Liwei Xu

Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field

In this paper, central discontinuous Galerkin methods are developed for solving ideal magnetohydrodynamic (MHD) equations. The methods are based on the original central discontinuous Galerkin methods designed for hyperbolic conservation laws on overlapping meshes, and use different discretization for magnetic induction equations. The resulting schemes carry many features of standard central discontinuous Galerkin methods such as high order accuracy and being free of exact or approximate Riemann solvers. And more importantly, the numerical magnetic field is exactly divergence-free. Such property, desired in reliable simulations of MHD equations, is achieved by first approximating the normal component of the magnetic field through discretizing induction equations on the mesh skeleton, namely, the element interfaces. And then it is followed by an element-by-element divergence-free reconstruction with the matching accuracy. Numerical examples are presented to demonstrate the high order accuracy and the robustness of the schemes.

Numerical simulation of three-dimensional nonlinear water waves

We present an accurate and efficient numerical model for the simulation of fully nonlinear (non-breaking), three-dimensional surface water waves on infinite or finite depth. As an extension of the work of Craig and Sulem [19], the numerical method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving surface quantities alone. This is accomplished by introducing the Dirichlet-Neumann operator which is described in terms of its Taylor series expansion in homogeneous powers of the surface elevation. Each term in this Taylor series can be computed efficiently using the fast Fourier transform. An important contribution of this paper is the development and implementation of a symplectic implicit scheme for the time integration of the Hamiltonian equations of motion, as well as detailed numerical tests on the convergence of the Dirichlet-Neumann operator. The performance of the model is illustrated by simulating the long-time evolution of two-dimensional steadily progressing waves, as well as the development of three-dimensional (short-crested) nonlinear waves, both in deep and shallow water.

High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model

In this paper, we consider a one-dimensional fully nonlinear weakly dispersive Green-Naghdi model for shallow water waves over variable bottom topographies. Such model describes a large spectrum of shallow water waves, and it is thus of great importance to design accurate and robust numerical methods for solving it. The governing equations contain mixed spatial and temporal derivatives of the unknowns. They also have still-water stationary solutions which should be preserved in stable numerical simulations. In our numerical approach, we first reformulate the Green-Naghdi equations into balance laws coupled with an elliptic equation. We then propose a family of high order numerical methods which discretize the balance laws with well-balanced central discontinuous Galerkin methods and the elliptic part with continuous finite element methods. Linear dispersion analysis for both the (reformulated) Green-Naghdi system and versions of the proposed numerical scheme is performed when the bottom topography is flat. Numerical tests are presented to illustrate the accuracy and stability of the proposed schemes as well as the capability of the Green-Naghdi model to describe a wide range of shallow water wave phenomena.

Positivity-preserving DG and central DG methods for ideal MHD equations

Ideal MHD equations arise in many applications such as astrophysical plasmas and space physics, and they consist of a system of nonlinear hyperbolic conservation laws. The exact density @r and pressure p should be non-negative. Numerically, such positivity property is not always satisfied by approximated solutions. One can encounter this when simulating problems with low density, high Mach number, or much large magnetic energy compared with internal energy. When this occurs, numerical instability may develop and the simulation can break down. In this paper, we propose positivity-preserving discontinuous Galerkin and central discontinuous Galerkin methods for solving ideal MHD equations by following [X. Zhang, C.-W. Shu, Journal of Computational Physics 229 (2010) 8918-8934]. In one dimension, the positivity-preserving property is established for both methods under a reasonable assumption. The performance of the proposed methods, in terms of accuracy, stability and positivity-preserving property, is demonstrated through a set of one and two dimensional numerical experiments. The proposed methods formally can be of any order of accuracy.

Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations

Ideal magnetohydrodynamic (MHD) equations consist of a set of nonlinear hyperbolic conservation laws, with a divergence-free constraint on the magnetic field. Neglecting this constraint in the design of computational methods may lead to numerical instability or nonphysical features in solutions. In our recent work [F. Li, L. Xu, S. Yakovlev, Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, Journal of Computational Physics 230 (2011) 4828-4847], second and third order exactly divergence-free central discontinuous Galerkin methods were proposed for ideal MHD equations. In this paper, we further develop such methods with higher order accuracy. The novelty here is that the well-established H(div)-conforming finite element spaces are used in the constrained transport type framework, and the magnetic induction equations are extensively explored in order to extract sufficient information to uniquely reconstruct an exactly divergence-free magnetic field. The overall algorithm is local, and it can be of arbitrary order of accuracy. Numerical examples are presented to demonstrate the performance of the proposed methods especially when they are fourth order accurate.

A coupled BEM and FEM for the interior transmission problem in acoustics

Abstract The interior transmission problem (ITP) is a boundary value problem arising in inverse scattering theory, and it has important applications in qualitative methods. In this paper, we propose a coupled boundary element method (BEM) and a finite element method (FEM) for the ITP in two dimensions. The coupling procedure is realized by applying the direct boundary integral equation method to define the so-called Dirichlet-to-Neumann (DtN) mappings. We show the existence of the solution to the ITP for the anisotropic medium. Numerical results are provided to illustrate the accuracy of the coupling method.

Computation of Maxwell’s transmission eigenvalues and its applications in inverse medium problems

We present an iterative method to compute the Maxwell’s transmission eigenvalue problem which has importance in non-destructive testing of anisotropic materials. The transmission eigenvalue problem is first written as a quad-curl eigenvalue problem. Then we show that the real transmission eigenvalues are the roots of a nonlinear function whose value is the generalized eigenvalue of a related self-adjoint quad-curl eigenvalue problem which is computed using a mixed finite element method. A secant method is used to compute the roots of the nonlinear function. Numerical examples are presented to validate the method. Moreover, the method is employed to study the dependence of the transmission eigenvalue on the anisotropy and to reconstruct the index of refraction of an inhomogeneous medium.

Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations

Abstract In this paper, we propose and numerically investigate a family of locally divergence-free central discontinuous Galerkin methods for ideal magnetohydrodynamic (MHD) equations. The methods are based on the original central discontinuous Galerkin methods (SIAM Journal on Numerical Analysis 45 (2007) 2442–2467) for hyperbolic equations, with the use of approximating functions that are exactly divergence-free inside each mesh element for the magnetic field. This simple strategy is to locally enforce a divergence-free constraint on the magnetic field, and it is known that numerically imposing this constraint is necessary for numerical stability of MHD simulations. Besides the designed accuracy, numerical experiments also demonstrate improved stability of the proposed methods over the base central discontinuous Galerkin methods without any divergence treatment. This work is part of our long-term effort to devise and to understand the divergence-free strategies in MHD simulations within discontinuous Galerkin and central discontinuous Galerkin frameworks.

Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis

In this paper, we are concerned with the error analysis for the nite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish an explicit priori error estimates in H 1 and L 2 - norms including both the eect of the truncation of the DtN mapping and that of the numerical discretization. To apply the nite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modication due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results.

Recovering fluid-type motions using Navier-Stokes potential flow

The classical optical flow assumes that a feature point maintains constant brightness across the frames. For fluid-type motions such as smoke or clouds, the constant brightness assumption does not hold, and accurately estimating the motion flow from their images is difficult. In this paper, we introduce a simple but effective Navier-Stokes (NS) potential flow model for recovering fluid-type motions. Our method treats the image as a wavefront surface and models the 3D potential flow beneath the surface. The gradient of the velocity potential describes the motion flow at every voxel. We first derive a general brightness constraint that explicitly models wavefront (brightness) variations in terms of the velocity potential. We then use a series of partial differential equations to separately model the dynamics of the potential flow. To solve for the potential flow, we use the Dirichlet-Neumann Operator (DNO) to simplify the 3D volumetric velocity potential to 2D surface velocity potential. We approximate the DNO via Taylor expansions and develop a Fourier domain method to efficiently estimate the Taylor coefficients. Finally we show how to use the DNO to recover the velocity potential from images as well as to propagate the wavefront (image) over time. Experimental results on both synthetic and real images show that our technique is robust and reliable.