Fourth Order Exponential Time Differencing Method with Local Discontinuous Galerkin Approximation for Coupled Nonlinear Schrödinger Equations

This paper studies a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.     

A Discrete-Ordinate Discontinuous-Streamline Diffusion Method for the Radiative Transfer Equation

The radiative transfer equation (RTE) arises in many different areas of science and engineering. In this paper, we propose and investigate a discrete-ordinate discontinuous-streamline diffusion (DODSD) method for solving the RTE, which is a combination of the discrete-ordinate technique and the discontinuous-streamline diffusion method. Different from the discrete-ordinate discontinuous Galerkin (DODG) method for the RTE, an artificial diffusion parameter is added to the test functions in the spatial discretization. Stability and error estimates in certain norms are proved. Numerical results show that the proposed method can lead to a more accurate approximation in comparison with the DODG method.     

An Adaptive Moving Mesh Discontinuous Galerkin Method for the Radiative Transfer Equation

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, frequency, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical examples.     

An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation

An iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter λ=O(1), and a special initial guess for λ≪1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of λ=O(1) and λ≪1. The (m+1)th order of accuracy for L² and mth order of accuracy for H¹for P^m elements are numerically obtained.     

A Weighted Runge-Kutta Discontinuous Galerkin Method for 3D Acoustic and Elastic Wave-Field Modeling

Numerically solving 3D seismic wave equations is a key requirement for forward modeling and inversion. Here, we propose a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method for 3D acoustic and elastic wave-field modeling. For this method, the second-order seismic wave equations in 3D heterogeneous anisotropic media are transformed into a first-order hyperbolic system, and then we use a discontinuous Galerkin (DG) solver based on numerical-flux formulations for spatial discretization. The time discretization is based on an implicit diagonal Runge-Kutta (RK) method and an explicit iterative technique, which avoids solving a large-scale system of linear equations. In the iterative process, we introduce a weighting factor. We investigate the numerical stability criteria of the 3D method in detail for linear and quadratic spatial basis functions. We also present a 3D analysis of numerical dispersion for the full discrete approximation of acoustic equation, which demonstrates that the WRKDG method can efficiently suppress numerical dispersion on coarse grids. Numerical results for several different 3D models including homogeneous and heterogeneous media with isotropic and anisotropic cases show that the 3D WRKDG method can effectively suppress numerical dispersion and provide accurate wave-field information on coarse mesh.     

Modeling Magma Dynamics with a Mixed Fourier Collocation - Discontinuous Galerkin Method

A high-order discretization consisting of a tensor product of the Fourier collocation and discontinuous Galerkin methods is presented for numerical modeling of magma dynamics. The physical model is an advection-reaction type system consisting of two hyperbolic equations and one elliptic equation. The high-order solution basis allows for accurate and efficient representation of compaction-dissolution waves that are predicted from linear theory. The discontinuous Galerkin method provides a robust and efficient solution to the eigenvalue problem formed by linear stability analysis of the physical system. New insights into the processes of melt generation and segregation, such as melt channel bifurcation, are revealed from two-dimensional time-dependent simulations.     

Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids. In order to obtain a sufficiently stable higher order scheme with respect to the time and space coordinates, we develop a combination of the discontinuous Galerkin finite element (DGFE) method for the space discretization and the backward difference formulae (BDF) for the time discretization. Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step, we employ suitable linearizations of inviscid as well as viscous fluxes which give a linear algebraic problem at each time step. Finally, the resulting BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results are compared with reference data.     

A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations

A discontinuous Galerkin method for the ideal 5 moment two-fluid plasma system is presented. The method uses a second or third order discontinuous Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping scheme. The method is benchmarked against an analytic solution of a dispersive electron acoustic square pulse as well as the two-fluid electromagnetic shock [1] and existing numerical solutions to the GEM challenge magnetic reconnection problem [2]. The algorithm can be generalized to arbitrary geometries and three dimensions. An approach to maintaining small gauge errors based on error propagation is suggested.     

A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation

A weak Galerkin (WG) method is introduced and numerically tested for the Helmholtz equation. This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property. At the same time, the WG finite element formulation is symmetric and parameter free. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains. Challenging problems with high wave numbers are also examined. Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement, and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.     

Modeling 3D Magma Dynamics Using a Discontinuous Galerkin Method

Discontinuous Galerkin (DG) and matrix-free finite element methods with a novel projective pressure estimation are combined to enable the numerical modeling of magma dynamics in 2D and 3D using the library deal.II. The physical model is an advection-reaction type system consisting of two hyperbolic equations to evolve porosity and soluble mineral abundance at local chemical equilibrium and one elliptic equation to recover global pressure. A combination of a discontinuous Galerkin method for the advection equations and a finite element method for the elliptic equation provide a robust and efficient solution to the channel regime problems of the physical system in 3D. A projective and adaptively applied pressure estimation is employed to significantly reduce the computational wall time without impacting the overall physical reliability in the modeling of important features of melt segregation, such as melt channel bifurcation in 2D and 3D time dependent simulations.     

Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients

Extrapolation cascadic multigrid (EXCMG) method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linear finite element discretization. However, it is not trivial to generalize the vertex-centred EXCMG method to cell-centered finite volume (FV) methods for diffusion equations with strongly discontinuous and anisotropic coefficients, since a non-nested hierarchy of grid nodes are used in the cell-centered discretization. For cell-centered FV schemes, the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones (primary unknowns). One of the novelties is to propose a new gradient transfer (GT) method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposed method.     

A Constrained Finite Element Method Based on Domain Decomposition Satisfying the Discrete Maximum Principle for Diffusion Problems

In this paper, we are concerned with the constrained finite element method based on domain decomposition satisfying the discrete maximum principle for diffusion problems with discontinuous coefficients on distorted meshes. The basic idea of domain decomposition methods is used to deal with the discontinuous coefficients. To get the information on the interface, we generalize the traditional Neumann-Neumann method to the discontinuous diffusion tensors case. Then, the constrained finite element method is used in each subdomain. Comparing with the method of using the constrained finite element method on the global domain, the numerical experiments show that not only the convergence order is improved, but also the nonlinear iteration time is reduced remarkably in our method.     

Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme

We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes with the Runge-Kutta method. To illustrate the feasibility of the approach, we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters of the random variable in the uncertain differential equation even if discontinuities are present.     

A Sparse Grid Discrete Ordinate Discontinuous Galerkin Method for the Radiative Transfer Equation

The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case, leading to great difficulties in numerical simulation. To tackle this bottleneck, we first use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we make the spatial discretization by means of the discontinuous Galerkin (DG) method combined with the sparse grid method. The final linear system is solved by the block Gauss-Seidal iteration method. The computational complexity and error analysis are developed in detail, which show the new method is more efficient than the original discrete ordinate DG method. A series of numerical results are performed to validate the convergence behavior and effectiveness of the proposed method.     

Space-Time Discontinuous Galerkin Method for Maxwell's Equations

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.     

Picard-Newton Iterative Method with Time Step Control for Multimaterial Non-Equilibrium Radiation Diffusion Problem

For a new nonlinear iterative method named as Picard-Newton (P-N) iterative method for the solution of the time-dependent reaction-diffusion systems, which arise in non-equilibrium radiation diffusion applications, two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented. The non-equilibrium radiation diffusion problems with flux limiter are considered, which appends pesky complexity and nonlinearity to the diffusion coefficient. Numerical results are presented to demonstrate that compared with Picard method, for a desired accuracy, significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.     

Numerical Discretization of Variational Phase Field Model for Phase Transitions in Ferroelectric Thin Films

Phase field methods have been widely used to study phase transitions and polarization switching in ferroelectric thin films. In this paper, we develop an efficient numerical scheme for the variational phase field model based on variational forms of the electrostatic energy and the relaxation dynamics of the polarization vector. The spatial discretization combines the Fourier spectral method with the finite difference method to handle three-dimensional mixed boundary conditions. It allows for an efficient semi-implicit discretization for the time integration of the relaxation dynamics. This method avoids explicitly solving the electrostatic equilibrium equation (a Poisson equation) and eliminates the use of associated Lagrange multipliers. We present several numerical examples including phase transitions and polarization switching processes to demonstrate the effectiveness of the proposed method.     

Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation

In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L²error estimates are established for the linear and nonlinear equation using several projections for different parameter choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using piecewise k-th degree polynomials for many cases.     

A High-Order Cell-Centered Discontinuous Galerkin Multi-Material Arbitrary Lagrangian-Eulerian Method

In this paper, a high-order cell-centered discontinuous Galerkin (DG) multi-material arbitrary Lagrangian-Eulerian (MMALE) method is developed for compressible fluid dynamics. The MMALE method utilizes moment-of-fluid (MOF) interface reconstruction technology to simulate multi-materials of immiscible fluids. It is an explicit time-marching Lagrangian plus remap type. In the Lagrangian phase, an updated high-order discontinuous Galerkin Lagrangian method is applied for the discretization of hydrodynamic equations, and Tipton's pressure relaxation closure model is used in the mixed cells. A robust moment-of-fluid interface reconstruction algorithm is used to provide the information of the material interfaces for remapping. In the rezoning phase, Knupp's algorithm is used for mesh smoothing. For the remapping phase, a high-order accurate remapping method of the cell-intersection-based type is proposed. It can be divided into four stages: polynomial reconstruction, polygon intersection, integration, and detection of problematic cells and limiting. Polygon intersection is based on the "clipping and projecting" algorithm, and detection of problematic cells depends on a troubled cell marker, and a posteriori multi-dimensional optimal order detection (MOOD) limiting strategy is used for limiting. Numerical tests are given to demonstrate the robustness and accuracy of our method.     

A Scalable Domain Decomposition Method for Ultra-Parallel Arterial Flow Simulations

Ultra-parallel flow simulations on hundreds of thousands of processors require new multi-level domain decomposition methods. Here we present such a new two-level method that has features both of discontinuous and continuous Galerkin formulations. Specifically, at the coarse level the domain is subdivided into several big patches and within each patch a spectral element discretization (fine level) is employed. New interface conditions for the Navier-Stokes equations are developed to connect the patches, relaxing the C⁰continuity and minimizing data transfer at the patch interface. We perform several 3D flow simulations of a benchmark problem and of arterial flows to evaluate the performance of the new method and investigate its accuracy.