifold. We propose unconditionally stable first/second order numerical schemes to simulate this geometric motion of the droplet, which is described using motion by mean curvature coupled with moving contact lines. The schemes are based on (i) explicit moving boundaries, which decouple the dynamic updates of the contact lines and the capillary surface, (ii) a semi-Lagrangian method on moving grids and (iii) a predictor-corrector method with a nonlinear elliptic solver upto second order accuracy. For the case of quasi-static dynamics with continuous spatial variable in the numerical schemes, we prove the stability and convergence of the first/second order numerical schemes. To demonstrate the accuracy and long-time validation of the proposed schemes, several challenging computational examples - including breathing droplets, droplets on inhomogeneous rough surfaces and quasi-static Kelvin pendant droplets - are constructed and compared with exact solutions to quasi-static dynamics obtained by desingularized differential-algebraic system of equations (DAEs).

math.NA↗
arXiv2020

Multi-wave, breather wave and lump solutions of the Boiti-Leon-Manna-Pempinelli equation with variable coefficients

In this paper, a variable-coefficient Boiti-Leon-Manna-Pempinelli equation is to be investigated. We obtain abundant multi-wave, breather wave and lump solutions by using the three waves method, the homoclinic breather approach and the Hirota's bilinear method. All solutions have been verified to be correct with the help of Mathematica software. We study the propagation characteristics of these solutions by some three-dimensional images. The obtained results are useful for understanding fluid propagating and incompressible fluid.

nlin.PS↗
arXiv2020

Field model for complex ionic fluids: analytical properties and numerical investigation

In this paper, we consider the field model for complex ionic fluids with an energy variational structure, and analyze the well-posedness to this model with regularized kernels. Furthermore, we deduce the estimate of the maximal density function to quantify the finite size effect. On the numerical side, we adopt a finite volume scheme to the field model, which satisfies the following properties: positivity-preserving, mass conservation and energy dissipation. Besides, series of numerical experiments are provided to demonstrate the properties of the steady state and the finite size effect by showing the equilibrium profiles with different values of the parameter in the kernel.

math.AP↗
arXiv2020

A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration

In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. While the equations exhibit diversified solutions from various numerical observations, the properties of solutions are not yet completely understood, and by far there has been no rigorous numerical analysis work concerning such models. We propose a conservative and conditionally positivity preserving scheme for these Fokker-Planck equations, and we show that in the linear case, the semi-discrete scheme satisfies the discrete relative entropy estimate, which essentially matches the only known long time asymptotic solution property. We also provide extensive numerical tests to verify the scheme properties, and carry out several sets of numerical experiments, including finite-time blowup, convergence to equilibrium and capturing time-period solutions of the variant models.

math.NA↗
arXiv2020

Sensitivity analysis of Burgers' equation with shocks

Generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accuracy, PDE solutions need to have high regularity in the random space, but this is what hyperbolic type problems cannot provide. We provide a counter-argument in this paper, and show that even though the solution profile develops singularities in the random space, which destroys the spectral accuracy of gPC, the physical quantities (such as the shock emergence time, the shock location, and the shock strength) are all smooth functions of the uncertainties coming from both initial data and the wave speed: with proper shifting, the solution's polynomial interpolation approximates the real solution accurately, and the error decays as the order of the polynomial increases. Therefore this work provides a new perspective to "quantify uncertainties" and significantly improves the accuracy of the gPC method with a slight reformulation. We use the Burgers' equation as an example for the thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.

math.NA↗
arXiv2020

A Proximal-Gradient Algorithm for Crystal Surface Evolution

As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method based on the macroscopic partial differential equation, leveraging its formal structure as the gradient flow of the total variation energy, with respect to a weighted $H^{-1}$ norm. This gradient flow structure relates to several metric space gradient flows of recent interest, including 2-Wasserstein flows and their generalizations to nonlinear mobilities. We develop a novel semi-implicit time discretization of the gradient flow, inspired by the classical minimizing movements scheme (known as the JKO scheme in the 2-Wasserstein case). We then use a primal dual hybrid gradient (PDHG) method to compute each element of the semi-implicit scheme. In one dimension, we prove convergence of the PDHG method to the semi-implicit scheme, under general integrability assumptions on the mobility and its reciprocal. Finally, by taking finite difference approximations of our PDHG method, we arrive at a fully discrete numerical algorithm, with iterations that converge at a rate independent of the spatial discretization: in particular, the convergence properties do not deteriorate as we refine our spatial grid. We close with several numerical examples illustrating the properties of our method, including facet formation at local maxima, pinning at local minima, and convergence as the spatial and temporal discretizations are refined.

