Article
Mathematics, Applied
Yabing Sun, Quan Zhou
Summary: This work introduces a second order exponential time differencing Runge-Kutta (ETDRK) scheme for solving the epitaxial growth model without slope selection. The preservation of mass conservation and rigorous error estimate are proven for the ETDRK2 scheme, with several numerical experiments verifying its accuracy. Coarsening dynamics with small diffusion coefficients are also simulated to demonstrate theoretical energy decay rate and growth rates of surface roughness and mound width.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Yayun Fu, Zhuangzhi Xu
Summary: In this paper, we propose an explicit energy-preserving Runge-Kutta scheme for solving the nonlinear Schrodinger equation. By introducing an auxiliary variable and using the projection technique, we construct a stable discrete scheme and provide stability results. Extensive numerical examples demonstrate the high accuracy, energy preservation, and effectiveness of the proposed scheme in long time simulation.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
Zhaohui Fu, Jiang Yang
Summary: This paper proves that the second order ETDRK scheme unconditionally preserves the energy dissipation law for a family of phase field models, and demonstrates the accuracy and stability of the ETDRK2 scheme through numerical simulations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
S. Blanes, F. Casas, A. Escorihuela-Tomas
Summary: Different families of Runge-Kutta-Nystrom (RKN) symplectic splitting methods of order 8 are presented and tested for second-order systems of ordinary differential equations. They demonstrate better efficiency than state-of-the-art symmetric compositions of 2nd-order symmetric schemes and RKN splitting methods of orders 4 and 6, particularly for medium to high accuracy. In some specific examples, they are even more efficient than extrapolation methods for high accuracies and integrations over relatively short time intervals.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Yonglei Fang, Xianfa Hu, Jiyong Li
Summary: This paper introduces the explicit pseudo two-step exponential Runge-Kutta (EPTSERK) methods for numerical integration of first-order ordinary differential equations, analyzing the order conditions and global errors, and demonstrating convergence and efficiency through numerical experiments.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Jianguo Huang, Lili Ju, Yuejin Xu
Summary: In this paper, an efficient exponential integrator finite element method is proposed for solving a class of semilinear parabolic equations in rectangular domains. The method performs spatial discretization using finite element approximation and time integration using explicit exponential Runge-Kutta approach. The proposed method successfully derives error estimates in H1 norm and provides fast solution process through simultaneous diagonalization of mass and coefficient matrices.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Severiano Gonzalez-Pinto, Domingo Hernandez-Abreu, Maria S. Perez-Rodriguez, Arash Sarshar, Steven Roberts, Adrian Sandu
Summary: This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge-Kutta (GARK) methods. New IMIM-GARK splitting methods are developed and tested using parabolic systems. Classical splitting methods can be studied in this unified framework.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics
Shanqin Chen
Summary: This paper extends WENO methods with large time-stepping SSP integrating factor Runge-Kutta time discretization to solve general nonlinear two-dimensional (2D) problems. The efficient evaluation of the matrix exponential operator is addressed through the Krylov subspace projection method when applying IF temporal discretization for PDEs on high spatial dimensions. Numerical examples demonstrate the accuracy and large time-step size of the method.
Article
Computer Science, Interdisciplinary Applications
Mostafa Abbaszadeh, Mehdi Dehghan
Summary: This contribution presents a new high-order numerical algorithm for solving cubic-quintic complex Ginzburg-Landau equations, which is based on problem decomposition and the application of different numerical techniques to obtain numerical approximations.
ENGINEERING WITH COMPUTERS
(2022)
Article
Mathematics, Applied
Adam Preuss, Jessica Lipoth, Raymond J. Spiteri
Summary: Many mathematical models of natural phenomena are described by partial differential equations with additive contributions from different physical processes. This study compares the performance of two different additive splitting techniques and finds that dynamic linearization generally outperforms physics-based splitting.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics
Weaam Alhejaili, Alvaro H. Salas, Samir A. El-Tantawy
Summary: In this study, novel analytical approximations for both unforced and forced pendulum-cart system oscillators were obtained. The ansatz method with equilibrium point and the Krylov-Bogoliubov-Mitropolsky (KBM) method were implemented to analyze the problems. The obtained results showed good coincidence with the RK4 numerical approximation.
