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Hailiang Liu

Publications and source records attributed to Hailiang Liu.

At least 55 records · Page 3Linked to original sources
arXiv2020

Critical thresholds in a nonlocal Euler system with relaxation

We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.

math.AP↗
arXiv2020

On the SAV-DG method for a class of fourth order gradient flows

For a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty-free discontinuous Galerkin (DG) spatial discretization leads to SAV-DG schemes. These schemes are linear and shown unconditionally energy stable. But the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre-evaluate the auxiliary variable in the piecewise polynomial space. As a result, the computational complexity of $O(\mathcal{N}^2)$ reduces to $O(\mathcal{N})$ when exploiting the conjugate gradient (CG) solver. This hybrid SAV-DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV-DG schemes.

math.NA↗
arXiv2020

Selection dynamics for deep neural networks

This paper presents a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth. We study the wellposedness, the large time solution behavior, and the characterization of the steady states of the forward problem. Several useful time-uniform estimates and stability/instability conditions are presented. We state and prove optimality conditions for the inverse deep learning problem, using standard variational calculus, the Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between neural networks, PDE theory, variational analysis, optimal control, and deep learning.

math.AP↗
arXiv2020

Dynamics of many species through competition for resources

This paper is concerned with a mathematical model of competition for resource where species consume noninteracting resources. This system of differential equations is formally obtained by renormalizing the MacArthur's competition model at equilibrium, and agrees with the trait-continuous model studied by Mirrahimi S, Perthame B, Wakano JY [J. Math. Biol. 64(7): 1189-1223, 2012]. As a dynamical system, self-organized generation of distinct species occurs. The necessary conditions for survival are given. We prove the existence of the evolutionary stable distribution (ESD) through an optimization problem and present an independent algorithm to compute the ESD directly. Under certain structural conditions, solutions of the system are shown to approach the discrete ESD as time evolves. The time discretization of the system is proven to satisfy two desired properties: positivity and energy dissipation. Numerical examples are given to illustrate certain interesting biological phenomena.

math.DS↗
arXiv2020

An energy stable and positivity-preserving scheme for the Maxwell-Stefan diffusion system

We develop a new finite difference scheme for the Maxwell-Stefan diffusion system. The scheme is conservative, energy stable and positivity-preserving. These nice properties stem from a variational structure and are proved by reformulating the finite difference scheme into an equivalent optimization problem. The solution to the scheme emerges as the minimizer of the optimization problem, and as a consequence energy stability and positivity-preserving properties are obtained.

math.NA↗
arXiv2020

Critical thresholds in 1D pressureless Euler-Poisson systems with varying background

The Euler Poisson equations describe important physical phenomena in many applications such as semiconductor modeling and plasma physics. This paper is to advance our understanding of critical threshold phenomena in such systems in the presence of different forces. We identify critical thresholds in two damped Euler Poisson systems, with and without alignment, both with attractive potential and spatially varying background state. For both systems, we give respective bounds for subcritical and supercritical regions in the space of initial configuration, thereby proving the existence of a critical threshold for each scenario. Key tools include comparison with auxiliary systems, phase space analysis of the transformed system.

math.AP↗
arXiv2020

A positivity-preserving and energy stable scheme for a quantum diffusion equation

We propose a new fully-discretized finite difference scheme for a quantum diffusion equation, in both one and two dimensions. This is the first fully-discretized scheme with proven positivity-preserving and energy stable properties using only standard finite difference discretization. The difficulty in proving the positivity-preserving property lies in the lack of a maximum principle for fourth order PDEs. To overcome this difficulty, we reformulate the scheme as an optimization problem based on variational structure and use the singular nature of the energy functional near the boundary values to exclude the possibility of non-positive solutions. The scheme is also shown to be mass conservative and consistent.

math.NA↗
arXiv2020

Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems

In this paper, we design, analyze, and numerically validate positive and energy-dissipating schemes for solving the time-dependent multi-dimensional system of Poisson-Nernst-Planck (PNP) equations, which has found much use in the modeling of biological membrane channels and semiconductor devices. The semi-implicit time discretization based on a reformulation of the system gives a well-posed elliptic system, which is shown to preserve solution positivity for arbitrary time steps. The first order (in time) fully-discrete scheme is shown to preserve solution positivity and mass conservation unconditionally, and energy dissipation with only a mild $O(1)$ time step restriction. The scheme is also shown to preserve the steady-state. For the fully second order (in both time and space) scheme with large time steps, solution positivity is restored by a local scaling limiter, which is shown to maintain the spatial accuracy. These schemes are easy to implement. Several three-dimensional numerical examples verify our theoretical findings and demonstrate the accuracy, efficiency, and robustness of the proposed schemes, as well as the fast approach to steady states.