math.NA↗
arXiv2020

Inbetweening auto-animation via Fokker-Planck dynamics and thresholding

We propose an equilibrium-driven deformation algorithm (EDDA) to simulate the inbetweening transformations starting from an initial image to an equilibrium image, which covers images varying from a greyscale type to a colorful type on plane or manifold. The algorithm is based on Fokker-Planck dynamics on manifold, which automatically cooperates positivity, unconditional stability, mass conservation law, exponentially convergence and also the manifold structure suggested by dataset. The thresholding scheme is adapted for the sharp interface dynamics and is used to achieve the finite time convergence. Using EDDA, three challenging examples, (I) facial aging process, (II) coronavirus disease 2019 (COVID-19) invading/treatment process, and (III) continental evolution process are conducted efficiently.

math.NA↗
arXiv2020

Convergence of Random Batch Method for interacting particles with disparate species and weights

We consider in this work the convergence of Random Batch Method proposed in our previous work [Jin et al., J. Comput. Phys., 400(1), 2020] for interacting particles to the case of disparate species and weights. We show that the strong error is of $O(\sqrtτ)$ while the weak error is of $O(τ)$ where $τ$ is the time step between two random divisions of batches. Both types of convergence are uniform in $N$, the number of particles. The proof of strong convergence follows closely the proof in [Jin et al., J. Comput. Phys., 400(1), 2020] for indistinguishable particles, but there are still some differences: since there is no exchangeability now, we have to use a certain weighted average of the errors; some refined auxiliary lemmas have to be proved compared with our previous work. To show that the weak convergence of empirical measure is uniform in $N$, certain sharp estimates for the derivatives of the backward equations have been used. The weak convergence analysis is also illustrating for the convergence of Random Batch Method for $N$-body Liouville equations.

math.NA↗
arXiv2020

Breather wave and lump-type solutions of new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in incompressible fluid

Under investigation is a new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. The main results are listed as follows: (i) lump solutions; (ii) Interaction solutions between lump wave and solitary waves; (iii) Interaction solutions between lump wave and periodic waves; (iv) Breather wave solutions. Furthermore, graphical representation of all solutions is studied and 9shown in some 3D- and contour plots.

nlin.PS↗
arXiv2020

On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system

We rigorously justify the mean-field limit of a $N$-particle system subject to the Brownian motion and interacting through a Newtonian potential in $\mathbb{R}^3$. Our result leads to a derivation of the Vlasov-Poisson-Fokkker-Planck (VPFP) equation from the microscopic $N$-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and trajectories following the the mean-field is bounded by $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\leq\varepsilon<\frac{1}{36}$) for a system with blob size $N^{-δ}$ ($\frac{1}{3}\leqδ<\frac{19}{54}-\frac{2\varepsilon}{3}$) up to a probability $1-N^{-α}$ for any $α>0$. Moreover, we prove the convergence rate between the empirical measure associated to the particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates crucially rely on the randomness coming from the initial data and from the Brownian motion.

math.AP↗
arXiv2019

Large time behavior, bi-Hamiltonian structure and kinetic formulation for complex Burgers equation

We prove the existence and uniqueness of positive analytical solutions with positive initial data to the mean field equation (the Dyson equation) of the Dyson Brownian motion through the complex Burgers equation with a force term on the upper half complex plane. These solutions converge to a steady state given by Wigner's semicircle law. A unique global weak solution with nonnegative initial data to the Dyson equation is obtained and some explicit solutions are given by Wigner's semicircle laws. We also construct a bi-Hamiltonian structure for the system of the real and imaginary components of the complex Burgers equation (coupled Burgers system). We establish a kinetic formulation for the coupled Burgers system and prove the existence and uniqueness of entropy solutions. The coupled Burgers system in Lagrangian variable naturally leads to two interacting particle systems: Fermi-Pasta-Ulam-Tsingou model with nearest-neighbor interactions, and Calogero-Moser model. These two particle systems yield the same Lagrangian dynamics in the continuum limit.

math.AP↗
arXiv2019

Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects

We study a $4$th order PDE with a $2$nd order correction that is a modification of one suggested by Peter Smereka using the tools of Krug et al (Z. Phys. B., 1994). First, we provide a non-rigorous derivation of the full model derived with a quadratic interaction potential using the rates and methods of the last author with Jonathan Weare (PRE, 2013). Then, we prove the existence of a global in time weak solution by regularizing the equation in a way that allows us to apply the tools of Bernis-Friedman (JDE, 1990). The methods developed here can be applied to a large number of $4$th order degenerate models. In an appendix, we also discuss the global smooth solution with small data in the Weiner algebra framework following recent developments using tools of the second author with Robert Strain (IFB, 2018).

math.AP↗
arXiv2019

Variable-coefficient symbolic computation approach for finding multiple rogue wave solutions of nonlinear system with variable coefficients

In this paper, a variable-coefficient symbolic computation approach is proposed to solve the multiple rogue wave solutions of nonlinear equation with variable coefficients. As an application, a (2+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation is investigated. The multiple rogue wave solutions are obtained and their dynamics features are shown in some 3D and contour plots.

nlin.PS↗