Article
Mathematics, Applied
Raynold Tan, Andrew Ooi, Richard D. Sandberg
Summary: This study compares different combinations of spatial discretization methods in two-dimensional wavenumber space and examines the impact of a hybrid finite difference/Fourier spectral scheme on dispersion and dissipation properties in the linearized compressible Navier-Stokes Equations. The results show that the accuracy of the hybrid scheme varies compared to a full finite difference method depending on the wave propagation angle, and numerical anisotropy is quantified using group velocities and phase velocities.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jinwei Fang, Rui Zhan
Summary: This paper analyzes the order conditions of high order explicit exponential Runge-Kutta methods for stiff semilinear delay differential equations, deriving stiff order conditions up to order five. It is further demonstrated that the method can stiffly converge of order p even if the order conditions hold in a weak form. Numerical tests confirm the superiority of high order methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Julius O. Ehigie, Vu Thai Luan, Solomon A. Okunuga, Xiong You
Summary: A new class of exponentially fitted two-derivative diagonally implicit Runge-Kutta (EFTDDIRK) methods is constructed and derived for the numerical solution of differential equations with oscillatory solutions. The methods exhibit superconvergent properties and have been optimized for improved accuracy and efficiency. Numerical experiments demonstrate the superiority of the newly derived methods compared to existing trigonometric/exponential fitting implicit Runge-Kutta methods.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Hao Chen, Hai-Wei Sun
Summary: This paper presents a numerical method for solving the multidimensional Allen-Cahn equations with theoretical proof of preserving the discrete maximum principle and error estimation. In practical computations, the algorithm can be implemented by computing linear systems and matrix exponentials of Toeplitz matrices to reduce complexity. Numerical examples demonstrate the effectiveness and efficiency of the proposed scheme in two and three spatial dimensions.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Software Engineering
Xiao Li, Lili Ju, Thi-Thao-Phuong Hoang
Summary: This paper introduces a localized exponential time differencing method based on overlapping domain decomposition, which has been successfully applied to parallel computations for extreme-scale numerical simulations. By studying the numerical solutions of a class of semilinear parabolic equations and conducting numerical experiments, the effectiveness and accuracy of the proposed algorithm have been demonstrated.
BIT NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Jingwei Li, Lili Ju, Yongyong Cai, Xinlong Feng
Summary: Compared with the Cahn-Hilliard equation, the classic Allen-Cahn equation satisfies the maximum bound principle but does not conserve mass over time. This paper proposes a modified Allen-Cahn equation with a nonlocal Lagrange multiplier term to enforce mass conservation and introduces first and second order stabilized exponential time differencing schemes for time integration. The schemes are shown to preserve the maximum bound principle at the time discrete level and are validated through various numerical experiments in two and three dimensions.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Physics, Mathematical
Wei Leng, Lili Ju
Summary: In this paper, a novel diagonal sweeping domain decomposition method with source transfer is proposed and tested for solving the high-frequency Helmholtz equation. The method improves the efficiency of solving the system by employing a more efficient subdomain solving order. Numerical experiments demonstrate the effectiveness of the proposed method in achieving the exact solution of the global PML problem in a constant medium.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Physics, Mathematical
Xucheng Meng, Thi-Thao-Phuong Hoang, Zhu Wang, Lili Ju
Summary: This paper investigates the performance of the ETD method applied to the rotating shallow water equations and proposes a localized approach to accelerate ETD simulations. By dividing the original problem into smaller subdomain problems and solving them locally, the proposed approach speeds up the calculation of matrix exponential vector products, showing great potential for high-performance computing.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Lili Ju, Xiao Li, Zhonghua Qiao, Jiang Yang
Summary: This paper investigates high-order MBP-preserving time integration schemes using the integrating factor Runge-Kutta (IFRK) method and shows that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems, although not strong stability preserving. The efficiency and convergence of these numerical schemes are theoretically proved and numerically verified, with simulations conducted on 2D and 3D long-time evolutional behaviors, including a model that is not a typical gradient flow as the Allen-Cahn type of equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Artificial Intelligence
Yuesong Wang, Keyang Luo, Zhuo Chen, Lili Ju, Tao Guan
Summary: This paper proposes a simple deep learning based method, DeepFusion, to improve the results from traditional multi-view stereo methods. By designing a neural network to predict depth confidences and using plane sweeps to obtain matching costs, the method effectively fuses depth maps and demonstrates its effectiveness and generalization ability through experiments on multiple datasets.
