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Workshop on Nonlinear Partial Differential Equations: Analysis, Computation and Applications
(7 - 10 Mar 2012)
~ Abstracts ~
Multiscale methods and analysis for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime
Weizhu Bao, National University of Singapore
In this talk, I will review our recent works on numerical methods and analysis for solving the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the KG equation. Rigorious error estimates show that the EWI spectral method show much better temporal resolution than the FDTD methods for the KG equation in the nonrelativistic limit regime. In order to design a multiscale method for the KG equation, we establish error estimates of FDTD and EWI spectral methods for the nonlinear Schrodinger equation perturbed with a wave operator. Finally, a multiscale method is presented for discretizing the nonlinear KG equation in the nonrelativistic limit regime based on large-small amplitude wave decompostion. This multiscale method converges uniformly in spatial/temporal discretization with respect to the small parameter for the nonlinear KG equation in the nonrelativistic limite regime. Finally, applications to several high oscillatory dispersive partial differential equations will be discussed.
« Back... Dimension reduction for dipolar Gross-Pitaevskii equation
Yongyong Cai, National University of Singapore
Bose-Einstein condensate (BEC) with dipole-dipole interaction has received considerable research interests recently. At zero temperature, the dipolar BEC is well-described by a three dimensional (3D) Gross-Pitaevskii equation (GPE) with a nonlocal dipole-dipole interaction term. With strongly anisotropic confining potentials, the three dimensional dipolar GPE will result in effective two-dimensional (2D) equation for disk-shaped BEC or effective one-dimensional (1D) equation for cigar-shaped BEC . Upon a new formulation of the 3D dipolar GPE, we obtain the corresponding effective lower dimensional equations. Ground state and dynamics for the 3D and lower dimensional equations are discussed. Extensions to multi-layered dipolar condensate will be also discussed.
« Back... Some recent results on simulating quantum systems with Coulomb interaction
Xuan Chun Dong, National University of Singapore
I begin by the computation for the Schrödinger-Poisson-Slater (SPS) system, which serves as a local single particle approximation of the Hartree-Fock equations for a quantum system of N electrons interacting via Coulomb potential. The focus lies on the comparisons of different numerical methods for computing the ground state and dynamics, and the emphasis is put on the various approaches for the Hartree potential, with a conclusion that the spectral approach based on sine bases is the best choice in 3D. Also, simplified spectral methods for spherically symmetric SPS system are presented, which reduce the memory and computational load significantly. Next, an application of the results obtained for SPS system to compute the relativistic Hartree equation, which models a quantum system of N bosons with relativistic dispersion interacting through Coulomb potential, is presented. Some intriguing phenomena observed in the numerical experiments are shown.
« Back... Optimization, adaptation, and initialization of biological transport networks
Dan Hu, Shanghai Jiao Tong University, China
Blood vessel systems and leaf venations are typical biological transport networks. The energy consumption for such a system to perform its biological functions is determined by the network structure. In the first part, I will talk about the optimized structure of the network, and show how the blood vessel system adapts itself to an optimized structure. Mathematical models are used to predict pruning vessels in the experiments of zebra fish. In the second part, I will discuss our recent discovery and our mathematical model on the initialization of transport networks. Simulation results can illustrate how a tree-like structure is obtained from a continuum adaptation equation set, and how loops can exist in our model. Possible further application of this model will also be discussed.
« Back... A conservative scheme for solving the coupled surface-bulk convection-diffusion equations
Ming-Chih Lai, National Chiao Tung University, Taiwan
Many problems in biological, physical and material sciences involve solving partial differential equations in complex domains or deformable interfaces. In particular, the underlying material quantities in the bulk domain may couple with the one in the interface through adsorption of mass from the bulk to the interface and the desorption of mass from the interface to the bulk. In this talk, we present a conservative scheme for solving the coupled bulk-surface concentration equations and test it for different setups. The total mass is well conserved numerically comparing with the literature. The effect of solubility of surfactant on drop deformation in a shear flow is investigated in detail.
« Back... One field formulation and a simple stable explicit interface advancing scheme for fluid structure interaction
Jie Liu, National University of Singapore
We develop a one field formulation for the fluid structure (FS) interaction problem. The unknowns are (u;p;v), being the fluid velocity, fluid pressure and solid velocity. This one field formulation uses Arbitrary Lagrangian Eulerian (ALE) description for the fluid and Lagrangian description for the solid. It automatically enforces the simultaneous continuities of both velocity and traction along the FS interface. We present a first order in time fully discrete scheme when the flow is incompressible Navier-Stokes and when the solid is elastic. The interface position is determined by first order extrapolation so that the generation of the fluid mesh and the computation of (u;p;v) are decoupled. This explicit interface advancing enables us to save half of the unknowns comparing with traditional monolithic schemes. When the solid has convex strain energy (e.g. linear elastic), we prove that the total energy of the fluid and the solid at time t^{n} is bounded by the total energy at time t^{n-1}. Like in the continuous case, the fluid mesh velocity which is used in ALE description does not enter into the stability bound. Surprisingly, the nonlinear convection term in the Navier-Stokes equations plays a crucial role to stabilize the scheme and the stability result does not apply to Stokes flow. As the nonlinear convection term is treated semi-implicitly, in each time step, we only need to solve a linear system (and only once) which involves merely (u;p;v) if the solid is linear elastic. Two numerical tests including the benchmark test of Navier-Stokes flow past a Saint Venant-Kirchhoff elastic bar are performed. In addition to the stability, we also confirm the first order temporal accuracy of our explicit interface advancing scheme.
