Table of Content

20 March 2019, Volume 1 Issue 1

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Foreword by the Editor-in-Chief

Chi-Wang Shu

2019, 1(1):  1-1.  doi:10.1007/s42967-019-0010-2

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Preface

Benqi Guo, Heping Ma, Jie Shen, Chi-Wang Shu, Li-Lian Wang

2019, 1(1):  3-4.  doi:10.1007/s42967-019-0011-1

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Analysis and Approximation of Gradient Flows Associated with a Fractional Order Gross-Pitaevskii Free Energy

Mark Ainsworth, Zhiping Mao

2019, 1(1):  5-19.  doi:10.1007/s42967-019-0008-9

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We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross-Pitaevskii free energy functional and some basic properties of the solution. The equation reduces to the Allen-Cahn or Cahn-Hilliard equations in the case where the mass tends to zero and an integer order derivative is used in the energy. We study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard case. In particular, we show that, analogous to the Cahn-Hilliard case, the solutions consist of regions in which the solution is a piecewise constant (whose value depends on the mass and the fractional order) separated by an interface whose width is independent of the mass and the fractional derivative. However, if the average value of the initial data exceeds some threshold (which we determine explicitly), then the solution will tend to a single constant steady state.

A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions

Zachary Grant, Sigal Gottlieb, David C. Seal

2019, 1(1):  21-59.  doi:10.1007/s42967-019-0001-3

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High-order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multi-derivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second-derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second-derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufcient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is p=6, and defne an optimization procedure that allows us to fnd such SSP methods. Several types of these methods are presented and their efciency compared. Finally, these methods are tested on several PDEs to demonstrate the beneft of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.

Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model

Ying He, Jie Shen

2019, 1(1):  61-80.  doi:10.1007/s42967-019-0004-0

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We consider in this paper numerical approximation of a nonlinear fluid-structure interaction (FSI) model with a fixed interface. We construct a new class of pressure-correction schemes for the FSI problem, and prove rigorously that they are unconditionally stable. These schemes are computationally very efficient, as they lead to, at each time step, a coupled linear elliptic system for the velocity and displacement in the whole region and a discrete Poisson equation in the fluid region.

A Cubic H³-Nonconforming Finite Element

Jun Hu, Shangyou Zhang

2019, 1(1):  81-100.  doi:10.1007/s42967-019-0009-8

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The lowest degree of polynomial for a finite element to solve a 2kth-order elliptic equation is k. The Morley element is such a finite element, of polynomial degree 2, for solving a fourth-order biharmonic equation. We design a cubic H³-nonconforming macro-element on two-dimensional triangular grids, solving a sixth-order tri-harmonic equation. We also write down explicitly the 12 basis functions on each macro-element. A convergence theory is established and verified by numerical tests.

Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

Yong Liu, Chi-Wang Shu, Mengping Zhang

2019, 1(1):  101-116.  doi:10.1007/s42967-019-0006-y

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In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in[18] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be k + 2 when piecewise P^k polynomials with k ≥ 1 are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise P^k polynomials with arbitrary k ≥ 1. Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of k + 1 and k + 2, respectively. We also prove, under suitable choice of initial discretization, a (2k + 1)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results.

Multiscale Radiative Transfer in Cylindrical Coordinates

Wenjun Sun, Song Jiang, Kun Xu

2019, 1(1):  117-139.  doi:10.1007/s42967-019-0007-x

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The radiative transfer equations in cylindrical coordinates are important in the application of inertial confinement fusion. In comparison with the equations in Cartesian coordinates, an additional angular derivative term appears in the cylindrical case. This term adds great difficulty for a numerical scheme to keep the conservation of total energy. In this paper, based on weighting factors, the angular derivative term is properly discretized, and the interface fluxes in the radial r-direction depend on such a discretization as well. A unified gas kinetic scheme (UGKS) with asymptotic preserving property for the gray radiative transfer equations is constructed in cylindrical coordinates. The current UGKS can naturally capture the radiation diffusion solution in the optically thick regime with the cell size being much larger than photon's mean free path. At the same time, the current UGKS can present accurate solutions in the optically thin regime as well. Moreover, it is a finite volume method with total energy conservation. Due to the scale-dependent time evolution solution for the interface flux evaluation, the scheme can cover multiscale transport mechanism seamlessly. The cylindrical hohlraum tests in inertial confinement fusion are used to validate the current approach, and the solutions are compared with implicit Monte Carlo result.

Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

Hui Xie, Xuejun Xu

2019, 1(1):  141-165.  doi:10.1007/s42967-019-0005-z

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In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale H(div) problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this H(div) problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart-Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale H(div) system is bounded by C(1 + (H²/δ²))(1 + log⁴(H/h)), where δ denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.