KNOWLEDGE-BASED SYSTEMS
(2021)
Article
Mathematics, Applied
Anthony Gruber, Max Gunzburger, Lili Ju, Yuankai Teng, Zhu Wang
Summary: The study introduces a dimension reduction method based on Nonlinear Level set Learning, which is effective in handling sparsely sampled data and provides a higher accuracy and information richness in the reduced representation.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Kun Jiang, Lili Ju, Jingwei Li, Xiao Li
Summary: The paper discusses numerical solutions of the modified Allen-Cahn equation with nonlocal and local effects with a Lagrange multiplier that satisfies the maximum bound principle and conserves mass. Linear stabilizing techniques are used to reformulate the model equation, and first- and second-order exponential time differencing schemes are constructed for time integration. The proposed schemes are proven to unconditionally preserve the maximum bound principle and mass conservation in the time discrete sense, with error estimates derived under certain regularity assumptions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Engineering, Multidisciplinary
Rihui Lan, Wei Leng, Zhu Wang, Lili Ju, Max Gunzburger
Summary: This study investigates the parallel performance of two ocean models using exponential time differencing (ETD) methods, which show potential for improving computational efficiency in numerical simulations. Benchmark tests demonstrate the effectiveness of ETD methods for simulating real-world geophysical flows in ocean modeling.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Jingwei Li, Xiao Li, Lili Ju, Xinlong Feng
Summary: This paper studies a stabilized IFRK (sIFRK) method by combining the IFRK method with linear stabilization technique, successfully deriving sufficient conditions for MBP preservation and constructing high-order accurate sIFRK schemes. Extensive numerical experiments demonstrate the unconditionally MBP-preserving properties of these schemes.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Qiang Du, Lili Ju, Xiao Li, Zhonghua Qiao
Summary: The paper explores a time-invariant maximum bound principle for a class of semilinear parabolic equations, studying conditions on L and f that lead to this principle for dynamic systems of infinite or finite dimensions. The development of first- and second-order accurate temporal discretization schemes that unconditionally satisfy the maximum bound principle in the time-discrete setting is discussed, along with error estimates and energy stability analyses. The abstract framework and analysis techniques provide an effective approach to studying the maximum bound principle of the abstract evolution equation and its numerical discretization schemes, which are demonstrated through numerical experiments.
Proceedings Paper
Computer Science, Artificial Intelligence
Keyang Luo, Tao Guan, Lili Ju, Yuesong Wang, Zhuo Chen, Yawei Luo
2020 IEEE/CVF CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION (CVPR)
(2020)
Proceedings Paper
Computer Science, Artificial Intelligence
Yuesong Wang, Tao Guan, Zhuo Chen, Yawei Luo, Keyang Luo, Lili Ju
2020 IEEE/CVF CONFERENCE ON COMPUTER VISION AND PATTERN RECOGNITION (CVPR)
(2020)
Article
Mathematics, Applied
Lili Ju, Wei Leng, Zhu Wang, Shuai Yuan
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2020)
Article
Mathematics, Applied
Thi-Thao-Phuong Hoang, Lili Ju, Wei Leng, Zhu Wang
MATHEMATICS OF COMPUTATION
(2020)