« Back... The string method for the study of rare events
Weiqing Ren, National University of Singapore and Institute for High Performance Computing
Many problems arising from applied sciences can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimensions. Well known examples include nucleation events during phase transitions, conformational changes of bio-molecules, chemical reactions, etc. The system is confined in metastable states for long times before making transitions from one metastable state to another. The disparity of time scales makes the study of transition pathways and transition rates a very challenging task. In this talk, I will present the string method for the study of complex energy landscapes and the associated transition events.
« Back... An Arnoldi-based method for model order reduction of delay systems
Yangfeng Su, Fudan University, China
For large scale time-delay systems, Michiels, Jarlebring, and Meerbergen gave an efficient Arnoldi-based model order reduction method. To reduce the order from $n$ to $k$, their method needs $nk^2/2$ memory. In this talk, we propose a new implementation for the Arnoldi process, which is numerical stable and needs only $nk$ memory.
This is a joint work with Yujie Zhang.
« Back... Analysis of wetting and contact angle hysteresis on rough surfaces
Xiao-Ping Wang, The Hong Kong University of Science and Technology, Hong Kong
We analyze the wetting hysteresis on rough and chemically patterned surfaces from a phase-field model for immiscible two phase fluid. In the slow motion, the dynamic equations of the interface as well as the contact angle can be derived from the matched asymptotic expansions. The contact angle hysteresis can then be studied from these equations.
« Back... Nonlinear integro-differential equations in Laminar-turbulent transition and combustion instability
Xuesong Wu, Imperial College, UK
Transition of a laminar shear flow to a turbulent state is a complex nonlinear phenomenon. Crucial to its understanding and prediction is the nonlinear evolution of initially small-amplitude perturbations, or the so-called instability modes. In the past two decades, some interesting integro-differential equations were derived from the Navier-Stokes equations by using sophisticated asymptotic analysis. Further new equations arise in recent studies of acoustic instability of premixed combustion. These equations generalise well-known Ginzbug-Landau and Michelson-Sivashinsky equations. A common novel feature is that they all consist of history dependent nonlinear terms. The physical context of these equations and their main mathematical properties will be reviewed with a call to applied and numerical analysts for more attention to analytical and computational aspects of these equations. The focus of the talk will be on a set of equations which describe the interaction and coupling of the small- and large-scale motions. Numerical solutions will be presented to demonstrate that they are able to capture spectral broadening and randomization of free shear layers as observed in experiments.
« Back... A Cartesian grid method for the reaction-diffusion equation on complex domains
Wenjun Ying, Shanghai Jiao Tong University, China
In this talk, I will describe a Cartesian grid method for the time-dependent nonlinear reaction-diffusion partial differential equation on complex domains. In this work, the grid lines are not required to be aligned with the boundary of the computational domain while the method has second-order accuracy and does not have time step restriction in time integration since the evolution equation is advanced implicitly. The Cartesian grid method is especially suited for moving interface problems. This work is an extension and further development of the kernel free boundary integral method for elliptic partial differential equations.
« Back... Some recent developments of the Mumford-Shah image segmentation model
Andy Yip, National University of Singapore
The Mumford-Shah and Chan-Vese models have been successfully applied to solve various image segmentation problems. In this talk, I will present some extensions for solving texture segmentation and additive segmentation problems
« Back... Phase and large deformation of hydrogel
Hui Zhang, Beijing Normal University, China
Here we investage phase and large deformation of hydrogel of soft matter. It is shown that the time-dependment Ginzburg-Landau equation with special Flory-Huggins free energy to simulatie the phase of MMC(Macromolecular microsphere compositegel) gel. On the other hand, we will set up the model of large deformation of general hydrogel. Numerical experiments are presented to show large deformation rate of cylindrcal gel.
« Back... Recent numerical and theoretical developments in inverse obstacle scattering
Jun Zou, The Chinese University of Hong Kong, Hong Kong
In this talk we will discuss some recent developments in inverse obstacle scattering problems. We will first introduce some important mathematical advances on the unique identifiability of scatterers, then present several new numerical methods for the problems and their numerical simulations. The work was substantially supported by Hong Kong RGC grants (projects 405110 and 404611).
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