arXiv subjects

Lili Ju

Publications and source records attributed to Lili Ju.

64 records · Page 4Linked to original sources

arXiv2019

High Order Explicit Local Time-Stepping Methods For Hyperbolic Conservation Laws

In this paper we present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta discontinuous Galerkin (RKDG) methods and design LTS algorithms based on strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow spatially variable time step sizes to be used for time integrations in different regions. The proposed algorithms are of predictor-corrector type, in which the interface information along the time direction is first predicted based on the SSP-RK approximations and Taylor expansions, and then the fluxes over the region of interface are corrected to conserve mass exactly at each time step. Following the proposed framework, we detail the corresponding LTS schemes with accuracy up to the fourth order, and prove their conservation property and nonlinear stability for the scalar conservation laws. Numerical experiments are also presented to demonstrate excellent performance of the proposed LTS algorithms.

math.NA↗

arXiv2019

Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation

The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete maximum principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadrature-based finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.

math.NA↗

arXiv2018

Interactive Binary Image Segmentation with Edge Preservation

Binary image segmentation plays an important role in computer vision and has been widely used in many applications such as image and video editing, object extraction, and photo composition. In this paper, we propose a novel interactive binary image segmentation method based on the Markov Random Field (MRF) framework and the fast bilateral solver (FBS) technique. Specifically, we employ the geodesic distance component to build the unary term. To ensure both computation efficiency and effective responsiveness for interactive segmentation, superpixels are used in computing geodesic distances instead of pixels. Furthermore, we take a bilateral affinity approach for the pairwise term in order to preserve edge information and denoise. Through the alternating direction strategy, the MRF energy minimization problem is divided into two subproblems, which then can be easily solved by steepest gradient descent (SGD) and FBS respectively. Experimental results on the VGG interactive image segmentation dataset show that the proposed algorithm outperforms several state-of-the-art ones, and in particular, it can achieve satisfactory edge-smooth segmentation results even when the foreground and background color appearances are quite indistinctive.

cs.CV↗

arXiv2018

An Additive Overlapping Domain Decomposition Method for the Helmholtz Equation

In this paper, we propose and analyze an additive domain decomposition method (DDM) for solving the high-frequency Helmholtz equation with the Sommerfeld radiation condition. In the proposed method, the computational domain is partitioned into structured subdomains along all spatial directions, and each subdomain contains an overlapping region for source transferring. At each iteration all subdomain PML problems are solved completely in parallel, then all horizontal, vertical and corner directional residuals on each subdomain are passed to its corresponding neighbor subdomains as the source for the next iteration. This DDM method is highly scalable in nature and theoretically shown to produce the exact solution for the PML problem defined in ${\mathbb{R}}^2$ in the constant medium case. A slightly modified version of the method for bounded truncated domains is also developed for its use in practice and an error estimate is rigorously proved. Various numerical experiments in two and three dimensions are conducted on the supercomputer "Tianhe-2 Cluster" to verify the theoretical results and demonstrate excellent performance of the proposed method as an iterative solver or a preconditioner.

math.NA↗

arXiv2018

Conservative finite volume schemes for the quasi-geostrophic equation on coastal-conforming unstructured primal-dual meshes

In this paper we propose finite volume schemes for solving the inviscid and viscous quasi-geostrophic equations on coastal-conforming unstructured primal-dual meshes. Several approaches for enforcing the boundary conditions are also explored along with these schemes. The pure transport part in these schemes are shown to conserve the potential vorticity along fluid paths in an averaged sense, and conserve the potential enstrophy up to the time truncation errors. Numerical tests based on the centroidal Voronoi-Delaunay meshes are performed to confirm these properties, and to distinguish the dynamical behaviors of these schemes. Finally some potential applications of these schemes in different situations are discussed.

math.NA↗

arXiv2018

An Exponential Time-Integrator Scheme for Steady and Unsteady Inviscid Flows

An exponential time-integrator scheme of second-order accuracy based on the predictor-corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge-Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.

physics.comp-ph↗

arXiv2018

Fast Spherical Centroidal Voronoi Mesh Generation: A Lloyd-preconditioned LBFGS Method in Parallel

Centroidal Voronoi tessellation (CVT)-based mesh generation is a very effective technique for creating high-quality Voronoi meshes and their dual Delaunay triangulations that often play a crucial role in applications, including ocean and atmospheric simulations using finite volume schemes. In the next generation climate models, the spacing scales change dramatically across the whole sphere and require ultra-high resolution and smooth transitions from coarse to fine grid regions. Thus fast and robust spherical CVT (SCVT) meshing algorithms become highly desirable. In this paper, we first propose a Lloyd-preconditioned limited-memory BFGS method for constructing SCVTs that is also applicable to the construction of CVTs of general domains. This method is then parallelized based on overlapping domain decomposition, enabling excellent scalability on distributed systems. Results of several computational experiments show that the new method could incur computational time costs one order of magnitude smaller compared with some existing methods for generating large-scale highly variable-resolution meshes, while also providing significantly improvements in mesh quality.

math.NA↗

arXiv2017

Overlapping Localized Exponential Time Differencing Methods for Diffusion Problems

The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are proposed to solve the fully discrete problem: the first method is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second method is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergence of the associated iterative solutions to the corresponding fully discrete multidomain solution and to the exact semi-discrete solution is rigorously proved. Numerical experiments are carried out to confirm theoretical results and to compare the performance of the two methods.

math.NA↗

arXiv2017

Linear and Unconditionally Energy Stable Schemes for the binary Fluid-Surfactant Phase Field Model

In this paper, we consider the numerical solution of a binary fluid-surfactant phase field model, in which the free energy contains a nonlinear coupling entropy, a Ginzburg-Landau double well potential, and a logarithmic Flory-Huggins potential. The resulting system consists of two coupled, nonlinear Cahn-Hilliard type equations. We develop a set of first and second order time marching schemes for this system using the "Invariant Energy Quadratization" approach, in particular, the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi-explicitly. Both schemes are linear and lead to symmetric positive definite systems at each time step, thus they can be efficiently solved. We further prove that these schemes are unconditionally energy stable in the discrete sense. Various 2D and 3D numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.

math.NA↗

arXiv2015

An Overlapping Domain Decomposition Preconditioner for the Helmholtz equation

In this paper, based on the overlapping domain decomposition method (DDM) proposed in \cite{Leng2015}, an one step preconditioner is proposed to solve 2D high frequency Helmholtz equation. The computation domain is decomposed in both $x$ and $y$ directions, and the local solution on each subdomain is updated simultaneously in one iteration, thus there is no sweeping along certain directions. In these ways, the overlapping DDM is similar to the popular DDM for Poisson problem. The one step preconditioner simply take the restricted source on each subdomain, solve the local problems and summarize the local solutions on all subdomains including their PML area. The complexity of solving the problem with the preconditioner is $O(N n_{\text{iter}})$, where $n_{\text{iter}}$ is the number of iteration, and it is shown numerically that $n_{\text{iter}}$ is proportional to the number of subdomains in one direction. 2D Helmholtz problem with nearly a billion unknowns are solved efficiently with the preconditioner on massively parallel machines.

math.NA↗