MA3: Recent Advances in Numerical PDE: Fast Algorithms and Applications

Room: Old Main Academic Center 3070

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OrganizersShan Zhao, University of Alabama, and Chuan Li, West Chester University of Pennsylvania

Shan Zhao, University of Alabama

Time: 10:00 am - 10:25 am (CST)

Title: High Order Central Difference FFT Poisson Solvers

Abstract: In this talk, we will introduce an augmented matched interface and boundary (AMIB) method for solving elliptic interface and boundary value problems. The AMIB method utilizes the fictitious values generated by the matched interface and boundary (MIB) scheme to correct the fourth order central difference near interfaces and boundaries. By introducing jump corrected Taylor series and an enlarged linear system in the AMIB, the discrete Laplacian can be efficiently inverted by the fast Fourier transform (FFT) algorithm. For elliptic boundary value problems over cubic domains in 2D and 3D, the AMIB method is the first high order Poisson solver in the literature that can handle Dirichlet, Neumann, and Robin boundary conditions, and their mix combinations. For elliptic interface problems in 2D with curved interfaces, the FFT-based AMIB not only achieves a fourth order convergence in dealing with interfaces and boundaries, but also produces an overall complexity of O(n^2 log n) for a n-by-n uniform grid. Moreover, the AMIB method can provide fourth order accurate approximations to solution gradients and fluxes.

Chuan Li, West Chester University of Pennsylvania

Time: 10:25 am - 10:50 am (CST)

Title: A FFT Accelerated High-order Method for Solving Parabolic equations in Irregular Domains

Abstract: In this work, a recently developed high-order finite difference method is present to solve parabolic equations over irregularly shaped domains in two-and three-dimensions. The unique features of this method consist of a set of corrected finite differences to account for possible jump conditions across the boundary at various types of grids near the boundary for maintaining desired high order of accuracy, an augmented system with fitting structure which allows the system to be solved via the FFT for accelerated matrix-vector multiplication and matrix inversion, and the capability of solving the interested problem with any commonly used boundary conditions (Dirichlet, Neumann and Robin) and their mixed combinations. As a consequence, the proposed method is numerically verified to be unconditionally stable, second order of accuracy in time and fourth order of accuracy in space, and effective to solve a variety of problems with complex boundaries in two-and three-dimensions.

Sanghyun Lee, Florida State University

Time: 10:50 am - 11:05 am (CST)

Title: Physics Preserving Enriched Galerkin Methods

Abstract: In this talk, we consider new finite element methods for solving two different problems. One is coupled flow and transport systems in porous media and the other one is linear elasticity (mechanics) equation. The primary purpose of the study is to develop computationally efficient and robust numerical methods that could be free of both oscillations due to lack of local conservation and locking effects. The locally conservative enriched Galerkin (LF-EG), which will be utilized for solving flow problem is constructed by adding a constant function to each elements based on the classical continuous Galerkin methods. The locking-free enriched Galerkin (LC-EG) adds a vector to the displacement space. These EG methods employs the well-known discontinuous Galerkin (DG) techniques, but the approximation spaces have fewer degrees of freedom than those for the typical DG methods, thus offering an efficient alternative to DG methods. We present a priori error estimates of optimal order. We also demonstrate through some numerical examples that the new method is free of oscillations and locking.

Xiaoming He, Missouri University of Science and Technology

Time: 11:05 am - 11:40 am (CST)

Title: PIFE-PIC: Parallel Immersed-Finite-Element Particle-in-Cell for 3-D Kinetic Simulations of Plasma-Material Interactions

Abstract: In this presentation, we present a recently developed particle simulation to PIFE-PIC, which is a novel three-dimensional (3-D) Parallel Immersed-Finite-Element (IFE) Particle-in-Cell (PIC) simulation model for particle simulations of plasma-material interactions. This framework is based on the recently developed non-homogeneous electrostatic IFE-PIC method, which is designed to handle complex plasma-material interface conditions associated with irregular geometries using a Cartesian-mesh-based PIC. Three-dimensional domain decomposition is utilized for both the electrostatic field solver with IFE and the particle operations in PIC to distribute the computation among multiple processors. A simulation of the orbital-motion-limited (OML) sheath of a dielectric sphere immersed in a stationary plasma is carried out to validate PIFE-PIC and profile the parallel performance of the code package. Parallel efficiency up to approximately 110 super linear speed up was achieved for strong scaling test. Furthermore, a large-scale simulation of plasma charging at a lunar crater containing 2 million PIC cells (10 million FE/IFE cells) and about 1 billion particles, running for 20,000 PIC steps in about 154 wall-clock hours, is presented to demonstrate the high-performance computing capability of PIFE-PIC.

Wenyuan Liao, University of Calgary, Alberta

Time: 11:40 am - 12:05 pm (CST)

Title: A Fast Operator Splitting Method for Solving the Nonlinear Reaction-diffusion Equation with Applications in Mathematical Biology

Abstract: An efficient finite-difference algorithm is presented in this talk for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms, which has been widely used in modeling species populations. The new method is fourth-order accurate in spatial dimension and second-order in time and can be improved to fourth-order in time by Richardson extrapolation. The higher efficiency of the method is obtained through a novel operator splitting technique that converts the multi-dimension finite-difference operator into a sequence of locally one-dimensional problems. The method preserves the unconditional stability as the time integration is originated from the conventional Crank-Nicolson algorithm. Numerical examples are presented to demonstrate the efficiency and accuracy of the new algorithm.