math.NA↗
arXiv2020

Radially symmetric solutions of the ultra-relativistic Euler equations

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure $p$, the spatial part $\underline{u} \in \R^3$ of the dimensionless four-velocity and the particle density $n$. Radially symmetric solutions of these equations are studied. Analytical solutions are presented for the linearized system. For the original nonlinear equations we design and analyze a numerical scheme for simulating radially symmetric solutions in three space dimensions. The good performance of the scheme is demonstrated by numerical examples. In particular, it was observed that the method has the capability to capture accurately the pressure singularity formation caused by shock wave reflections at the origin.

math.NA↗
arXiv2019

Unconditionally energy stable DG schemes for the Swift-Hohenberg equation

The Swift-Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift-Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in [H. Liu and P. Yin, J. Sci. Comput., 77: 467--501, 2018] for the spatial discretization, and the "Invariant Energy Quadratization" method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.

math.NA↗
arXiv2019

A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are $L^2$ stable even without interior penalty. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal $L^2$ error estimate of $O(h^{k+1})$ for polynomials of degree $k$ for semi-discrete DG schemes, and the $L^2$ error of $O(h^{k+1} +(Δt)^2)$ for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift-Hohenberg equation endowed with a decay free energy is presented.

math.NA↗
arXiv2019

Positive and free energy satisfying schemes for diffusion with interaction potentials

In this paper, we design and analyze second order positive and free energy satisfying schemes for solving diffusion equations with interaction potentials. The semi-discrete scheme is shown to conserve mass, preserve solution positivity, and satisfy a discrete free energy dissipation law for nonuniform meshes. These properties for the fully-discrete scheme (first order in time) remain preserved without a strict restriction on time steps. For the fully second order (in both time and space) scheme, we use a local scaling limiter to restore solution positivity when necessary. It is proved that such limiter does not destroy the second order accuracy. In addition, these schemes are easy to implement, and efficient in simulations over long time. Both one and two dimensional numerical examples are presented to demonstrate the performance of these schemes.

math.NA↗
arXiv2019

Unconditional positivity-preserving and energy stable schemes for a reduced Poisson-Nernst-Planck system

The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. In this paper, we design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme, as well as the fast approach to steady states.

math.NA↗
arXiv2019

THIRD ORDER MAXIMUM-PRINCIPLE-SATISFYING DG SCHEMES Third Order Maximum-Principle-Satisfying DG schemes for Convection-Diffusion problems with Anisotropic Diffusivity DIFFUSIVITY

For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection-diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in [Liu and Yu, SIAM J. Sci. Comput. 36(5): A2296{A2325, 2014] when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.

math.NA↗
arXiv2019

Critical Thresholds in One Dimensional Damped Euler-Poisson Systems

This paper is concerned with the critical threshold phenomenon for one dimensional damped, pressureless Euler-Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J., 50:109--157, 2001]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line.

math.AP↗
arXiv2019

General superpositions of Gaussian beams and propagation errors

Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of dimension $m$ in phase space, and show that the tools recently developed in [ H. Liu, O. Runborg, and N. M. Tanushev, Math. Comp., 82: 919--952, 2013] can be applied to obtain the propagation error of order $k^{1- \frac{N}{2}- \frac{d-m}{4}}$, where $N$ is the order of beams and $d$ is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.

math.NA↗
arXiv2019

Mass- and energy-conserved numerical schemes for nonlinear Schrödinger equations

In this paper, we propose a family of time-stepping schemes for approximating general nonlinear Schrödinger equations. The proposed schemes all satisfy both mass conservation and energy conservation. Truncation and dispersion error analyses are provided for each proposed scheme. Efficient fixed-point iterative solvers are also constructed to solve the resulting nonlinear discrete problems. As a byproduct, an efficient one-step implementation of the BDF schemes is obtained as well. Extensive numerical experiments are presented to demonstrate the convergence and the capability of capturing the blow-up phenomenon of the proposed schemes.

math.NA↗